Equidistribution of polynomially bounded o-minimal curves in homogeneous spaces

Michael Bersudsky, Nimish A. Shah, and Hao Xing The Ohio State University, Columbus, OH 43210, USA [email protected], [email protected], and [email protected]

Abstract.

We extend Ratner’s theorem on equidistribution of individual orbits of unipotent flows on finite volume homogeneous spaces of Lie groups to trajectories of non-contracting curves definable in polynomially bounded o-minimal structures.

To be precise, let $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ be a continuous map whose coordinate functions are definable in a polynomially bounded o-minimal structure; for example, rational functions. Suppose that $\varphi$ is non-contracting; that is, for any linearly independent vectors $v_{1},\ldots,v_{k}$ in $\mathbb{R}^{n}$ , $\varphi(t).(v_{1}\wedge\cdots\wedge v_{k})\not\to 0$ as $t\to\infty$ . Then, there exists a unique smallest subgroup $H_{\varphi}$ of $\text{SL}(n,\mathbb{R})$ generated by unipotent one-parameter subgroups such that $\varphi(t)H_{\varphi}\to g_{0}H_{\varphi}$ in $\text{SL}(n,\mathbb{R})/H_{\varphi}$ as $t\to\infty$ for some $g_{0}\in\text{SL}(n,\mathbb{R})$ .

Let $G$ be a closed subgroup of $\text{SL}(n,\mathbb{R})$ and $\Gamma$ be a lattice in $G$ . Suppose that $\varphi([0,\infty))\subset G$ . Then $H_{\varphi}\subset G$ , and for any $x\in G/\Gamma$ , the trajectory $\{\varphi(t)x:t\in[0,T]\}$ gets equidistributed with respect to the measure $g_{0}\mu_{Lx}$ as $T\to\infty$ , where $L$ is a closed subgroup of $G$ such that $\overline{Hx}=Lx$ and $Lx$ admits a unique $L$ -invariant probability measure, denoted by $\mu_{Lx}$ .

A crucial new ingredient in this work is proving that for any finite-dimensional representation $V$ of $\text{SL}(n,\mathbb{R})$ , there exist $T_{0}>0$ , $C>0$ , and $\alpha>0$ such that for any $v\in G$ , the map $t\mapsto\lVert\varphi(t)v\rVert$ is $(C,\alpha)$ -good on $[T_{0},\infty)$ .

1. Introduction

Let $G$ be a Lie group and $\Gamma$ be a lattice in $G$ . We say that a probability measure $\mu$ on $X:=G/\Gamma$ is homogeneous if there exists an $x\in X$ such that the orbit $Hx$ is closed in $X$ and $\mu$ is the (unique) $H$ -invariant probability measure on the orbit $Hx$ . We also call the orbit $Hx$ a periodic orbit of $H$ and $\mu=:\mu_{Hx}$ the homogeneous probability measure supported on that periodic orbit. Marina Ratner, in proving Raghunathan’s conjectures stated in [Dan81], showed that a Borel probability measure $\mu$ which is invariant and ergodic under the action of a $\text{Ad}_{G}$ -unipotent one-parameter subgroup is homogeneous, see [Rat91]. Using this measure classification and non-divergence property of unipotent orbits due to Margulis and Dani [Mar71, Dan86], Ratner [Rat91a] proved the following equidistribution result for unipotent flows: Let $U:=\{u(t):t\in\mathbb{R}\}$ be a one-parameter $\text{Ad}_{G}$ -unipotent subgroup of $G$ and $x\in X$ . Then there exists a closed subgroup $H$ of $G$ containing $U$ such that $H_{x}$ is a periodic orbit, and for any $f\in C_{c}(X)$ , $\frac{1}{T}\int_{0}^{T}f(u(t)x)dt\to\int_{X}f\,d\mu_{Hx}$ as $T\to\infty$ .

More generally, for a continuous curve $\varphi:[0,\infty)\to G$ , $x\in X$ , and $T>0$ , consider the probability measure $\mu_{T,\varphi,x}$ on $G/\Gamma$ defined by:

(1.1)

\mu_{T,\varphi,x}(f)=\frac{1}{T}\int_{0}^{T}f(\varphi(t)x)\,dt,~{}\forall f\in C_{c}(X).

In [Sha96] it was shown that the equidistribution of unipotent flows due to Ratner generalizes to polynomial curves in $G$ . Namely, if $G\leq\text{SL}(n,\mathbb{R})$ and $\varphi:\mathbb{R}\to G$ is a map whose coordinate functions are polynonmial, then the measures $\mu_{T,\varphi,x}$ converge to $\varphi(0)\mu_{Hx}$ as $T\to\infty$ , where $\mu_{Hx}$ is supported on a periodic orbit $Hx$ , for $H\leq G$ being the smallest closed subgroup for which $\{\varphi(0)^{-1}\varphi(t):t\in\mathbb{R}\}\subset H$ such that the orbit $Hx$ is periodic.

More recently, Peterzil and Strachenko [PS18] studied such questions in the following setting: Let $G$ be a closed subgroup of the group of upper triangle unipotent subgroup of $\text{SL}(n,\mathbb{R})$ and $\Gamma$ be a (cocompact) lattice in $G$ . They proved that if $S\subset G$ is definable in an o-minimal structure, then there exists a definable set $S^{\prime}\subset G$ whose image in $G/\Gamma$ is the closure of the image of $S$ in $G/\Gamma$ . Moreover, they proved the following [PS18, Theorem 1.6]: Let $\varphi:[0,\infty)\to G$ be a curve definable in a polynomially bounded o-minimal structure. Let $x\in G/\Gamma$ . If $\{\varphi(t)x:t\geq 0\}$ is dense in $G/\Gamma$ , then $\mu_{T,\varphi,x}$ converges to the $G$ -invariant probability on $G/\Gamma$ as $T\to\infty$ .

Our goal in this paper is to generalize the above equidistribution results for curves in an affine algebraic group such that the curves are definable in polynomially bounded o-minimal structures and satisfy an additional non-contraction condition.

We now proceed to give some basic definitions and describe our results.

1.1. O-minimal curves

We recall the basic definition of o-minimal structures. For more details on o-minimal structures, we refer the readers to [DM96, Dri98]. We will recall the required results when needed.

We only deal with o-minimal structures on the field of real numbers. The definition below is borrowed from an article of Dries and Miller [DM96].

Definition 1.1.

An o-minimal structure $\mathscr{S}$ on the real field is a sequence of families of sets

\mathscr{S}_{n}\subseteq\text{subsets of }\mathbb{R}^{n},~{}n\in\mathbb{N},

such that the following requirements are satisfied:

(1)

For each $n\in\mathbb{N}$ , $\mathscr{S}_{n}$ is a boolean algebra of sets, and $\mathbb{R}^{n}\in\mathscr{S}_{n}$ .
(2)

For each $n\in\mathbb{N}$ , $\mathscr{S}_{n}$ contains the diagonals $\{(x_{1},...,x_{n}):x_{i}=x_{j}\},$ for all $1\leq i<j\leq n$ .
(3)

For each $n\in\mathbb{N}$ , if $A\in\mathscr{S}_{n}$ , then $A\times\mathbb{R}\in\mathscr{S}_{n+1}$ and $\mathbb{R}\times A\in\mathscr{S}_{n+1}$ .
(4)

For each $n\in\mathbb{N}$ , if $A\in\mathscr{S}_{n+1}$ , then $\pi(A)\in\mathscr{S}_{n}$ , where $\pi:\mathbb{R}^{n+1}\to\mathbb{R}^{n}$ is the projection to the first $n$ -coordinates.
(5)

$\mathscr{S}_{3}$ contains the graph of addition $\{(x,y,x+y):x,y\in\mathbb{R}\}$ and the graph of multiplication $\{(x,y,xy):x,y\in\mathbb{R}\}$ .
(6)

$\mathscr{S}_{1}$ consists exactly of the finite unions of intervals (unbounded included) and singeltons.

We say that a set $A\subset\mathbb{R}^{n}$ is definable in $\mathscr{S}$ if $A\in\mathscr{S}_{n}$ . A function $f:B\to\mathbb{R}^{m}$ with $B\subseteq\mathbb{R}^{n}$ is said to be definable if its graph is in $\mathscr{S}_{n+m}$ .

•

An o-minimal structure is polynomially bounded, if for every definable function $f:[0,\infty)\to\mathbb{R}$ there exists an $r\in\mathbb{R}$ such that $f(x)=O(x^{r})$ as $x\to\infty$ .

In many regards, functions definable in a polynomially bounded o-minimal structures behave like rational functions. We note the following result which we will use throughout the paper:

•

By [Mil94a]: If $f:[0,\infty)\to\mathbb{R}$ definable in a polynomially bounded o-minimal structure $\mathscr{S}$ , then either $f$ is eventually constantly zero, or there exists $r\neq 0$ such that $\lim_{t\to\infty}f(t)/t^{r}=c\neq 0$ . In the latter case, we define

(1.2) $\deg(f):=r.$

If $f$ is eventually constantly zero, we define $\deg(f)=-\infty$ .

The following allows for constructing many examples of functions definable in a polynomially bounded o-minimal structure.

•

By [Mil94], there is a polynomially bounded o-minimal structure $\mathscr{S}$ such that the functions $f(t)=t^{r}$ ( $t>0$ ), for $r\in\mathbb{R}$ , and all real analytic functions restricted to a compact set are definable in $\mathscr{S}$ .
•

If a finite collection of functions is definable, then every function in the algebra they generate is also definable in the same structure.
•

If $f,g$ are definable functions, then $h(t):=f(t)/g(t)$ , where $t$ is such that $g(t)\neq 0$ , is definable in the same structure.
•

Suppose that $I,J\subseteq\mathbb{R}$ and $f:I\to\mathbb{R}$ and $g:J\to I$ are definable. Then, the composition $f\circ g$ is definable in the same structure.
•

Let $I\subseteq\mathbb{R}$ . If $f:I\to\mathbb{R}$ is definable and injective, then its inverse function is definable in the same structure.

For example, using the above one sees that

x\mapsto\left(\tan\left(\frac{\pi}{2}\frac{x}{x+1}\right)\frac{x^{\sqrt{2}}+2}{x+1}+\arcsin\left(\frac{x^{2}}{x^{2}+3}\right)x^{4}\right)^{\pi},x>0,

is definable in a polynomially bounded o-minimal structure.

Remark 1.2.

For our equidistribution results, we will be mainly concerned with polynomially bounded o-minimal structures. As was noted in [PS18], the class of polynomially bounded o-minimal structures is a natural class to study equidistribution in view of the following. By [Mil94a], if an o-minimal structure is not polynomially bounded, then the exponential function is definable. As a consequence, $\log(t)$ will also be definable. Notice that the measures on the circle defined by

\frac{1}{T}\int_{0}^{T}f(\log(t+1)+\mathbb{Z})\,dt,~{}f\in C(\mathbb{R}/\mathbb{Z}),

do not converge as $T\to\infty$ . Namely, for

\varphi(t):=\begin{bmatrix}1&\log(t+1)\\ 0&1\end{bmatrix},

and $x$ being the identity coset in $\text{SL}(2,\mathbb{R})/\text{SL}(2,\mathbb{Z})$ , the measures $\mu_{T,\varphi,x}$ will not converge.

1.2. o-minimal curves in affine algebraic groups

In this paper we will denote by $\mathbf{G}$ an affine algebraic group defined over $\mathbb{R}$ , and by $\mathbf{G}(\mathbb{R})$ the $\mathbb{R}$ -points of $\mathbf{G}$ . We view $\mathbf{G}(\mathbb{R})$ as a Lie group in the standard way via the subspace topology obtained by embedding algebraically $\mathbf{G}(\mathbb{R})\hookrightarrow\text{SL}(n,\mathbb{R})\subset\text{M}(n,\mathbb{R})$ .

Definition 1.3.

Let $G:=\mathbf{G}(\mathbb{R})$ , and let $\mathbb{R}[G]$ be the coordinate ring of real polynomial functions on $G$ . We say that a curve $\varphi:[0,\infty)\to G$ is definable in an o-minimal structure $\mathscr{S}$ , if the map $t\mapsto f(\varphi(t))$ on $[0,\infty)$ is definable in $\mathscr{S}$ , for all $f\in\mathbb{R}[G]$ .

We observe that if $V$ is a finite dimensional rational representation, and $\varphi:[0,\infty)\to G$ is definable in $\mathscr{S}$ , then the curve $t\mapsto\varphi(t).v$ is definable in $\mathscr{S}$ for all $v\in V$ . In particular, if $G$ is embedded in $\text{SL}(n,\mathbb{R})\subset\mathrm{M}(n,\mathbb{R})$ , namely

(1.3)

\varphi(t)=[\varphi_{i,j}(t)]_{1\leq i,j\leq n},~{}t\in[0,\infty),

then $\varphi$ is definable in $\mathscr{S}$ if and only if each $\varphi_{i,j}:[0,\infty)\to\mathbb{R}$ is definable in $\mathscr{S}$ .

1.2.1. The non-contraction property

The following introduces a certain “non-contraction” property which narrows down the polynomially bounded o-minimal curves to a class of curves whose orbits on rational linear representations of $G$ have the $(C,\alpha)$ -good growth property introduced by Kleinbock and Margulis [KM98].

Definition 1.4.

We say that a definable curve $\varphi:[0,\infty)\to G$ in an affine algebraic group $G=\mathbf{G}(\mathbb{R})$ is non-contracting in $G$ if for all finite dimensional representations $V$ of $G$ defined over $\mathbb{R}$ , it holds that $\lim_{t\to\infty}\varphi(t).v\neq 0$ for all $v\in V\smallsetminus\{0\}$ .

Remark 1.5.

Suppose that $G$ is a unipotent group and $\varphi:[0,\infty)\to G$ is a definable curve. Then $\varphi$ is non-contracting. This is so because, for any algebraic action of an algebraic unipotent group on a finite-dimensional vector space, every orbit of a nonzero vector is Zariski closed, and in particular, the closure of the orbit does not contain the origin, see [Bir71, Theorem 12.1].

We now give a practical criterion that allows one to verify the non-contraction property for curves in $\text{SL}(n,\mathbb{R})$ . Consider the exterior representation of $\text{SL}(n,\mathbb{R})$ on $\bigwedge^{k}\mathbb{R}^{n}$ defined by

g.(v_{1}\wedge v_{2}\wedge\cdots\wedge v_{k}):=gv_{1}\wedge gv_{2}\wedge\cdots\wedge gv_{k}.

Proposition 1.6.

Suppose that $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ is a continuous curve definable curve in an o-minimal structure. Then $\varphi$ is non-contracting if and only if for all $1\leq k\leq n-1$ and linearly independent vectors $v_{1},...,v_{k}\in\mathbb{R}^{n}$ , it holds that

(1.4)

\lim_{t\to\infty}\varphi(t).(v_{1}\wedge\cdots\wedge v_{k})\neq 0.

Due to the previous result, the following provides criteria for the non-contraction of definable curves in algebraic groups generated by unipotent one-parameter subgroups.

Proposition 1.7.

Let $G=\mathbf{G}(\mathbb{R})$ be such that the identity component $G^{0}$ is generated by unipotent one-parameter subgroups. Suppose $\rho:G\to\text{SL}(n,\mathbb{R})$ is rational homomorphism with a finite kernel. Then, for any polynomially bounded definable curve $\varphi:[0,\infty)\to G$ , if $\rho\circ\varphi$ is non-contracting in $\text{SL}(n,\mathbb{R})$ , then $\varphi$ is non-contracting in $G$ .

1.3. The main equidistribution result

In Section 2.2, we will show that for a given continuous definable non-contracting curve $\varphi:[0,\infty)\to G$ in an algebraic group $G=\mathbf{G}(\mathbb{R})$ , there exists a unique subgroup generated by one-parameter unipotent subgroups of $G$ , denoted by $H_{\varphi}\leq G$ and called the hull of $\varphi$ , such that $\varphi([0,\infty))/H_{\varphi}$ is contained in a compact subset of $G/H_{\varphi}$ . Also, we can choose a bounded definable curve $\beta:[0,\infty)\to G$ , which we refer to as a correcting curve such that $\beta(t)\varphi(t)\in H_{\varphi}$ for all $t$ ; see Theorem 2.4 and Definition 2.5. We note that $\lim_{t\to\infty}\beta(t)=\beta_{\infty}\in G$ by the Monotonicity Theorem [DM96, Theorem 4.1].

Theorem 1.8.

\lim_{T\to\infty}\mu_{T,\beta\varphi,x_{0}}=\mu_{\overline{H_{\varphi}x_{0}}},

in the weak- $\ast$ topology, where, in view of Ratner’s theorem, $\mu_{\overline{H_{\varphi}x_{0}}}$ is the $L$ -invariant probability measure on $\overline{H_{\varphi}x_{0}}=Lx_{0}$ for a closed connected subgroup $L$ of $\mathcal{G}$ . In particular,

\lim_{T\to\infty}\mu_{T,\varphi,x_{0}}=\beta_{\infty}^{-1}\mu_{\overline{H_{\varphi}x_{0}}},

where $\beta_{\infty}=\lim_{t\to\infty}\beta(t)$ .

Remark 1.9.

the non-contraction assumption is necessary for Theorem 1.8 to hold in the above generality. We note that if $\varphi$ is a contracting curve in $\text{SL}(n,\mathbb{R})$ , then there exists $x\in\text{SL}(n,\mathbb{R})/\text{SL}(n,\mathbb{Z})$ such that $\lim_{t\to\infty}\varphi(t)x=\infty$ . To see the existence of such $x$ , since $\varphi$ is contracting, by Proposition 1.6, there exists linearly independent vectors $\{v_{1},...,v_{k}\}\subseteq\mathbb{R}^{n}$ such that $\varphi(t).(v_{1}\wedge\cdots\wedge v_{k})\to 0$ as $t\to\infty$ . Identify $\text{SL}(n,\mathbb{R})/\text{SL}(n,\mathbb{Z})$ with the space of unimodular lattices, and let $x$ be a lattice which includes $\{v_{1},...,v_{k}\}$ . Then by Minkowski’s theorem, the length of the shortest nonzero vector(s) of the discrete subgroup $\text{Span}_{\mathbb{Z}}\{\varphi(t)v_{1},...,\varphi(t)v_{k}\}\subseteq\varphi(t)x_{0}$ goes to zero as $t\to\infty$ . By Mahler’s criterion, the curve $\varphi(t)x_{0}$ diverges to $\infty$ in the space of unimodular lattices.

In the next result, proved in Section 7, we give an example of a definable curve in $\text{SL}(n+1,\mathbb{R})$ whose trajectory diverges in every irreducible representation of $\text{SL}(n+1,\mathbb{R})$ ; in other words, it is non-contracting, and its hull is $\text{SL}(n+1,\mathbb{R})$ . As a consequence, every trajectory of such a curve in the space of unimodular lattice in $\mathbb{Z}^{n+1}$ gets equidistributed (Corollary 1.11).

Proposition 1.10.

Let $f_{0},...,f_{n}$ , and $h_{1},\ldots,h_{n}$ be continuous real functions on $[0,\infty)$ definable in a polynomially bounded o-minimal structure. For $t\geq 0$ , define

(1.5)

\varphi(t):=\begin{bmatrix}f_{0}(t)&f_{1}(t)&\cdots&f_{n}(t)\\ &h_{1}(t)^{-1}&&\\ &&\ddots&\\ &&&h_{n}(t)^{-1}\end{bmatrix}

Suppose that the functions $h_{i}$ ’s and $f_{i}$ ’s defining $\varphi$ satisfy the following conditions:

(1)

$f_{0}=h_{1}\cdots h_{n}$ , $\deg f_{0}=n$ , and $\deg h_{1}\geq\cdots\geq\deg h_{n}>0$ .
(2)

$\deg(f_{0}+g)\geq n$ for any $g$ in the linear span of $f_{1},\ldots,f_{n}$ .
(3)

For any $i\in\{1,\ldots,n\}$ , $\deg(f_{i}+g)>n-i$ for any $g$ in the liner span of $f_{i+1},\ldots,f_{n}$ .

Then the hull of $\varphi$ is $H_{\varphi}=\text{SL}(n+1,\mathbb{R})$ .

For example, let $f_{0}(t)=t^{n}$ . Suppose $f_{1},\dots,f_{n}$ are rational functions such that $\deg f_{1}>\cdots>\deg f_{n}$ , and $\deg f_{i}\neq n$ and $\deg f_{i}>n-i$ for all $1\leq i\leq n$ . Let $h_{i}(t)=t^{r_{i}}$ for some $r_{1}\geq\cdots\geq r_{n}>0$ such that $\sum_{i=1}^{n}r_{i}=n$ . Then, these functions satisfy the conditions of Proposition 1.10.

Using Theorem 1.8, we obtain the following:

Corollary 1.11.

Let $G$ be a Lie group and $\Gamma$ be a lattice in $G$ . Suppose that $\rho:\text{SL}(n+1,\mathbb{R})\to G$ is a continuous homomorphism. Let $\varphi:[0,\infty)\to\text{SL}(n+1,\mathbb{R})$ be a curve satisfying the conditions of Proposition 1.10. Then for any $x\in G/\Gamma$ , and any bounded continuous function $f$ on $G/\Gamma$ ,

\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}f(\varphi(t)x_{0})dt=\mu_{X}(f),

where $\mu_{X}$ is the homogeneous probability measure on the homogeneous space $\overline{\rho(\text{SL}(n+1,\mathbb{R}))x}$ .

1.4. Growth of o-minimal functions

We now describe the $(C,\alpha)$ -good property concerning the growth of certain families of functions definable in polyonomially bounded o-minimal structures. The property is well known for polynomials, and it is new in the o-minimal setting. It is key for the technique used in this paper.

We need the following definition.

Definition 1.12.

Let $\mathcal{V}$ be finite dimensional vector space spanned over $\mathbb{R}$ by real functions definable in a polynomially bounded o-minimal structure. Identify $\mathcal{V}$ with $\mathbb{R}^{N}$ by choosing a basis for $\mathcal{V}$ . A subset $\mathscr{F}\subseteq\mathcal{V}$ is called a closed definable cone if it is a closed definable subset of $\mathbb{R}^{N}$ such that if $v\in\mathscr{F}$ then $\alpha v\in\mathscr{F}$ for all scalars $\alpha\geq 0$ .

Example 1.13.

Let $\psi:[0,\infty)\to\text{GL}(m,\mathbb{R})$ be a curve definable in a polynomially bounded o-minimal structure, and let $\|\cdot\|$ be the Euclidean norm. Then, the vector space

\mathcal{V}:=\text{Span}_{\mathbb{R}}\{\|\psi(t)\textbf{v}\|^{2}:\textbf{v}\in\mathbb{R}^{m}\}

is finite dimensional, and $\mathscr{F}:=\{\|\psi(t)\textbf{v}\|^{2}:\textbf{v}\in\mathbb{R}^{m}\}$ is a closed definable cone.

Theorem 1.14.

( $(C,\alpha)$ -good property). Let $\mathcal{V}$ be a finite-dimensional real vector space spanned by functions $f:[0,\infty)\to\mathbb{R}$ definable in a polynomially bounded o-minimal structure. Suppose that $\mathscr{F}\subset\mathcal{V}$ is a closed definable cone such that for all $f\in\mathscr{F}\smallsetminus\{0\}$ it holds that

\lim_{t\to\infty}f(t)\neq 0.

Then, there exist $C>0$ , $\alpha>0$ and $T_{0}\geq 1$ such that for any $f\in\mathscr{F}\smallsetminus\{0\}$ , $f$ is $(C,\alpha)$ -good in $[0,\infty)$ ; that is, $\forall\epsilon>0$ , and for every bounded interval $I\subseteq[T_{0},\infty)$

(1.6)

|\{t\in I:|f(t)|\leq\epsilon\}|\leq C\left(\frac{\epsilon}{\|f\|_{I}}\right)^{\alpha}\cdot|I|.

We note the following result is an immediate consequence of Theorem 1.14.

Proposition 1.15.

Let $G:=\mathbf{G}(\mathbb{R})$ and let $\varphi:[0,\infty)\to G$ be a non-contracting curve definable in a polynomially bounded o-minimal structure. Suppose that $V$ is a finite dimensional rational representation of $G$ over $\mathbb{R}$ . Fix a norm on $\|\cdot\|$ on $V$ . Then, there exist $T_{0}>0$ , $C>0$ , and $\alpha>0$ such that for all $0\neq v\in V\smallsetminus\{0\}$ the function $\Theta_{v}(t):=\|\varphi(t).v\|$ is $(C,\alpha)$ -good in $[T_{0},\infty)$ .

1.5. Proof ideas and structure of the paper

Our main theorem will be proved by the following strategy which is common in many works on equidistribution in homogeneous spaces, see e.g. [DM93, MS95]:

(1)

Proving that there is no escape of mass – that is, any limiting measure $\mu$ of the probability measures $\mu_{T}$ is a probability measure.
(2)

Showing that any limiting measure $\mu$ of the measures $\mu_{T}$ is invariant under a group generated by unipotent one-parameter subgroups.
(3)

Studying the ergodic decomposition of a limiting measure $\mu$ . Since $\mu$ is invariant under a unipotent group, every ergodic component appearing in the ergodic decomposition of $\mu$ is homogeneous by Ratner’s measure classification theorem. The theorem is proved once it is verified that $\mu$ equals to exactly one uniquely deteremined ergodic component. This will step will be treated using the linearization technique of [DM93].

To establish unipotent invariance, we will use an idea similar to the one used for polynomial curves [Sha94]. This idea generalizes to the o-minimal setting via the notion of the Peterzil-Steinhorn group (see Section 5), which was also used in [PS18].

The $(C,\alpha)$ -good property will be used to prove the non-escape of mass (Section 4) and to apply the linearization technique for the ergodic decomposition of our limiting measure (Section 6).

1.6. Some conventions about notation

We will denote the Zariski closure of a subset $A$ of an affine space by $\text{zcl}(A)$ . For a linear Lie group $H$ , we denote by $H_{u}$ the subgroup of $H$ generated by all one-parameter unipotent subgroups contained in $H$ . We will write $G:={\bf G}(\mathbb{R})$ to mean that $G$ is the set of $\mathbb{R}$ points of an algebraic group ${\bf G}$ defined over $\mathbb{R}$ .

Acknowledgement.

We thank Chris Miller for helpful discussions.

2. Consequences of the non-contracting property

2.1. The criterion for the non-contracting property

Our goal here is to prove Proposition 1.6, namely to reduce the verification of the non-contracting property for any vector in any representation to decomposable vectors in the exterior of the standard representation of $\text{SL}(n,\mathbb{R})$ . Our proof follows from the observations in [SY24, Section 2], which uses Kempf’s numerical criteria on geometric invariant theory [Kem78].

We recall the following preliminaries. Let S denote the full diagonal subgroup of $\text{SL}(n,\mathbb{R})$ . The group of strictly upper triangular matrices in $\text{SL}(n,\mathbb{R})$ , denoted by $N$ , is a maximal unipotent subgroup of $\text{SL}(n,\mathbb{R})$ normalized by S. Let $X^{\ast}(\textbf{S})$ denote the group of algebraic homomorphisms from S to $\mathbb{R}^{\ast}$ defined over $\mathbb{R}$ . Then $X^{\ast}(\textbf{S})$ is a free abelian group on $(n-1)$ -generators, and we treat it as an additive group. For each $i\in\{1,\ldots,n-1\}$ , let $\alpha_{i}\in X^{\ast}(\textbf{S})$ be defined by $\alpha_{i}(\text{diag}(t_{1},\ldots,t_{n}))=t_{i}/t_{i+1}$ . Then $\Delta=\{\alpha_{i}:1\leq i\leq n-1\}$ is the set of simple roots on S corresponding to the choice of the maximal unipotent subgroup $N$ . For each $1\leq i\leq n-1$ , let $\mu_{i}\in X^{\ast}(\textbf{S})$ be defined by $\mu_{i}(\text{diag}(t_{1},\ldots,t_{n}))=t_{1}\cdots t_{i}$ . Then $\{\mu_{i}:1\leq i\leq n-1\}$ is the set of fundamental characters (weights) of $\mathbf{S}$ with respect to our choice of the simple roots. For each $1\leq i\leq n-1$ , let $W_{i}$ denote the $i$ -th exterior of the standard representation of $\text{SL}(n,\mathbb{R})$ on $\mathbb{R}^{n}$ , called a fundamental representation of $\text{SL}(n,\mathbb{R})$ . So $W_{i}=\wedge^{i}\mathbb{R}^{n}$ , and for any $v_{1}\wedge\cdots\wedge v_{i}\in W_{i}$ and $g\in\text{SL}(n,\mathbb{R})$ , $g.(v_{1}\wedge\cdots\wedge v_{i})=gv_{1}\wedge\cdots\wedge gv_{i}$ . Let $w_{i}=e_{1}\wedge\cdots\wedge e_{i}$ . Then $w_{i}$ is fixed by $N$ , and S acts on the line $\mathbb{R}w_{i}$ via the fundamental character $\mu_{i}$ .

A dominant integral character (weight) is a non-negative integral combination of fundamental characters. Suppose $\chi=m_{1}\mu_{1}+\cdots+m_{n-1}\mu_{n-1}$ , where $m_{i}\in\mathbb{Z}_{\geq 0}$ . For each $i$ , let $W_{i}^{\otimes m_{i}}$ denote the tensor product of $m_{i}$ -copies of $W_{i}$ , and $w_{i}^{m_{i}}:=w_{i}\otimes\cdots\otimes w_{i}\in W_{i}^{\otimes m_{i}}$ . Consider the tensor product representation of $\text{SL}(n,\mathbb{R})$ on $W_{\chi}=W_{1}^{\otimes m_{1}}\otimes\cdots\otimes W_{n-1}^{\otimes m_{n-1}}$ , and let $w_{\chi}=w_{1}^{m_{1}}\otimes\cdots\otimes w_{n-1}^{m_{n-1}}$ . Then $w_{\chi}$ is fixed by $N$ , and S acts on the line $\mathbb{R}w_{\chi}$ via the character $\chi$ .

Proof of Proposition 1.6.

Let $V$ be a representation of $G:=\text{SL}(n,\mathbb{R})$ . Suppose $v\in V$ is a a nonzero such that $\varphi(t)v\to 0$ as $t\to\infty$ . We call such a $v$ a $G$ -unstable vector in $V$ . So, by [SY24, Remark 2.3], there exist $g_{0}\in G$ , a dominant integral character $\chi\in X^{\ast}(\textbf{S})$ , and constants $\beta>0$ and $C>0$ such that for any $g\in G$ ,

(2.1)

\lVert gg_{0}.w_{\chi}\rVert\leq C\lVert g.v\rVert^{\beta}.

Therefore, $\lim_{t\to\infty}\varphi(t)g_{0}.w_{\chi}=0$ . Let $A=\textbf{S}^{0}$ , the component of the identity in S. Let $K=\text{SO}(n)$ . Then by Iwasawa decomposition, $G=KAN$ . For each $t$ , we express $\varphi(t)g_{0}=k_{t}a_{t}n_{t}$ , where $k_{t}\in K$ , $a_{t}\in A$ and $n_{t}\in N$ . Without loss of generality, we may assume that the norm on $W_{\chi}$ is $K$ -invariant. Since, $N$ fixes $w_{\chi}$ ,

\lVert\varphi(t)g_{0}.w_{\chi}\rVert=\lVert a_{t}.w_{\chi}\rVert=\chi(a_{t})\lVert w_{\chi}\rVert.

Hence, since $\lim_{t\to\infty}\varphi(t)g_{0}.w_{\chi}=0$ , we get $\lim_{t\to\infty}\chi(a_{t})=0$ . Now

\chi(a_{t})=\mu_{1}(a_{t})^{m_{1}}\cdots\mu_{n-1}(a_{t})^{m_{n-1}}.

Since $m_{i}\geq 0$ , after passing to a subsequence, we can pick $i\in\{1,\ldots,n-1\}$ and a sequence $t_{l}\to\infty$ such that $\lim_{l\to\infty}\mu_{i}(a_{t_{l}})=0$ . Therefore, since $N$ fixes $w_{i}$ , and the norm is $K$ -invariant,

\lVert\varphi(t_{l})g_{0}.w_{i}\rVert=\lVert{a_{t_{l}}.w_{i}}\rVert=\mu_{i}(a_{t_{l}})\lVert w_{i}\rVert\to 0\text{, as $l\to\infty$.}

Since the coordinate functions of $t\to\varphi(t)g_{0}.w_{i}$ are definable in an o-minimal structure, we get $\lim_{t\to\infty}\varphi(t)g_{0}.w_{i}=0$ . Let $v_{j}=g_{0}e_{j}$ for each $j$ . Since $w_{i}=e_{1}\wedge\cdots\wedge e_{i}$ , $\varphi(t).(g_{0}w_{i})=\varphi(t).(v_{1}\wedge\cdots\wedge v_{i})\to 0$ as $t\to\infty$ . This contradicts our assumption 1.4. ∎

The following statement will be used in the proof of Lemma 7.1, which will needed for proving Propositions 1.10.

Lemma 2.1.

Suppose that $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ is a continuous curve such that

(2.2)

\lim_{t\to\infty}\varphi(t).(v_{1}\wedge\dots\wedge v_{k})=\infty,

for all $1\leq k\leq n-1$ and linearly independent vectors $v_{1},...,v_{k}\in\mathbb{R}^{n}$ . Then for every finite dimensional rational representation $V$ and an $v\in V$ such that the curve $\varphi(t).v,t\geq 0$ is bounded in $V$ , it holds that $\text{SL}(n,\mathbb{R}).v$ is Zariski closed.

Proof.

We will use the notations above. Suppose that $V$ is a finite dimensional rational representation of $\text{SL}(n,\mathbb{R})$ and let $v\in V$ such that $\text{SL}(n,\mathbb{R}).v$ is not Zariski closed. By [SY24, Corollary 2.5] and by [SY24, Remark 2.3], there exist a $g_{0}\in\text{SL}(n,\mathbb{R})$ , a dominant integral character $\chi\in X^{\ast}(\textbf{S})$ , and a constant $\beta>0$ such that for any $R>0$ there is a constant $C>0$ with the following property:

(2.3)

\displaystyle\text{if $\|g.v\|\leq R$, then }\lVert gg_{0}.w_{\chi}\rVert\leq C\lVert g.v\rVert^{\beta}.

Now suppose that $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ is a continuous curve satisfying the diverging property (2.2) and assume for contradiction that $\varphi(t).v,~{}t\geq 0$ is bounded in $V$ . Then, by (2.3), we get that $\lVert\varphi(t)g_{0}.w_{\chi}\rVert,~{}t>0$ is bounded. As in the proof of Proposition 1.6, we use Iwasawa decomposition to express $\varphi(t)g_{0}=k_{t}a_{t}n_{t},t>0$ , where $k_{t}\in K$ , $a_{t}\in A$ and $n_{t}\in N$ , and observe that

\lVert\varphi(t)g_{0}.w_{\chi}\rVert=\lVert a_{t}.w_{\chi}\rVert=\chi(a_{t})\lVert w_{\chi}\rVert.

We claim that $\lim_{t\to\infty}\chi(a_{t})=\infty,$ which is a contradiction. Indeed, we have

\chi(a_{t})=\mu_{1}(a_{t})^{m_{1}}\cdots\mu_{n-1}(a_{t})^{m_{n-1}},

for $m_{i}\geq 0$ , and

\lVert\varphi(t)g_{0}.w_{i}\rVert=\lVert{a_{t}.w_{i}}\rVert=\mu_{i}(a_{t})\lVert w_{i}\rVert,

where $w_{i}=e_{1}\wedge\cdots\wedge e_{i}$ and $1\leq i\leq n-1$ . Finally, due to (2.2), we have $\lim_{t\to\infty}\lVert\varphi(t)g_{0}.w_{i}\rVert=\infty.$ ∎

2.2. On the hull and the correcting curve

In order to define the hull and to obtain its usesful properties we will need the following notion.

Definition 2.2.

Let $G:=\mathbf{G}(\mathbb{R})$ . We say that a closed subgroup $H$ is observable in $G$ if there is a finite-dimensional rational representation $V$ and an $v\in V$ such that $H=\{g\in G:g.v=v\}$ .

The notion of observable groups was introduced [BHM63]. More precisely, [BHM63] defines observable groups as the algebraic subgroups of algebraic groups for which every finite dimensional rational representation extends to a finite dimensional rational representation of the ambient group. The above definition turns out equivalent to the extension property [BHM63, Theorem 8] (see [TB05, Theorem 9] for this statement for non-algebraically closed fields).

We note:

•

[Gro97, Corollary 2.8]: If $H$ is an algebraic subgroup over $\mathbb{R}$ whose radical is unipotent, then $H$ is observable.

Definition 2.3.

We say that a curve $\varphi\subset G$ is bounded modulo a closed subgroup $H\leq G$ if there exists a compact subset $K\subseteq G$ such that $\varphi\subseteq KH$ . Alternatively, $\varphi$ is bounded modulo $H$ if its image in $G/H$ is contained in a compact subset of $G/H$ .

Here is our main theorem of this section.

Theorem 2.4.

Let $G:=\mathbf{G}(\mathbb{R})$ . Suppose that $\varphi:[0,\infty)\to G$ is a continuous, non-contracting curve definable in an o-minimal structure $\mathscr{S}$ . Then, there exists a unique closed connected (with respect to the real topology) subgroup $H_{\varphi}\leq G$ of the smallest dimension such that $H_{\varphi}$ is generated by one-parameter unipotent subgroup and such that $\varphi$ is bounded modulo $H_{\varphi}$ . Moreover, the following properties hold:

(1)

The subgroup $H_{\varphi}$ is minimal in the following sense: if $H^{\prime}$ is an observable group such that $\varphi$ is bounded modulo $H^{\prime}$ , then $H_{\varphi}\leq H^{\prime}$ . Moreover, if $V$ is a rational representation, and $v\in V$ is such that $\{\varphi(t).v:t\geq 0\}$ is bounded in $V$ , then $\varphi$ is bounded modulo the isotropy group of $v$ and $H_{\varphi}.v=\{v\}$ .
(2)

There is a bounded, continuous curve $\beta:[0,\infty)\to G$ definable in $\mathscr{S}$ such that $\beta(t)\varphi(t)\in H_{\varphi}$ for all $t\geq 0$ .
(3)

If $V$ is a rational finite dimensional representation of $G$ over $\mathbb{R}$ , then for $v\in V$ either $H_{\varphi}.v=\{v\}$ or $\lim_{t\to\infty}\beta(t)\varphi(t).v=\infty$ .

In view of the above statement, we can now define the hull.

Definition 2.5.

Let $G=\mathbf{G}(\mathbb{R})$ and and let $\varphi:[0,\infty)\to G$ be a continuous non-contracting curve definable in an o-minimal structure $\mathscr{S}$ . We define the hull $H_{\varphi}$ to be the smallest connected closed subgroup generated by unipotent elements such that $\varphi$ is bounded modulo $H_{\varphi}$ , and we call a curve $\beta:[0,\infty)\to G$ a correcting curve if $\beta(t)\varphi(t)\in H_{\varphi}$ for all $t\geq 0$ .

Remark 2.6.

We note that if $\varphi$ is as in the above theorem and $\beta$ is a definable correcting curve, then the hull of $\psi(t):=\beta(t)\varphi(t)$ is the same as the hull of $\varphi$ , and the correcting curve of $\psi$ can be chosen to be the constant identity matrix. This follows from the uniqueness of the hull.

Remark 2.7.

Suppose that $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ is a curve whose entry functions $\varphi_{i,j}$ generate an algebra which only consists of either constant functions or functions diverging to $\infty$ , e.g., when $t\mapsto\varphi_{i,j}(t)$ are polynomials for all $i,j$ . Then it follows by Theorem 2.4 that $\varphi(0)^{-1}\varphi([0,\infty))\subset H_{\varphi}$ . A correcting curve is given by the constant curve $\beta(t)=\varphi(0)^{-1},\forall t\geq 0$ .

Before proving Theorem 2.4, we establish that the hull is intrinsically defined.

Lemma 2.8.

Let $G_{1},G_{2}$ be the real points of algebraic groups over $\mathbb{R}$ , and let $f:G_{1}\to G_{2}$ be a rational homomorphism. Suppose that $\varphi:[0,\infty)\to G_{1}$ is a non-contracting curve definable in an o-minimal structure $\mathscr{S}$ . Then $f(H_{\varphi})=H_{f\circ\varphi}$ .

Proof.

We start with showing that $H_{f\circ\varphi}\subseteq f(H_{\varphi})$ . Let $\beta:[0,\infty)\to G_{1}$ be a correcting curve of $\varphi$ . Since $H_{\varphi}$ is generated by unipotents, we get that $f(H_{\varphi})$ is generated by unipotents. Moreover, the curve $f\circ\varphi$ is bounded modulo $f(H_{\varphi})$ . In fact, $t\mapsto f(\beta(t))$ is bounded, and for all $t\geq 0$ :

f(\beta(t)\varphi(t))=f(\beta(t))f(\varphi(t))\in f(H_{\varphi}).

Thus, by minimality of the hull, we conclude that $H_{f\circ\varphi}\subseteq f(H_{\varphi})$ .

We now show the other inclusion. Since $\text{zcl}(H_{f\circ\varphi})$ is observable, we may choose a rational representation $\rho:G_{2}\to\text{GL}(V)$ and a $v\in V$ such that

\text{zcl}(H_{f\circ\varphi})=\{g\in G_{2}:\rho(g)v=v\}.

Since $f\circ\varphi$ is bounded modulo $H_{f\circ\varphi}$ , we get that the trajectory

\rho((f\circ\varphi)(t))v=(\rho\circ f)(\varphi(t))v,~{}t\geq 0,

is bounded in $V$ . Now, $\eta:=\rho\circ f:G_{1}\to\text{GL}(V)$ is a representation of $G_{1}$ such that $t\mapsto\eta(\varphi(t))v,~{}t\geq 0$ is bounded. By Theorem 2.4(1), we obtain that $\rho(f(H_{\varphi}))v=\eta(H_{\varphi})v=\{v\}$ . That is, $f(H_{\varphi})$ is contained in the isotropy group of $v$ which equals to $\text{zcl}(H_{f\circ\varphi})$ . ∎

2.3. Proving Theorem 2.4

We begin with the following “curve correcting” lemma.

Lemma 2.9.

Consider an algebraic action of $G:=\mathbf{G}(\mathbb{R})$ on an affine or projective variety $\mathcal{Z}$ . Let $\varphi:[0,\infty)\to G$ be a continuous curve definable in an o-minimal structure $\mathscr{S}$ . Suppose that $x\in\mathcal{Z}$ and $g_{0}\in G$ are such that

\lim_{t\to\infty}\varphi(t).x=g_{0}.x.

Then there exists a continuous $\mathscr{S}$ -definable curve $\delta:[0,\infty)\to G$ such that $\lim_{t\to\infty}\delta(t)=e$ , and

\delta(t)\varphi(t).x=g_{0}.x,\forall t\geq 0.

Proof.

Identify $G$ as a closed subgroup of $\text{SL}(n,\mathbb{R})$ . Consider the following definable set:

\displaystyle\mathcal{D}

\displaystyle:=\{(t,g):t>0,g\in G,\|g-I_{n}\|<1\text{ and }g_{0}^{-1}\varphi(t).x=g.x\}.

Here $\|\cdot\|$ is a definable norm (e.g. the sum of squares norm). Since the orbit map is an open map (maps open sets in $G$ to open sets in $G.x$ with respect to the induced topology from $\mathcal{Z}$ , see [PRR23, Corollary 3.9]), and since

\lim_{t\to\infty}g_{0}^{-1}\varphi(t).x=x,

we get that for all $t$ large enough, say $t\geq T_{0}$ , there exists $g\in G$ such that $(t,g)\in\mathcal{D}$ . Using the choice function theorem [DM96, Theorem 4.5], there is a definable curve $\tilde{\delta}(t):[T_{0},\infty)\to G$ such that $(t,\tilde{\delta}(t))\in\mathcal{D},~{}\forall t\geq T_{0}$ . In particular,

\tilde{\delta}(t).x=g_{0}^{-1}\varphi(t).x,\forall t\geq T_{0},

and $\tilde{\delta}$ is bounded. Since the curve $\tilde{\delta}(t)$ is bounded, by the Monotonicity Theorem [DM96, Theorem 4.1] we have that $\lim_{t\to\infty}\tilde{\delta}(t)=\delta_{0}$ . Note that $\delta_{0}.x=x$ . Consider, $\delta(t):=g_{0}(\tilde{\delta}(t)\delta_{0}^{-1})^{-1}g_{0}^{-1}$ for $t\geq T_{0}$ . Then,

\delta(t)\varphi(t).x=g_{0}.x,\forall t\geq T_{0}.

By cell decomposition [DM96, Theorem 4.2], $\delta(t)$ is continuous in $[T^{\prime}_{0},\infty)$ . We now extend $\delta$ to the interval $[0,T^{\prime}_{0}]$ by putting

\delta(t):=\delta(T^{\prime}_{0})\varphi(T^{\prime}_{0})\varphi(t)^{-1},t\leq T^{\prime}_{0}.

Then $\delta(t)\varphi(t).x=g_{0}.x,\forall t\geq 0$ . Since $\varphi$ is continuous and definable, $\delta:[0,\infty)\to G$ is continuous, definable and bounded. ∎

We now observe the following.

Lemma 2.10.

Let $G:=\mathbf{G}(\mathbb{R})$ . Suppose that $\varphi:[0,\infty)\to G$ is a non-contracting, continuous curve, definable is an o-minimal structure $\mathscr{S}$ . Suppose that $V$ is a finite dimensional rational representation and assume that there is an $v\in V\smallsetminus\{0\}$ such that $\varphi(t).v$ is bounded. Then, there exists a $g_{0}\in G$ such that

\lim_{t\to\infty}\varphi(t).v=g_{0}.v.

In particular, there exists a bounded, continuous curve $\beta:[0,\infty)\to\text{SL}(n,\mathbb{R})$ definable in $\mathscr{S}$ , such that $\beta(t)\varphi(t).v=v,\forall t\geq 0$ .

Proof.

Since $\varphi:[0,\infty)\to G$ is definable, and since $V$ is a rational representation, it follows that $\varphi(t).v$ is a bounded definable curve. By the monotonicity theorem (see [DM96, Section 4]), it follows that there is an $v^{\prime}\in V$ such that $\lim_{t\to\infty}\varphi(t).v=v^{\prime}$ . We claim that $v^{\prime}\in G.v$ . Denote $S:=\text{zcl}(G.v)\smallsetminus G.v$ , and suppose for contradiction that $v^{\prime}\in S$ . Then, by Lemma A.1 which is a slight modification of [Kem78, Lemma 1.1(b)], there exists a rational representation $W$ of $G$ and an $G$ -equivariant polynomial map $P:V\to W$ such that $P(v)\neq 0$ and $P(S)=0$ . But then

\lim_{t\to\infty}\varphi(t).P(v)=P(\varphi(t).v)=P(v^{\prime})=0,

which is a contradiction since $\varphi$ is non-contracting. Thus, we conclude that there is an $g_{0}\in G$ such that

\lim_{t\to\infty}\varphi(t).v=g_{0}.v.

The existence of the bounded curve $\beta$ is obtained by Lemma 2.9. ∎

Lemma 2.11.

Let $G:=\mathbf{G}(\mathbb{R})$ . Suppose that $\varphi:[0,\infty)\to G$ is a non-contracting, continuous curve, definable in an o-minimal structure. Let $H_{1},H_{2}\leq G$ be two observable groups such that the image of $\varphi$ is bounded in $G/H_{i}$ for $i\in\{1,2\}$ . Then, the image of $\varphi$ is bounded in $G/(H_{1}\cap H_{2})$ .

Proof.

For $i\in\{1,2\}$ , let $V_{i}$ be a finite dimensional rational representations such that $H_{i}=\{g\in G:g.v_{i}=v_{i}\}$ where $v_{i}\in V_{i}$ . Consider the direct-sum representation $V_{1}\oplus V_{2}$ , and notice that

H_{1}\cap H_{2}=\{g\in G:g.(v_{1},v_{2})=(v_{1},v_{2})\}.

Since the image of $\varphi$ is bounded in $G/H_{i}$ , $\varphi(t).v_{i}$ is bounded for all $i\in\{1,2\}$ . In particular, $\varphi(t).(v_{1},v_{2})$ is bounded in $V_{1}\oplus V_{2}$ . Then, by Lemma 2.10 there is a bounded continuous curve $\delta:[0,\infty)\to G$ such that $\delta(t)\varphi(t).(v_{1},v_{2})=(v_{1},v_{2})$ for all $t\geq 0$ . Namely, $\delta(t)\varphi(t)\in H_{1}\cap H_{2}$ for all $t\geq 0$ , which shows that claim. ∎

Corollary 2.12.

Let $G:=\mathbf{G}(\mathbb{R})$ . Suppose that $\varphi:[0,\infty)\to G$ is a non-contracting, continuous curve definable in an o-minimal structure. Then there exists a unique observable subgroup $\tilde{H}_{\varphi}\leq G$ of the smallest dimension such that $\varphi$ is bounded in $G/\tilde{H}_{\varphi}$ , and if $H$ is an observable subgroup such that $\varphi$ is bounded in $G/H$ , then $\tilde{H}_{\varphi}\leq H$ .

Proof.

Let $H_{0}$ be a Zariski connected observable group of the smallest dimension such that $\varphi$ is bounded in $G/H_{0}$ . If $H$ is any connected observable group such that $\varphi$ is bounded in $G/H$ , then by Lemma 2.11, we get that $\varphi$ is bounded in $G/(H_{0}\cap H)$ . If $H$ does not include $H_{0}$ , then $H\cap H_{0}$ is a proper subgroup of dimension strictly smaller than $\dim H_{0}$ , which is impossible. ∎

Corollary 2.13.

Let $G:=\mathbf{G}(\mathbb{R})$ . Suppose that $\varphi:[0,\infty)\to G$ is a non-contracting, continuous curve, definable in an o-minimal structure. Let $\tilde{H}_{\varphi}\leq G$ be the minimal observable group such that $\varphi$ is bounded modulo $\tilde{H}_{\varphi}$ . Denote $H_{\varphi}:=\tilde{H}_{\varphi}^{0}$ , the identity component of $\tilde{H}_{\varphi}$ .

Then, there exists a bounded, continuous curve $\beta:[T_{0},\infty)\to G$ such that $\beta(t)\varphi(t)\in H_{\varphi}$ for all $t\geq 0$ . Moreover, if $V$ is a finite-dimensional rational representation of $G$ and $v\in V$ is such that $\varphi(t).v$ is bounded, then $\beta(t)\varphi(t).v=v,\forall t\geq T_{0}$ .

Proof.

By Lemma 2.10, we obtain a continuous definable curve $\tilde{\beta}:[0,\infty)\to\text{SL}(n,\mathbb{R})$ such that $\tilde{\beta}(t)\varphi(t)\in\tilde{H}_{\varphi}$ for all $t\geq 0$ . Since $\varphi$ is continuous, it is contained in a connected component of $H_{\varphi}$ . By choosing a suitable $h_{0}\in H_{\varphi}$ we get that for $\beta(t):=h_{0}\tilde{\beta}(t)$ it holds $\beta(t)\varphi(t)\in H_{\varphi}$ for all $t\geq 0$ . Now, if $V$ is a finite-dimensional rational representation such that $\varphi(t).v$ is bounded, then by Lemma 2.10, we conclude that $\varphi$ is bounded modulo the observable group $H:=\{g\in G:g.v=v\}$ . Hence, by minimality of $\tilde{H}_{\varphi}$ , we get that $\beta(t)\varphi(t)\in H_{\varphi}\leq H$ . ∎

In order to establish Theorem 2.4 it remains to prove that the identity component of the above minimal observable group is generated by one-parameter unipotent subgroups.

Lemma 2.14.

Let $G:=\mathbf{G}(\mathbb{R})$ . Suppose that $\varphi:[0,\infty)\to G$ be a non-contracting, continuous curve, definable in an o-minimal structure, and let $\tilde{H}_{\varphi}$ be the smallest observable subgroup such that $\varphi$ is bounded modulo $\tilde{H}_{\varphi}$ . Then $H_{\varphi}=\tilde{H}_{\varphi}^{0}$ is generated by one-parameter unipotent subgroups.

Proof.

Denote $\psi(t):=\beta(t)\varphi(t),t\geq 0$ , where $\beta$ is a correcting curve of $\varphi$ . Let $L\leq\tilde{H}_{\varphi}$ be the Zariski closure of the subgroup generated by all one-parameter unipotent subgroups of $\tilde{H}_{\varphi}$ . We will now show that the image of $\varphi$ in $\tilde{H}_{\varphi}/L$ is bounded, and since $L$ is observable [Gro97, Corollary 2.8], we will get that $\tilde{H}_{\varphi}=L$ , which proves the claim.

Now, note that $L$ is a normal subgroup, so that $\tilde{H}_{\varphi}/L$ is an affine algebraic group. Since $L$ contains the unipotent radical of $\tilde{H}_{\varphi}$ , we get that $\tilde{H}_{\varphi}/L$ is reductive. Then, there is an almost direct product decomposition $\tilde{H}_{\varphi}/L=ZD$ , where $Z$ is the center, which is a torus, and $D$ is the derived group, which is a normal semi-simple group. Notice that each simple factor of $D$ must be compact since a non-compact simple group is generated by unipotents. Namely, $D$ is compact. Let $q:\tilde{H}_{\varphi}\to\tilde{H}_{\varphi}/L$ be the natural quotient map. To finish the proof, we show below that for every character $\chi:\tilde{H}_{\varphi}/L\to\mathbb{R}^{\times}$ it holds that $\chi\circ q(\psi(t))=1,\forall t\geq 0$ . This will complete the proof since then $q\circ\psi([0,\infty))\subseteq Z_{a}D$ , where $Z_{a}\leq Z$ is the maximal anisotropic compact torus subgroup of $Z$ . Let $\chi:\tilde{H}_{\varphi}/L\to\mathbb{R}^{\times}$ be a character, and consider the representation of $\tilde{H}_{\varphi}/L$ on $\mathbb{R}^{2}$ defined by

\overline{h}.(x,y):=\left(\chi(\overline{h})x,\chi(\overline{h})^{-1}y\right),~{}(x,y)\in\mathbb{R}^{2},~{}\overline{h}\in\tilde{H}_{\varphi}/L.

Then $\tilde{H}_{\varphi}$ acts on $\mathbb{R}^{2}$ be pre-composing with $q$ . Namely, $h.(x,y):=q(h).(x,y)$ . Since $\tilde{H}_{\varphi}\leq G$ is observable, the above representation comes from a representation of $G$ , see [TB05, Theorem 9]. Now, fix two non-zero real numbers $x_{0},y_{0}$ . Because $\psi$ is non-contracting, we get that $\psi(t).(x_{0},0)$ and $\psi(t).(0,y_{0})$ are bounded away from $0$ . But this is only possible if $\psi(t).(x_{0},y_{0})$ is bounded. So by Corollary 2.13 we obtain that $\psi(t).(x_{0},y_{0})=(x_{0},y_{0})$ . Thus, $\chi\circ q(\psi(t))=1,~{}\forall t\geq 0$ . ∎

2.4. The non-contraction property under homomorphisms

In this section we show that the homomorphic image of a definable non-contracting curve is non-contracting, Lemma 2.15, and also we show that if a definable curve in a homomorphic image is non-contracting , then it has a definable non-contracting lift, Lemma 2.19. Finally, we establish that the non-contracting property is intrinsic for curves in groups whose radical is unipotent, Proposition 2.20.

Lemma 2.15.

Let $G_{1},G_{2}$ be the real points of two affine algebraic groups over $\mathbb{R}$ , and let $f:G_{1}\to G_{2}$ be a homomorphism of algebraic groups. Let $\varphi:[0,\infty)\to G_{1}$ be a continuous, non-contracting curve in $G_{1}$ definable in $\mathscr{S}$ . Then $f\circ\varphi$ is a continuous, non-contracting curve in $G_{2}$ definable in $\mathscr{S}$ .

Proof.

Let $\rho:G_{2}\to\text{GL}(V)$ be a rational representation of $G_{2}$ . Let $v\in V$ . Then $\rho\circ f:G_{1}\to\text{GL}(V)$ is a rational representation of $G_{1}$ . Let $v\in V$ . If $\rho(f\circ\varphi(t))v\to 0$ as $t\to\infty$ , then $(\rho\circ f)(\varphi(t))v\to 0$ as $t\to\infty$ . Since $\varphi$ is non-contracting, we get $v=0$ . ∎

Lemma 2.16.

Let $G:=\mathbf{G}(\mathbb{R})$ and suppose that $U\leq G$ is a normal, unipotent algebraic subgroup. Let $q:G\to G/U$ be the natural map. Then, a curve $\varphi:[0,\infty)\to G$ is not contracting in $G$ if and only if $q\circ\varphi$ is not contracting in $G/U$ .

Proof.

In view of Lemma 2.15, it suffices to show that if $q\circ\varphi$ is non-contracting, then $\varphi$ is non-contracting. Suppose for contradiction that $q\circ\varphi$ is non-contracting, but $\varphi$ is contracting. Let $V$ be a rational representation of the smallest dimension such that there exists a $v\in V\smallsetminus\{0\}$ with $\lim_{t\to\infty}\varphi(t).v=0$ . Since $U$ is a unipotent group, the subspace of $U$ -fixed points $V^{U}:=\{x\in V:u.x=x,\forall u\in U\}$ is of positive dimension. Since $U$ is a normal subgroup, $V^{U}$ is $G$ -invariant. Then $G$ acts on $V/V^{U}$ , and the natural quotient map $\pi:V\to V/V^{U}$ is $G$ -equivariant. Now,

\lim_{t\to\infty}\varphi(t).\pi(v)=\lim_{t\to\infty}\pi(\varphi(t).v)=0.

Since $\dim(V/V^{U})<\dim(V)$ , we get that $\pi(v)=0$ . Namely, $v\in V^{U}$ , and again, since $V$ is of smallest dimension with a $\varphi(t)$ -contracting nonzero vector, we get $V=V^{U}$ . Since $U$ acts trivially on $V$ , the action of $G$ on $V$ factors through the action of $G/U$ . Therefore, $\varphi(t)v=(q\circ\varphi(t))v$ . Since $\varphi(t)v\to 0$ as $t\to\infty$ , and $q\circ\varphi$ is non-contracting, we get a contradiction. ∎

Lemma 2.17.

Let $G_{1},G_{2}$ be the real points of algebraic groups over $\mathbb{R}$ . Suppose that $f:G_{1}\to G_{2}$ is a homomorphism and $\varphi:[0,\infty)\to f(G_{1})$ is a curve definable in an o-minimal structure $\mathscr{S}$ . Then there exists a definable lift $\psi:[0,\infty)\to G_{1}$ . Namely, $\psi$ is definable in $\mathscr{S}$ , and $f\circ\psi=\varphi$ .

Proof.

Consider the definable set

\mathcal{D}:=\{(t,g):g\in G_{1},~{}t\geq 0,~{}f(g)=\varphi(t)\}.

By the choice function theorem [DM96, Theorem 4.5], there exists a definable curve $\psi:[0,\infty)\to G_{1}$ such that $(t,\psi(t))\in\mathcal{D}$ for all $t\geq 0$ . ∎

Lemma 2.18.

Let $G_{1}$ , $G_{2}$ be the $\mathbb{R}$ -points of irreducible semi-simple groups over $\mathbb{R}$ , and let $f:G_{1}\to G_{2}$ be homomorphism with finite kernel. Let $\psi:[0,\infty)\to G_{1}$ be a definable curve. Suppose that $f\circ\psi$ is non-contracting for $G_{2}$ , then $\psi$ is non-contracting for $G_{1}$ .

Proof.

Let $V$ be a finite dimensional irreducible rational representation of $G_{1}$ . Then, by Schur’s lemma the center of $G_{1}$ acts via a character on $V\otimes\mathbb{C}$ . Then, $\ker f$ , which is central in $G_{1}$ , acts trivially on the $n$ -fold tensor product $W=\otimes^{n}(V\otimes\mathbb{C})$ , where $n$ is the order of $\ker f$ . Therefore, the action of $G_{1}$ on $W$ factors through an action of $G_{1}/(\ker f)=G_{2}$ on $W$ . Let $v\in V$ . If $\psi(t).v\to 0$ , then $f\circ\psi(t).(v\otimes\cdots\otimes v)\to 0$ as $t\to\infty$ . But, since $f\circ\psi$ is non-contracting, we get that $v=0$ . ∎

Lemma 2.19.

Let $G_{1},G_{2}$ be the real points of two affine algebraic groups over $\mathbb{R}$ , and let $f:G_{1}\to G_{2}$ be a surjective homomorphism of algebraic groups. Let $\varphi:[0,\infty)\to G_{2}$ be a non-contracting curve in $G_{2}$ definable in $\mathscr{S}$ . Then, $\varphi$ has a definable lift $\psi:[0,\infty)\to G_{1}$ that is a non-contracting in $G_{1}$ .

Proof.

In this proof, all subgroups considered are the open subgroups of $\mathbb{R}$ -points of algebraic groups defined over $\mathbb{R}$ .

Without loss of generality, since the hull is generated by one-parameter unipotent subgroups, by modifying $\varphi$ with a bounded correcting curve and replacing $G_{2}$ by $(G_{2})_{u}$ , the subgroup of $G_{2}$ generated by unipotent one-parameter subgroups, we may assume that the radical of $G_{2}$ is unipotent, call it $U_{2}$ . Let $\bar{G}_{2}=G_{2}/U_{2}$ and $q_{2}:G_{2}\to\bar{G}_{2}$ be the quotient homomorphism. Let $U_{1}$ be the unipotent radical of $G_{1}$ . Then, there exists a reductive subgroup $M_{1}$ of $G_{1}$ such that $G_{1}=M_{1}U_{1}$ and $M_{1}\cap U_{1}=\{e\}$ , see [Rag72, Page 11]. Since $f$ is surjective, $f(U_{1})$ is a unipotent normal subgroup of $G_{2}$ , and hence $f(U_{1})\subset U_{2}$ . In fact, $f(U_{1})=U_{2}$ . This can be proved as follows. $f^{-1}(U_{2})=(M_{1}\cap f^{-1}(U_{2}))U_{1}$ . Since $f^{-1}(U_{2})$ is a normal subgroup of $G_{1}$ , $M_{1}\cap f^{-1}(U_{2})$ is reductive. So $f(M_{1}\cap f^{-1}(U_{2}))$ being a reductive subgroup of $U_{2}$ , it is trivial. Since $f$ is surjective, we get $f(U_{1})=U_{2}$ . Therefore, $q_{2}\circ f:M_{1}\to\bar{G}_{2}$ is surjective. Since $\bar{G}_{2}$ is semisimple, $q_{2}\circ f([M_{1},M_{1}])=\bar{G}_{2}$ . Let $L_{2}=[M_{1},M_{1}]\cap\ker(q_{2}\circ f)$ . Since $L_{2}$ is a normal subgroup of the semisimple group $[M_{1},M_{1}]$ , there exists a semisimple normal subgroup $L_{1}$ of $[M_{1},M_{1}]$ such that $[M_{1},M_{1}]=L_{1}L_{2}$ , where $L_{1}\cap L_{2}$ is a finite central subgroup of $L_{1}$ . Therefore, $q_{2}\circ f:L_{1}\to\bar{G}_{2}$ is a surjective homomorphism whose kernel is $L_{1}\cap L_{2}$ . Also, since $\bar{G}_{2}$ is generated by unipotent one-parameter subgroups, it has no compact simple normal subgroup. Therefore, $L_{1}$ also does not have a compact normal subgroup, and $L_{1}$ is generated by unipotent one-parameter subgroups.

Let $\tilde{G}_{1}=L_{1}U_{1}$ . Since $q_{2}\circ f(L_{1})=G_{2}/U_{2}$ and $f(U_{1})=U_{2}$ , we have $f:\tilde{G}_{1}\to G_{2}$ is a surjective map. Therefore, by Lemma 2.17, there exists a definable $\psi:[0,\infty)\to\tilde{G}_{1}$ such that $f\circ\psi=\varphi$ . Let $q_{1}:\tilde{G}_{1}\to\tilde{G}_{1}/U_{1}$ be the natural quotient map. Since $q_{1}:L_{1}\to\tilde{G}_{1}/U_{1}$ is surjective (it is an isomorphism), let $\tilde{\psi}:[0,\infty)\to L_{1}$ be a definable curve such that $q_{1}\circ\tilde{\psi}=q_{1}\circ\psi$ . Since, $U_{1}\subset\ker q_{2}\circ f$ , $q_{2}\circ f$ factors through $q_{1}$ . Therefore, we have $(q_{2}\circ f)\circ\tilde{\psi}=(q_{2}\circ f)\circ\psi=q_{2}\circ\varphi$ . Now $q_{2}\circ f:L_{1}\to\bar{G}_{2}$ is surjective with finite central kernel, and $q_{2}\circ\varphi$ is definable and non-contracting in $\bar{G}_{2}$ . Therefore, by Lemma 2.18, $\tilde{\psi}$ is non-contracting in $L_{1}$ . Therefore, $q_{1}\circ\psi=q_{1}\circ\tilde{\psi}$ is non-contracting in $\tilde{G}_{1}/U_{1}$ . Therefore, by Lemma 2.16, $\psi$ is non-contracting in $\tilde{G}_{1}$ . Finally, by Lemma 2.15 we get that $\psi$ is non-contracting in $G_{1}$ . ∎

The following observation extends Lemma 2.18. Notice that Proposition 1.7 from the introduction is a direct consequence of the proposition below.

Proposition 2.20.

Suppose that $G_{1}$ is a real algebraic group generated by unipotent one-parameter subgroups. Let $f:G_{1}\to G_{2}$ be a homomorphism of real algebraic groups with finite kernel. Then a definable curve $\varphi:[0,\infty)\to G_{1}$ is non-contracting in $G_{1}$ if and only if $f\circ\varphi$ is non-contracting in $G_{2}$ .

Proof.

If $\varphi$ is non-contracting in $G_{1}$ , then by Lemma 2.15, it follows that $f\circ\varphi$ is non-contracting in $G_{2}$ .

Now suppose that $f\circ\varphi$ is non-contracting in $G_{2}$ . Since $G_{1}$ is generated by unipotent one-parameter subgroups, $f(G_{1})$ is a subgroup of $G_{2}$ generated by unipotent one-parameter subgroups. Therefore, the solvable radical of $f(G_{1})$ is unipotent. Hence, $f(G_{1})$ is an observable subgroup of $G_{2}$ , see [Gro97, Corollary 2.8]. Therefore, given any finite-dimensional rational representation of $V$ of $f(G_{1})$ , $V$ is a subrepresentation of a finite-dimensional representation $W$ of $G_{2}$ restricted to $f(G_{1})$ . Let $v\in V$ . Suppose that $f\circ\varphi(t)v\to 0$ . Since $v\in V\subset W$ and $f\circ\varphi$ is non-contracting in $G_{2}$ , we conclude that $v=0$ . This proves that $f\circ\varphi$ is non-contracting for $f(G_{1})$ . Therefore, replacing $G_{2}$ by $f(G_{1})$ , we may assume that $f$ is surjective. Therefore, by Lemma 2.19 there exists a lift $\psi:[0,\infty)\to G_{1}$ of $f\circ\varphi$ which is definable and non-contracting in $G_{1}$ . By the Monotonicty Theorem [DM96, Theorem 4.1], there is a $T_{0}\geq 0$ such that $\varphi$ and $\psi$ are continuous on $[T_{0},\infty)$ . Let $z=\varphi(T_{0})\psi(T_{0})^{-1}\in\ker f$ . So, $t\mapsto z\psi(t)$ and $t\mapsto\varphi(t)$ are two lifts of $f\circ\varphi$ from $[T_{0},\infty)$ to $G_{1}$ from the point $\varphi(T_{0})\in G_{1}$ . Since $\ker f$ is finite, $f$ is a covering map, we get $\varphi(t)=z\psi(t)$ for all $t\in[0,\infty)$ . Since $\psi$ is non-contracting, so is $\varphi$ . ∎

3. $(C,\alpha)$ -good property

In our setting, a property more fundamental than the $(C,\alpha)$ -good property (Theorem 1.14) is the following.

Definition 3.1.

Let $\delta\in(0,1]$ . A family $\mathscr{F}$ of real functions defined on $[T_{0},\infty)$ is called $\delta$ -good if there exists a constant $M(\delta)$ depending only on $\delta$ such that

(3.1)

\frac{\|f\|_{I}}{\|f\|_{I_{\delta}}}\leq M(\delta),

for all $f\in\mathscr{F}$ and all bounded sub-intervals $I_{\delta}\subset I\subseteq[T_{0},\infty]$ satisfying $|I_{\delta}|=\delta|I|$ .

Such inequalities are well-known in the literature as Remez-type inequalities; see [Rem36]. We have the following theorem, which will imply Theorem 1.14.

Theorem 3.2.

Let $\mathcal{V}$ be a finite-dimensional vector space of real functions defined on $[0,\infty)$ which are definable in a polynomially bounded o-minimal structure. Suppose that $\mathscr{F}\subseteq\mathcal{V}$ is a closed definable cone such that for all $f\in\mathscr{F}\smallsetminus\{0\}$ ,

\lim_{t\to\infty}f(t)\neq 0.

Then, there exists $T_{0}>0$ such that for all $\delta\in(0,1]$ and all $f\in\mathscr{F}\smallsetminus\{0\}$ it holds that $f$ restricted to $[T_{0},\infty)$ is $\delta$ -good.

We now proceed to prove Theorem 1.14 by assuming Theorem 3.2, and the rest of the section will be dedicated to proving Theorem 3.2.

3.1. Proving Theorem 1.14 via $\delta$ -goodness

We first note the following Corollary from Theorem 3.2.

Corollary 3.3.

Let $\mathcal{V}$ be a finite dimensional vector space of real functions on $[0,\infty)$ which are definable in a polynomially bounded o-minimal structure. Suppose that $\mathscr{F}\subseteq\mathcal{V}$ is a closed definable cone such that for all $f\in\mathscr{F}\smallsetminus\{0\}$ ,

\lim_{t\to\infty}f(t)\neq 0.

Let $T_{0}\geq 1$ such that the outcome of Theorem 3.2 holds. Then, there exists $\mathrm{m},r>0$ such that for all $x\geq 1$ , $f\in\mathscr{F}\smallsetminus\{0\}$ and $I^{\prime}\subseteq I\subseteq[T_{0},\infty)$ such that $\frac{|I|}{|I^{\prime}|}\leq x$ it holds that

(3.2)

\|f\|_{I}\leq\mathrm{m}x^{r}\|f\|_{I^{\prime}}.

Proof.

Consider the definable set:

	$\displaystyle S:=\{(x,y):$	$\displaystyle x,y>0\text{, such that }\frac{\\|f\\|_{I}}{\\|f\\|_{I^{\prime}}}\leq y,\,\forall 0\neq f\in\mathscr{F},$
		$\displaystyle I^{\prime}\subseteq I\subseteq[T_{0},\infty),\,\frac{\|I\|}{\|I^{\prime}\|}\leq x\}.$

By Theorem 3.2, the projection of $S$ to the first coordinate includes $[1,\infty).$ Then, by the definable choice theorem (see [DM96, Section 4]), there exists a function $\phi:[1,\infty)\to\mathbb{R}^{2}$ definable in the same polynomially bounded o-minimal structure, such that

(x,\phi(x))\in S,~{}\forall x\geq 1.

Namely for all $x>1$ , $f\in\mathscr{F}\smallsetminus\{0\}$ and $I^{\prime}\subseteq I\subseteq[T_{0},\infty)$ such that $\frac{|I|}{|I^{\prime}|}\leq x$ it holds that

\|f\|_{I}\leq\phi(x)\|f\|_{I^{\prime}}.

Since $\phi$ is polynomially bounded, there exists $r>0$ and $T_{1}\geq 1$ such that $\frac{\phi(x)}{x^{r}}\leq c_{0}$ for all $x\geq T_{1}$ , for some $c_{0}>0$ . Finally, since $\frac{\phi(x)}{x^{r}}$ is bounded in $[1,T_{1}]$ , the result follows. ∎

We are now ready to prove the $(C,\alpha)$ -good property. Our proof is based on [KM98, proof of Proposition 3.2].

Proof of Theorem 1.14.

By o-minimality, the number of connected components of the family of definable (in parameters) sub-level sets

\{t\in I:|f(t)|\leq\epsilon\},~{}\epsilon>0,~{}I\subset[T_{0},\infty),~{}f\in\mathscr{F}\smallsetminus\{0\},

is bounded uniformly, say by $K$ . Now fix $I\subset[T_{0},\infty),~{}\epsilon>0$ and an $f\in\mathscr{F}\smallsetminus\{0\}$ . Let

I^{\prime}\subseteq\{t\in I:|f(t)|\leq\epsilon\}

be an interval of maximum length. Then

L\leq K|I^{\prime}|,

where

L:=|\{t\in I:|f(t)|\leq\epsilon\}|.

The latter inequality implies,

\frac{|I|}{|I^{\prime}|}\leq\frac{|I|}{L/K}.

By Corollary 3.3, we get

\|f\|_{I}\leq\mathrm{m}\left(\frac{|I|}{L/K}\right)^{r}\|f\|_{I^{\prime}}\leq\mathrm{m}\left(\frac{|I|}{L/K}\right)^{r}\epsilon.

Reordering the latter inequality, we get that

|\{t\in I:|f(t)|\leq\epsilon\}|=L\leq\mathrm{m}^{\frac{1}{r}}K\left(\frac{\epsilon}{\|f\|_{I}}\right)^{\frac{1}{r}}|I|.

∎

3.2. Proving Theorem 3.2

The space of polynomials of bounded degrees is the basic example of $\delta$ -good functions. We provide a proof for the sake of completeness.

Proposition 3.4 ([DM93, Lemma 4.1]).

Fix $n\in\mathbb{N}$ , and consider $\mathscr{F}:=\text{Span}_{\mathbb{R}}\{1,x,...,x^{n}\}\smallsetminus\{0\}$ . Then $\mathscr{F}$ is $\delta$ -good on $\mathbb{R}$ . More precisely:

(3.3)

\frac{\|c_{0}+c_{1}x+\cdots+c_{n}x^{n}\|_{I}}{\|c_{0}+c_{1}x+\cdots+c_{n}x^{n}\|_{I_{\delta}}}\leq(n+1)\frac{n^{n}}{\delta^{n}},

for all $c_{0},c_{1},...,c_{n}\in\mathbb{R}$ not all zero, interval $I\subset\mathbb{R}$ and sub-interval $I_{\delta}\subset I$ satisfying $|I_{\delta}|=\delta|I|>0.$

Proof.

Let $f(x):=c_{0}+c_{1}x+\cdots+c_{n}x^{n}$ and $I=[a,a+T]$ . Let $a\leq t_{0}\leq a+T-\delta T$ , and denote $t_{i}:=t_{0}+\frac{i\delta T}{n}$ for $i=0,1,...,n$ , $a\leq t_{0}<t_{1}<\cdots<t_{n}\leq a+T$ , and let $I_{\delta}:=[t_{0},t_{n}]$ . Then by polynomial interpolation of $f$ at points $t_{0},t_{1},...,t_{n}$ , we have

(3.4)

f(t)=\sum_{j=0}^{n}\frac{\prod_{i\neq j}(t-t_{i})}{\prod_{i\neq j}(t_{j}-t_{i})}f(t_{j}).

Now,

\left|\frac{\prod_{i\neq j}(t-t_{i})}{\prod_{i\neq j}(t_{j}-t_{i})}\right|\leq\frac{n^{n}}{\delta^{n}},~{}\forall t\in I,

and thus, by triangle inequality,

\left|f(t)\right|\leq(n+1)\frac{n^{n}}{\delta^{n}}\max{\left|f(t_{i})\right|}\leq(n+1)\frac{n^{n}}{\delta^{n}}\|f\|_{I_{\delta}},~{}\forall t\in I.

∎

We will need the following facts:

•

For any definable function $f:[0,\infty)\to\mathbb{R}$ , there exits $t_{0}\geq 0$ such that $f$ differentiable on $[t_{0},\infty)$ and $f^{\prime}$ is continuous and definable on $[t_{0},\infty)$ (see [DM96, Cell decompsition, Section 4]).
•

For a definable function $f:[0,\infty)\to\mathbb{R}$ in a polynomially bounded o-minimal structure with $f(t)\sim t^{r}$ where $r\neq 0$ , we have $f^{\prime}(t)\sim rt^{r-1},\text{ as }t\to\infty$ , see [Mil94b, Proposition 3.1]. In particular, if $\deg(f)\neq 0,$ then $\deg(f^{\prime})=\deg(f)-1$ .

Lemma 3.5.

For $i=0,1,\ldots,N$ , let $F_{i}:[0,\infty)\to\mathbb{R}$ be a function definable in a polynomially bounded o-minimal structure such that $F_{i}$ is not eventually the zero function. Suppose that the degrees $\deg(F_{0}),\deg(F_{1}),\ldots,\deg(F_{N})$ are all distinct. Then, there exists a $T_{0}>0$ such that for any interval $I\subset[T_{0},\infty)$ it holds that

\|c_{0}F_{0}+c_{1}F_{1}+...+c_{N}F_{N}\|_{I}>0

for all $(c_{0},c_{1},...,c_{N})\neq 0$ .

Proof.

By o-minimality, the definable set $F_{0}^{-1}(0)$ has finitely many connected components. Since $F_{0}$ is not eventually the zero function, for some $T_{0}>0$ , $F_{0}(t)\neq 0$ for all $t\geq T_{0}$ . Thus for all $I\subset[T_{0},\infty)$ we have

\left\|c_{0}F_{0}+c_{1}F_{1}+...+c_{N}F_{N}\right\|_{I}>0\iff\left\|c_{0}+c_{1}\frac{F_{1}}{F_{0}}+...+c_{N}\frac{F_{N}}{F_{0}}\right\|_{I}>0.

Clearly, since the degrees of $F_{0},F_{1},...,F_{N}$ are distinct, the degrees of $1,F_{1}/F_{0},F_{2}/F_{0},\cdots F_{N}/F_{0}$ are distinct. Thus, it is sufficient to prove the statement under the assumption that $F_{0}(t)\equiv 1$ .

We now prove the statement for $N=1$ , and then argue by induction. Since $\deg(F_{1})\neq\deg(F_{0})=0$ , we let $T_{0}>0$ be large enough such that $F_{1}^{\prime}(t)\neq 0$ for all $t\geq T_{0}$ . Assume that for an interval $I\subset[T_{0},\infty)$ we have $\|c_{0}+c_{1}F_{1}\|_{I}=0$ . Since $c_{0}+c_{1}F_{1}(t)=0$ for all $t\in I$ , we get that

\frac{d}{dt}\left(c_{0}+c_{1}F_{1}(t)\right)=c_{1}F_{1}^{\prime}(t)=0,~{}\forall t\in I,

and as $F_{1}^{\prime}(t)\neq 0$ for all $t\in[T_{0},\infty)$ , it follows that $c_{1}=0$ . As a consequence, $c_{0}=0$ .

Now let $N\geq 2$ , and let $T_{1}$ such that the functions $F_{1},F_{2},...,F_{n}$ are differentiable in $[T_{1},\infty)$ . Note that since the degrees of the functions $1,F_{1},F_{2},...$ $,F_{n}$ are distinct, we have that $\deg(F_{i})\neq 0$ for all $1\leq i\leq N$ . In particular, $\deg(F_{i}^{\prime})=\deg(F_{i})-1$ , and so the degrees of $F_{1}^{\prime},F_{2}^{\prime},\cdots,F_{N}^{\prime}$ are all distinct.

Now, if

\|c_{0}+c_{1}F_{1}+...+c_{N}F_{N}\|_{I}=0,

then

\|c_{1}F_{1}^{\prime}+...+c_{N}F_{N}^{\prime}\|_{I}=0.

The claim now follows by induction. ∎

Note that if $F_{0}(t),F_{1}(t),...,F_{N}(t)$ are definable functions, then they are continuously differentiable $(N+1)$ times for all $t\geq T$ for some $T$ . Hence, the Wronskian matrix $W(F_{0},...,F_{N})(t)$ which is the $(N+1)\times(N+1)$ matrix whose $k$ -th row is $(F^{(k)}_{0}(t),F_{1}^{(k)}(t),\ldots,F_{N}^{(k)}(t))$ for $0\leq k\leq N$ is well defined for all $t\geq T$ .

Lemma 3.6.

Suppose that $F_{i}:[0,\infty)\to\mathbb{R},~{}i=0,1,...,N$ , are functions definable in a polynomially bounded o-minimal structure such that the degrees $\deg(F_{i})$ , for $0\leq i\leq N$ are distinct. Then, there exists a $T_{0}>0$ such that the functions $F_{0},...,F_{N}$ are $(N+1)$ -time continuously differentiable and $W(F_{0},...,F_{N})(t)$ is non-singular for all $t\geq T_{0}$ .

Proof.

Since $\det(W(F_{0},...,F_{N})(t))$ is a definable function, then either

|\det(W(F_{0},...,F_{N})(t))|>0,

for all large $t$ or $\det(W(F_{0},...,F_{N})(t))=0$ for all large $t$ . Suppose for contradiction the latter case, say $\det(W(F_{0},...,F_{N})(t))=0,\forall t\geq T_{1}$ . Then, by a classical result of Bôcher [Bôc01], we get that $F_{0},F_{1},...,F_{N}$ are linearly dependent on any sub-interval of $[T_{1},\infty)$ . This contradicts Lemma 3.5. ∎

Lemma 3.7.

Suppose that $\mathcal{V}$ is a finite dimensional vector space of functions $f:[0,\infty)\to\mathbb{R}$ definable in a polynomially bounded o-minimal structure. Suppose that $\mathcal{V}$ does not contain non-zero functions, which are constantly zero eventually. Then there exist a basis $\{F_{1},...,F_{N}\}$ of $\mathcal{V}$ , where

(3.5)

\deg(F_{1})<\cdots<\deg(F_{N}).

Proof.

We argue by induction. The claim is trivial for $N=1$ . Now choose an arbitrary basis $\{H_{1},...,H_{N}\}$ of $\mathcal{V}$ , and suppose without loss of generality that $\deg(H_{i})\leq\deg(H_{N})$ for all $i$ . For each $i$ such that $\deg(H_{i})=\deg(H_{N})$ , we have $\deg(H_{i}-c_{i}H_{N})<\deg(H_{N})$ for some $c_{i}\neq 0$ , and we replace $H_{i}$ by $H_{i}-c_{i}H_{N}$ . Now, the modified basis $\{H_{1},...,H_{N}\}$ is such that $\deg(H_{i})<\deg(H_{N}),\forall i<N$ . The vector space $\mathcal{U}:=\text{Span}_{\mathbb{R}}\{H_{1},...,H_{N-1}\}$ is $(N-1)$ -dimensional, and $\deg(f)<\deg(H_{N})$ for all $f\in\mathcal{U}$ . By induction, there is a basis $\{F_{1},...,F_{N-1}\}$ for $\mathcal{U}$ such that $\deg(F_{i})<\deg(F_{i+1})$ for all $i$ . In particular, $\{F_{1},...,F_{N-1},F_{N}\}$ with $F_{N}:=H_{N}$ is the required basis for $\mathcal{V}$ . ∎

Remark 3.8.

If $\mathcal{V}$ is a finite dimensional vector spanned by definable functions, then the subset of functions which are eventually constantly zero forms a finite dimensional subspace $\mathcal{V}_{0}\leq\mathcal{V}$ . This implies that there exists a uniform $T_{0}>0$ such that for all $f\in\mathcal{V}_{0}$ , it holds that $f(t)=0,\forall t\geq T_{0}$ . Thus, upon restricting the functions in $\mathcal{V}$ to $[T_{0},\infty)$ , the subspace $\mathcal{V}_{0}$ is the trivial vector space.

We note the following fact:

•

A definable function $f:[0,\infty)\to\mathbb{R}$ either converges as $t\to\infty$ or diverges to $\infty$ or to $-\infty$ as $t\to\infty,$ see [DM96, Montonicity theorem, Section 4].

Proof of Theorem 3.2.

Let $\mathcal{V}$ be a finite dimensional vector space of functions $f:[0,\infty)\to\mathbb{R}$ definable in a polynomially bounded o-minimal structure. Because of Remark 3.8, there is no loss in generality in assuming that the only eventually constantly zero function is the zero function. So we may choose a basis $\{F_{1}^{-},...,F_{n}^{-},F_{0}^{+},F_{1}^{+},...,F_{m}^{+}\}$ for $\mathcal{V}$ such that

\deg(F_{1}^{-})<\cdots<\deg(F_{n}^{-})<0\leq\deg(F^{+}_{0})<\cdots<\deg(F_{m}^{+})

, where $m,n\in\mathbb{N}\cup\{0\}$ . Let $N=m+n+1$ .

We will denote

r_{i}=\deg(F^{+}_{i}),~{}-\kappa_{j}:=\deg(F^{-}_{j})

where $r_{i},\kappa_{j}\geq 0$ . For

\underline{c}:=(c_{1}^{-},\ldots,c_{n}^{-},c_{0}^{+},\ldots,c^{+}_{m})\in\mathbb{R}^{N}

we denote

(3.6)		$\displaystyle f_{\underline{c}}:=c^{-}_{1}F^{-}_{1}+\cdots+c^{-}_{n}F^{-}_{n}+c^{+}_{0}F_{0}^{+}\cdots+c^{+}_{m}F^{+}_{m},$
(3.7)		$\displaystyle f^{-}_{\underline{c}}:=c^{-}_{1}F^{-}_{1}+...+c^{-}_{n}F^{-}_{n},\text{ and }$
(3.8)		$\displaystyle f^{+}_{\underline{c}}:=c^{+}_{0}F_{0}^{+}+...+c^{+}_{m}F^{+}_{m}.$

Let $\mathscr{F}\subseteq\mathcal{V}$ be a closed definable cone such that for all $0\neq f\in\mathscr{F}$ we have

\lim_{t\to\infty}f(t)\neq 0.

•

Let $\tilde{\mathscr{F}}=\{\underline{c}\in\mathbb{R}^{n+m+1}:f_{\underline{c}}\in\mathscr{F}\}$ , which is a closed cone in $\mathbb{R}^{m+n+1}$ .

In addition, we assume (without loss of generality) that

\lim_{t\to\infty}\frac{F^{+}_{i}(t)}{t^{r_{i}}}=1,~{}\lim_{t\to\infty}\frac{F^{-}_{j}(t)}{t^{-\kappa_{j}}}=1.

We choose $T_{0}\geq 1$ such that the outcome of Lemma 3.6 holds. Fix $\delta\in(0,1)$ , and assume for contradiction that $\mathscr{F}\smallsetminus\{0\}$ is not $\delta$ -good on $[T_{0},\infty)$ . Consider the following definable subset:

(3.9)

\mathcal{A}:=\left\{(s,\underline{c},a,l,\alpha):\begin{array}[]{c}s\geq 1,~{}\underline{c}\in\tilde{\mathscr{F}}\smallsetminus\{0\},~{}a>T_{0},~{}l>0,\\ a\leq\alpha\leq a+l-\delta l,~{}\frac{\|f_{\underline{c}}\|_{[a,a+l]}}{\|f_{\underline{c}}\|_{[\alpha,\alpha+\delta l]}}>s\\ \end{array}\right\}.

By the assumption for contradiction, the projection of $\mathcal{A}$ to the first coordinate is $\mathbb{R}_{\geq 1}$ . By the choice function theorem (cf. [DM96, Section 4]), there exists a polynomially bounded definable curve

\phi(s):=(\underline{c}(s),a(s),l(s),\alpha(s))\text{ for $s\geq 1$},

such that $(s,\phi(s))\in\mathcal{A},~{}\forall s\geq 1.$ In particular, there exists an $C_{0}\neq 0$ and an $\kappa\in\mathbb{R}$ such that

(3.10)

\frac{a(s)}{l(s)}\sim C_{0}s^{\kappa},\text{ as $s\to\infty$}.

Let $\lVert\cdot\rVert_{1}$ denote the $\ell_{1}$ -norm on $\mathbb{R}^{N}$ , which is the sum of absolute values of the coordinates. Let

(3.11)

\hat{\underline{c}}(s):=\frac{\underline{c}(s)}{\|\underline{c}(s)\|_{1}},\,\forall s\geq 1.

We observe that by o-minimality, since $\lVert\hat{\underline{c}}(s)\rVert_{1}=1$ , $\hat{\underline{c}}(s)$ converges to some vector v with $\|\textbf{v}\|_{1}=1$ . Since $\underline{c}(s)\in\tilde{\mathscr{F}}\setminus\{0\}$ and $\tilde{\mathscr{F}}$ is a closed cone, we get $v\in\tilde{\mathscr{F}}\setminus\{0\}$ .

Case 1: $\kappa\leq 0$ . In this case, we treat two sub-cases in which $l(s)$ is bounded or not.

Case 1.1. $l(s)$ is bounded in $s$ . Then $a(s)$ is also bounded, and by o-minimality $l(s)$ and $a(s)$ converge as $s\to\infty$ . We first observe that $\lim_{s\to\infty}l(s)\neq 0$ . In fact, if otherwise $\lim_{s\to\infty}l(s)=0$ , then $\lim_{s\to\infty}\frac{a(s)}{l(s)}=\infty$ since $a(s)\geq T_{0}\geq 1,\forall s\geq 1$ , which is a contradiction to the assumption that $\kappa\leq 0$ . Also, the end-points of the intervals $[\alpha(s),\alpha(s)+\delta l(s)]\subset[a(s),a(s)+l(s)]$ converge to the end-points of intervals $I_{\delta}\subset I\subseteq[T_{0},\infty)$ of positive length, where $\frac{|I_{\delta}|}{|I|}=\delta$ . In particular, $\|f_{\hat{\underline{c}}(s)}\|_{[a(s),a(s)+l(s)]}$ is uniformly bounded in $s$ . Now

	$\displaystyle\frac{\\|f_{\hat{\underline{c}}(s)}\\|_{[a(s),a(s)+l(s)]}}{\\|f_{\hat{\underline{c}}(s)}\\|_{[\alpha(s),\alpha(s)+\delta l(s)]}}$	$\displaystyle=\frac{\\|f_{\underline{c}(s)}/\lVert\underline{c}(s)\rVert_{1}\\|_{[a(s),a(s)+l(s)]}}{\\|f_{\underline{c}(s)}/\lVert\underline{c}(s)\rVert_{1}\\|_{[\alpha(s),\alpha(s)+\delta l(s)]}}$
		$\displaystyle=\frac{\\|f_{\underline{c}(s)}\\|_{[a(s),a(s)+l(s)]}}{\\|f_{\underline{c}(s)}\\|_{[\alpha(s),\alpha(s)+\delta l(s)]}}>s.$

Namely, we have

\frac{\|f_{\hat{\underline{c}}(s)}\|_{[a(s),a(s)+l(s)]}}{s}\geq\|f_{\hat{\underline{c}}(s)}\|_{[\alpha(s),\alpha(s)+\delta l(s)]}.

By taking $s\to\infty$ we get

0=\|f_{\textbf{v}}\|_{I_{\delta}}.

This is a contradiction to Lemma 3.5.

Case 1.2. $l(s)$ is unbounded in $s$ . Then, by o-minimality $l(s)\to\infty$ .

For $s\geq 1$ , we define

(3.12)

\displaystyle C^{+}(s)=\sum_{i=0}^{m}|c^{+}_{i}(s)|l(s)^{r_{i}},\,C^{-}(s)=\sum_{j=1}^{n}|c^{-}_{j}(s)|\text{, and }C(s)=C^{+}(s)+C^{-}(s).

Then, the by properties of bounded definable functions, the following limit exists in $\mathbb{R}^{m+1}$ .

(3.13)

(w^{+}_{0},...,w^{+}_{m})=\lim_{s\to\infty}\frac{1}{C(s)}(c^{+}_{0}(s)l(s)^{r_{0}},...,c^{+}_{m}(s)l(s)^{r_{m}}).

We denote

\varphi_{{\underline{c}}(s)}(t):=\frac{f_{\underline{c}(s)}(a(s)+tl(s))}{\lambda(s)},\,\forall t\in[0,1].

We first want to verify the following claim to be used in further arguments.

Claim 1.

(3.14)

\sup_{s\geq S_{0}}\lVert\varphi_{{\underline{c}}(s)}\rVert_{[0,1]}<\infty.

Moreover, for any $t_{0}\in(0,1)$ .

(3.15)

\lim_{s\to\infty}\sup_{t\in[t_{0},1]}\lvert\varphi_{\underline{c}(s)}(t)-w^{+}_{0}\left(x_{0}+t\right)^{r_{0}}+\cdots+w^{+}_{m}\left(x_{0}+t\right)^{r_{m}}\rvert=0.

Proof of Claim 1.

For each $0\leq i\leq m$ , there exists $\nu_{i}>0$ such that for all $t\in[0,1]$ , we have

	$\displaystyle\frac{F^{+}_{i}(a+tl)}{l^{r_{i}}}$	$\displaystyle=\frac{\left(a+lt\right)^{r_{i}}}{l^{r_{i}}}+O\left(\frac{\left(a+lt\right)^{r_{i}-\nu_{i}}}{l^{r_{i}}}\right)$
(3.16)			$\displaystyle=\left(\frac{a}{l}+t\right)^{r_{i}}+O\left(l^{-\nu_{i}}\left(\frac{a}{l}+t\right)^{r_{i}-\nu_{i}}\right).$

Since $\frac{a(s)}{l(s)}$ is bounded, we have $\lim_{s\to\infty}\frac{a(s)}{l(s)}=x_{0}\in[0,\infty)$ . We now make the following observations:

•

We can pick $S_{0}\geq 1$ such that $l(s)\geq 1$ for all $s\geq S_{0}$ and the following two statements hold.

(1)

$\sup_{t\in[0,1]}\left(\frac{a(s)}{l(s)}+t\right)^{r_{i}}$ is bounded for all $s\geq S_{0}$ and for all $0\leq i\leq m$ .

This is immediate since $r_{i}\geq 0$ , $\lim_{s\to\infty}\frac{a(s)}{l(s)}=x_{0}$ and $t\in[0,1]$ .

(2)

$\sup_{t\in[0,1]}l(s)^{-\nu_{i}}\left(\frac{a(s)}{l(s)}+t\right)^{r_{i}-\nu_{i}}$ is bounded for all $s\geq S_{0}$ and for all $0\leq i\leq m$ .

This is clear if $r_{i}-\nu_{i}\geq 0$ . Now suppose that $r_{i}-\nu_{i}<0$ . Then, for all $t\in[0,1]$ , since $a(s)\geq T_{0}\geq 1$ and $l(s)\geq 1$ ,

\displaystyle l(s)^{-\nu_{i}}\left(\frac{a(s)}{l(s)}+t\right)^{r_{i}-\nu_{i}}\leq l(s)^{-\nu_{i}}\left(\frac{a(s)}{l(s)}\right)^{r_{i}-\nu_{i}}=\frac{a(s)^{r_{i}-\nu_{i}}}{l(s)^{r_{i}}}\leq 1.

•

Since, by definability, $\lvert a(s)/l(s)-x_{0}\rvert=O(l(s)^{-\epsilon})$ for some $\epsilon>0$ , for each $0\leq i\leq m$ , there exists $\epsilon_{i}>0$ such that as $s\to\infty$

$\sup_{t\in[0,1]}\left|\left(\frac{a(s)}{l(s)}+t\right)^{r_{i}}-\left(x_{0}+t\right)^{r_{i}}\right|=O(l(s)^{\epsilon_{i}}).$
•

For any $0\leq i\leq m$ , as $s\to\infty$ ,

$\sup_{t\in[0,1]}l(s)^{-\nu_{i}}\left|\frac{a(s)}{l(s)}+t\right|^{r_{i}-\nu_{i}}=O(l(s)^{-\nu_{i}}).$
•

Let $\nu:=\min\{\epsilon_{0},\ldots,\epsilon_{m},\nu_{0},\ldots,\nu_{n}\}>0$ . Then, for all $0\leq i\leq m$ ,

$\sup_{t\in[0,1]}\left|\frac{F^{+}_{i}(a(s)+tl(s))}{l(s)^{r_{i}}}-(x_{0}+t)^{r_{i}}\right|=O(l(s)^{-\nu}),$

Thus, as $s\to\infty$ , uniformly for $t\in[0,1]$ ,

(3.17)

\displaystyle f^{+}_{\underline{c}(s)}(a(s)+tl(s))=\sum_{i=0}^{m}(c_{i}^{+}(s)l(s)^{r_{i}})\left(\frac{a(s)}{l(s)}+t\right)^{r_{i}}+O(C^{+}(s)l(s)^{-\nu}).

We also observe that as $s\to\infty$ , we have

(3.18)

\sup_{t\in[0,1]}\left|f_{\underline{c}(s)}^{-}(a(s)+tl(s))\right|=O(C^{-}(s)),

and for any $t_{0}\in(0,1)$ ,

(3.19)

\sup_{t\in[t_{0},1]}\left|f_{\underline{c}(s)}^{-}(a(s)+tl(s))\right|=O(C^{-}(s)l(s)^{-\kappa_{n}}).

Now (3.14) follows from (3.17) and (3.18), and (3.15) follows from (3.17) and (3.19). This completes the proof of the Claim 1.

By (3.9) and the definition of the choice function $\phi$ ,

(3.20)

s<\frac{\|f_{\underline{c}(s)}\|_{[a(s),a(s)+l(s)]}}{\|f_{\underline{c}(s)}\|_{[\alpha(s),\alpha(s)+\delta l(s)]}}=\frac{\|\varphi_{{\underline{c}}(s)}\|_{[0,1]}}{\|\varphi_{{\underline{c}}(s)}\|_{I_{\delta}(s)}}

where $I_{\delta}(s):=\left[\frac{\alpha(s)-a(s)}{l(s)},\frac{\alpha(s)-a(s)+\delta l(s)}{l(s)}\right]\subseteq[0,1]$ having length $|I_{\delta}(s)|=\delta$ for all $s\geq 1.$ Therefore, by (3.14),

(3.21)

\lim_{s\to\infty}\|\varphi_{{\underline{c}}(s)}\|_{I_{\delta}(s)}\leq\lim_{s\to\infty}\frac{\|\varphi_{{\underline{c}}(s)}\|_{[0,1]}}{s}=0.

And, since the end-points of $I_{\delta}(s)$ are bounded, they converge to endpoints of a sub-interval $I_{\delta}$ of $[0,1]$ of length $\delta$ . Therefore, from (3.15) and (3.21), we deduce that

w^{+}_{0}\left(x_{0}+t\right)^{r_{0}}+\cdots+w^{+}_{m}\left(x_{0}+t\right)^{r_{m}}=0,

for all $t\in I_{\delta}\cap[t_{0},1]$ for any $t_{0}\in(0,1)$ ; and hence for all $t\in I_{\delta}$ . Since $I_{\delta}$ is an interval of length $\delta$ , by Lemma 3.5, we obtain that $w^{+}_{0}=\cdots=w^{+}_{m}=0$ . Therefore, by (3.13), we get

	$\displaystyle\lim_{s\to\infty}\frac{(c_{1}^{-}(s),...,c_{n}^{-}(s),c^{+}_{0}(s)l(s)^{r_{0}},...,c^{+}_{m}(s)l(s)^{r_{m}})}{\\|(c^{-}_{1}(s),...,c^{-}_{n}(s),c^{+}_{0}(s)l(s)^{r_{0}},...,c_{m}^{+}(s)l(s)^{r_{m}})\\|_{1}}$
	$\displaystyle=(w^{-}_{1},...,w^{-}_{n},w^{+}_{0},...,w^{+}_{m})=(w^{-}_{1},...,w^{-}_{n},0,...,0).$

Therefore, $n\geq 1$ and

(3.22)

\|(w^{-}_{1},...,w^{-}_{n})\|_{1}=1.

In view of (3.11), we have

	$\displaystyle\textbf{v}=(v^{-}_{1},...,v^{-}_{n},v^{+}_{0},...,v^{+}_{m})$	$\displaystyle:=\lim_{s\to\infty}\hat{}\underline{c}(s)$
		$\displaystyle=\lim_{s\to\infty}\frac{(c_{1}^{-}(s),...,c_{n}^{-}(s),c^{+}_{0}(s),...,c^{+}_{m}(s))}{\\|(c^{-}_{1}(s),...,c^{-}_{n}(s),c^{+}_{0}(s),...,c^{+}_{m}(s))\\|_{1}}.$

Then, $\|\textbf{v}\|_{1}=1$ and $\textbf{v}\in\tilde{\mathscr{F}}\smallsetminus\{0\}$ .

By (3.22), we have

	$\displaystyle 1$	$\displaystyle=\\|(w_{1}^{-},...,w_{n}^{-})\\|_{1}$
		$\displaystyle=\lim_{s\to\infty}\frac{\\|(c_{1}^{-}(s),...,c_{n}^{-}(s))\\|_{1}}{\\|(c^{-}_{1}(s),...,c^{-}_{n}(s),c^{+}_{0}(s)l(s)^{r_{0}},...,c^{+}_{m}(s)l(s)^{r_{m}})\\|_{1}}$
		$\displaystyle\leq\lim_{s\to\infty}\frac{\\|(c_{1}^{-}(s),...,c_{n}^{-}(s))\\|_{1}}{\\|(c^{-}_{1}(s),...,c^{-}_{n}(s),c^{+}_{0}(s),...,c^{+}_{m}(s))\\|_{1}}\text{, as $l(s)\to\infty$ and $r_{i}\geq 0$}$
		$\displaystyle=\\|(v_{1}^{-},...,v_{n}^{-})\\|_{1}\leq 1.$

Therefore, $\|(v_{1}^{-},...,v_{n}^{-})\|_{1}$ . As a consequence, we get $v^{+}_{0}=\cdots=v^{+}_{m}=0$ . Therefore, $f_{\textbf{v}}(t)\to 0$ as $t\to\infty$ , where $\textbf{v}\in\tilde{\mathscr{F}}\smallsetminus\{0\}$ , and hence $f_{\textbf{v}}\in\mathscr{F}\smallsetminus\{0\}$ . This contradicts our assumption that no nonzero function in $\mathscr{F}$ decays to zero.

Remark 3.9.

This is the only place in the proof of Theorem 1.14 where we use the non-decaying function assumption for the closed cone $\mathscr{F}$ . And this is the main reason we need the non-contraction assumption on the curves in this article.

Case 2: $\kappa>0$ .

Let $\lambda(s)=\|\underline{c}(s)\|_{1}$ and denote: $\hat{\underline{c}}:=\frac{\underline{c}}{\lambda}$ . As above, note that

\frac{\|f_{\hat{\underline{c}}(s)}\|_{[a(s),a(s)+l(s)]}}{\|f_{\hat{\underline{c}}(s)}\|_{[\alpha(s),\alpha(s)+\delta l(s)]}}=\frac{\|f_{\underline{c}(s)}\|_{[a(s),a(s)+l(s)]}}{\|f_{\underline{c}(s)}\|_{[\alpha(s),\alpha(s)+\delta l(s)]}}>s.

We now consider two cases: $a(s)$ is bounded or $a(s)\to\infty$ .

Case 2.1. $a(s)$ is bounded. We observe that $\lim_{s\to\infty}l(s)=0$ . This follows since $a(s)\geq T_{0}\geq 1$ and $\kappa>0$ , we get $\frac{a(s)}{l(s)}\sim C_{0}s^{\kappa}\to\infty$ . We have that $\lim_{s\to\infty}a(s)=x_{0}\geq T_{0}$ , and $\lim_{s\to\infty}\hat{\underline{c}}(s)=\textbf{v}$ with $\|\textbf{v}\|_{1}=1.$

By our choice of $T_{0}$ , the basis functions $F^{-}_{i},F^{+}_{j}$ are $(N+1)$ -times continuously differentiable where $N:=(n+m+1)$ . For any $t\geq T_{0}$ , let $W(t)$ denote the transpose of the Wronskian matrix $W(F_{-n}(t),\ldots,F_{m}(t))$ , whose $k$ -th row is $(\partial^{k}F_{-n}(t),\ldots,\partial^{k}F_{m}(t))$ for $k=0,\dots,N$ . For $h\in\mathbb{R}$ , define

(3.23)

Q_{N,a(s)}(h)=\hat{\underline{c}}(s)W(a(s))\cdot(h^{0},\ldots,h^{N}),

where $\cdot$ denotes the dot product on $\mathbb{R}^{N+1}$ . By Taylor’s theorem (mean value form for the remainder), for all $h\in[0,l(s)]$ , there exists $\xi\in a(s)+[0,l(s)]:=[a(s),a(s)+l(s)]$ such that

\displaystyle f_{\hat{\underline{c}}(s)}(a(s)+h)-Q_{N,a(s)}(h)=\frac{f_{\hat{\underline{c}}(s)}^{(N+1)}(\xi)}{(n+1)!}h^{N+1}.

Since $a(s)$ , $\hat{\underline{c}}(s)$ , and $l(s)$ are bounded, we get that $|f_{\hat{\underline{c}}(s)}^{(N+1)}(\xi)|$ is uniformly bounded in the range $\xi\in[a(s),a(s)+l(s)]$ .

Denote $I(s):=[0,l(s)]$ , and $I_{\delta}(s):=[0,\delta l(s)]$ . We have:

(By Taylor approximation)		$\displaystyle\\|f_{\hat{\underline{c}}(s)}\\|_{a(s)+I(s)}=$	$\displaystyle\\|Q_{N,a(s)}\\|_{I(s)}+O(l(s)^{N+1})$
(By Proposition 3.4)		$\displaystyle\leq$	$\displaystyle M_{N}(\delta)\\|Q_{N,a(s)}\\|_{I_{\delta}(s)}+O(l(s)^{N+1})$
(By Taylor approximation)		$\displaystyle\leq$	$\displaystyle M_{N}(\delta)\\|f_{\hat{\underline{c}}(s)}\\|_{a(s)+I_{\delta}(s)}+O(l(s)^{N+1})$
	$\displaystyle=$	$\displaystyle M_{N}(\delta)\\|f_{\hat{\underline{c}}(s)}\\|_{a(s)+I_{\delta}(s)}\left[1+O\left(\frac{l(s)^{N+1}}{\\|f_{\hat{\underline{c}}(s)}\\|_{a(s)+I_{\delta}(s)}}\right)\right].$

Thus,

\frac{\|f_{{\underline{c}}(s)}\|_{a(s)+I(s)}}{\|f_{{\underline{c}}(s)}\|_{a(s)+I_{\delta}(s)}}\ll 1+O\left(\frac{l(s)^{N+1}}{\|f_{\hat{\underline{c}}(s)}\|_{a(s)+I_{\delta}(s)}}\right).

Since $\|f_{\hat{\underline{c}}(s)}\|_{a(s)+I_{\delta}(s)}=\|Q_{N,a(s)}\|_{I_{\delta}(s)}+O(l(s)^{N+1})$ , we will show that

(3.24)

\lim_{s\to\infty}\frac{l(s)^{N+1}}{\|Q_{N,a(s)}\|_{I_{\delta}(s)}}=0.

This outcome gives a contradiction to our assumption that $s\leq\frac{\|f_{{\underline{c}}(s)}\|_{I(s)}}{\|f_{{\underline{c}}(s)}\|_{I_{\delta}(s)}}$ for all $s\geq 1$ .

To prove (3.24), we observe the following if $q(x)=c_{0}+c_{1}x+...+c_{k}x^{k}$ is a polynomial of degree bounded by $k$ and $J\subset[0,1]$ is an interval of length $|J|<1$ , then by Proposition 3.4,

(3.25)

\|q\|_{J}\geq(k+1)^{-1}k^{-k}\|q\|_{[0,1]}|J|^{k}\geq\|(c_{0},c_{1},...,c_{k})\|_{1}|J|^{k},

where the implied constant depends only on $k$ , because $\|q\|_{[0,1]}$ and $\|\underline{c}\|_{1}$ are two norms on the $(k+1)$ -dimensional space of polynomials of degree at most $k$ .

By (3.23), $Q_{N,a(s)}$ is a polynomial of degree bounded by $N$ whose coefficients are given by $W(a(s))\hat{\underline{c}}(s)$ . We have $W(a(s))\to W(x_{0})$ as $s\to\infty$ . And, by Lemma 3.6, $W(x_{0})$ is nonsingular. Therefore,

\lVert W(a(s))\hat{\underline{c}}(s)\rVert_{1}\geq\lVert W(a(s))^{-1}\rVert_{1}^{-1}\lVert\hat{\underline{c}}(s)\rVert_{1}\geq(1/2)\lVert W(x_{0})^{-1}\rVert_{1}^{-1}>0

for all $s\gg 1$ , as $\lVert\hat{\underline{c}}(s)\rVert_{1}=1$ . Then, by (3.25) applied to $J=[0,\delta l(s)]$ , we get

(3.26)

\|Q_{N,a(s)}\|_{I_{\delta}(s)}\gg\lVert W(x_{0})^{-1}\rVert_{1}^{-1}\delta^{N}l(s)^{N}.

Since $\delta$ is fixed, and $l(s)\to 0$ , we get (3.24).

Case 2.2. $a(s)$ is unbounded. Here $a(s)\asymp s^{\theta}$ , for $\theta>0$ . We recall that for each $\mathrm{n}\in\mathbb{N}$ there is $T_{\mathrm{n}}$ such that all basis functions $F^{-}_{i},F_{j}^{+}$ will be continuously differentiable $\mathrm{n}$ times in the ray $[T_{\mathrm{n}},\infty)$ for all $i,j$ , see [DM96]. By [Mil94b], it holds that

\left|\frac{d^{p}}{dt^{p}}F^{-}_{i}(x)\right|\ll x^{-\kappa_{i}-p},\text{ and }\left|\frac{d^{q}}{dt^{q}}F^{+}_{j}(x)\right|\ll x^{r_{j}-q},

where $p,q$ are non-negative integers.

Let $J(s)=[a(s),a(s)+l(s)]$ . Then we conclude that for an integer $\mathrm{n}$ such that $r_{i}-(\mathrm{n}+1)<0$ for all $0\leq i\leq m$ , we have

\sup_{t\in J(s)}\left\lvert\frac{d^{\mathrm{n}+1}}{dt^{\mathrm{n}+1}}f_{\hat{\underline{c}}(s)}(t)\right\rvert=O(a(s)^{r_{m}-(\mathrm{n}+1)}).

By Taylor’s theorem, for the Taylor polynomial $P_{\mathrm{n},a(s)}(t)$ of $f_{\hat{\underline{c}}(s)}(t)$ of degree $\mathrm{n}$ centered around $a(s)$ , we have

	$\displaystyle\sup_{t\in J(s)}\|f_{\hat{\underline{c}}(s)}(t)-P_{\mathrm{n},a(s)}(t)\|$	$\displaystyle=O(a(s)^{r_{m}-(\mathrm{n}+1)}l(s)^{\mathrm{n}+1})$
		$\displaystyle=O\left(a(s)^{r_{m}}\left(\frac{l(s)}{a(s)}\right)^{\mathrm{n}+1}\right)$
(3.27)			$\displaystyle=O\left(s^{r_{m}\theta}s^{-(\mathrm{n}+1)\kappa}\right),$

where we use the fact that $a(s)/l(s)\sim C_{0}s^{\kappa}$ for some $C_{0}\neq 0$ and $\kappa>0$ . Due to Lemma 3.5, the function $\Psi(s):=\|f_{\hat{\underline{c}}(s)}\|_{[\alpha(s),\alpha(s)+\delta l(s)]}$ is positive. Since $\Psi$ is also definable in a polynomially bounded o-minimal structure, we pick $\eta>0$ such that

(3.28)

\|f_{\hat{\underline{c}}(s)}\|_{J(s)}\gg s^{-\eta},~{}\text{as }s\to\infty.

We take $\mathrm{n}$ large enough such that

\nu:=r_{m}\theta-(\mathrm{n}+1)\kappa<-\eta.

Then, for $J_{\delta}(s)=[\alpha(s),\alpha(s)+\delta l(s)]$ ,

(by (3.27))		$\displaystyle\\|f_{\hat{\underline{c}}(s)}\\|_{J(s)}\leq$	$\displaystyle\\|P_{\mathrm{n},a(s)}\\|_{J(s)}+O(s^{\nu})$
(by Proposition 3.4)		$\displaystyle\leq$	$\displaystyle M_{\mathrm{n}}(\delta)\\|P_{\mathrm{n},a(s)}\\|_{J_{\delta}(s)}+O(s^{\nu})$
(by (3.27))		$\displaystyle\leq$	$\displaystyle M_{\mathrm{n}}(\delta)\\|f_{\hat{\underline{c}}(s)}\\|_{J_{\delta}(s)}+O(s^{\nu})$
	$\displaystyle=$	$\displaystyle M_{\mathrm{n}}(\delta)(\\|f_{\hat{\underline{c}}(s)}\\|_{J_{\delta}(s)}\left[1+O\left(\frac{s^{\nu}}{\\|f_{\hat{\underline{c}}(s)}\\|_{J_{\delta}(s)}}\right)\right]$
(by (3.28))		$\displaystyle\ll$	$\displaystyle M_{\mathrm{n}}(\delta)\\|f_{\hat{\underline{c}}(s)}\\|_{J_{\delta}(s)}\left[1+O\left(\frac{s^{\nu}}{s^{-\eta}}\right)\right]$

Namely, $\frac{\|f_{\hat{\underline{c}}(s)}\|_{J(s)}}{\|f_{\hat{\underline{c}}(s)}\|_{J_{\delta}(s)}}$ is bounded for all large $s$ , contradicting assumption $s<\frac{\|f_{\hat{\underline{c}}(s)}\|_{J(s)}}{\|f_{\hat{\underline{c}}(s)}\|_{J_{\delta}(s)}}$ . ∎

3.3. Relative time near varieties

Using the $(C,\alpha)$ -good property, we prove the following proposition. It is an analog of [DM93, Proposition 4.2], and it is key for the linearization technique.

Proposition 3.10.

Let $\psi:[0,\infty)\to\text{GL}(m,\mathbb{R})$ be a continuous, unbounded curve definable in a polynomially bounded o-minimal structure such that $\lim_{t\to\infty}\psi(t)v\neq 0,\forall v\in\mathbb{R}^{m}\smallsetminus\{0\}$ . Consider a non-zero polynomial $Q:\mathbb{R}^{m}\to\mathbb{R}$ , and let

\emph{\text{V}}:=\{x\in\mathbb{R}^{m}:Q(x)=0\}.

Then, there exists $T_{0}>0$ such that the following holds: for an $\epsilon>0$ and a compact subset $\emph{\text{K}}_{1}\subset\emph{\text{V}}$ , there exists a compact subset $\emph{\text{K}}_{2}$ with

\emph{\text{K}}_{1}\subset\emph{\text{K}}_{2}\subset\emph{\text{V}}

such that for every compact neighborhood $\Phi$ of $K_{2}$ in $\mathbb{R}^{m}$ , there exists a compact neighborhood $\Psi$ of $K_{1}$ in $\mathbb{R}^{m}$ , with $\Psi\subset\mathring{\Phi}$ , where $\mathring{\Phi}$ denotes the interior of $\Phi$ , such that:

(3.29)

|\{t\in[a,b]:\psi(t)v\in\Psi\}|\leq\epsilon(b-a).

for all $v\in\mathbb{R}^{m}$ and $[a,b]\subseteq[T_{0},\infty)$ , for which:

•

$\psi(t)v\in\Phi,~{}\forall t\in[a,b]$ , and
•

$\psi(b)v\in\Phi\smallsetminus\mathring{\Phi}$ .

Remark 3.11.

Compared to [DM93, Proposition 4.2], the statement of the above proposition is different in the assumption that the curve $\psi(t)v$ remains in $\overline{\Phi}$ throughout the whole interval $[a,b]$ , and exists $\Phi$ at $t=b$ . This is a stronger requirement than the one in [DM93, Proposition 4.2] which only asks for $\psi(t_{0})v\notin\Phi$ for some $t_{0}\in[a,b]$ .

Remark 3.12.

Notice that for any affine variety $\text{V}\subset\mathbb{R}^{m}$ there exists a polynomial $Q:\mathbb{R}^{m}\to\mathbb{R}$ such that V is the zero set of $Q$ . Indeed, by the Hilbert basis theorem, V is the zero set of finitely many polynomials $f_{1},...,f_{r}$ , and we put $Q:=f_{1}^{2}+\cdots+f_{r}^{2}$ .

Our proof of Proposition 3.10 builds on the following versions of the modified $(C,\alpha)$ -good property.

Proposition 3.13.

Let $\psi:[0,\infty)\to\text{GL}(m,\mathbb{R})$ be a continuous curve definable in a polynomially bounded o-minimal structure, and let $Q:\mathbb{R}^{m}\to\mathbb{R}$ be a polynomial. Then there exist constants $\nu,C,\alpha,T_{0}>0$ such that for any $v\in\mathbb{R}^{m}$ it holds that either:

(1)

$Q(\psi(t)v)=0,\forall t\geq T_{0}$ , or
(2)

$\Theta_{v}(t):=t^{\nu}Q(\psi(t)v)$ is $(C,\alpha)$ -good on $[T_{0},\infty)$ .

Proof.

Consider the vector space $\mathcal{V}$ , of functions from $[0,\infty)$ to $\mathbb{R}$ , defined by:

\mathcal{V}=\text{Span}_{\mathbb{R}}\{t\mapsto Q(\psi(t)v):v\in\mathbb{R}^{m}\}.

It is straightforward to verify that $\mathcal{V}$ is finite dimensional, and that all functions in $\mathcal{V}$ are definable. In view of Remark 3.8, there is a $T_{0}\geq 0$ such that if $f\in\mathcal{V}$ is eventually zero, then $f(t)=0,\forall t\in[T_{0},\infty)$ . We consider $\mathscr{F}$ to be the space of functions of $\mathcal{V}$ restricted to $[T_{0},\infty).$ By Lemma 3.7 that there is a basis $\{f_{0},f_{1},...,f_{n}\}$ for $\mathscr{F}$ such that $\deg(f_{0})<\deg(f_{1})<...<\deg(f_{n})$ . Let $\nu>0$ such that $\deg(t^{\nu}f_{0})\geq 0$ . Then, by Theorem 1.14, we pick $T_{1}>0$ such that any non-zero function

f\in t^{\nu}\otimes\mathscr{F}=\text{Span}_{\mathbb{R}}\{t\mapsto t^{\nu}Q(\psi(t)v):v\in\mathbb{R}^{m}\}

is $(C,\alpha)$ -good in $[T_{1},\infty)$ . ∎

We now note the following corollary.

Corollary 3.14.

Let $\psi:[0,\infty)\to\text{GL}(m,\mathbb{R})$ be a continuous curve, definable in a polynomially bounded o-minimal structure. Fix a polynomial $Q:\mathbb{R}^{m}\to\mathbb{R}$ . Then there exist $T_{0},C,\alpha>0$ such that for any $v\in\mathbb{R}^{m}$ and any interval $[x,y]\subseteq[T_{0},\infty)$ , if $\sup_{t\in[x,y]}|Q(\psi(t)v)|=|Q(\psi(y)v)|$ , then

(3.30)

\left|\{t\in[x,y]:|Q(\psi(t)v)|\leq\eta\}\right|\leq C\left(\frac{\eta}{|Q(\psi(y)v)|}\right)^{\alpha}(y-x),

for all $\eta>0$ ; we may say that $t\mapsto\lvert Q(\psi(t)v)\rvert$ is right-max- $(C,\alpha)$ -good in $[T_{0},\infty)$ .

Proof.

Let $\nu,C,\alpha,T_{0}>0$ be as in Proposition 3.13. Let $[x,y]\subseteq[T_{0},\infty)$ and consider a vector $v\in\mathbb{R}^{m}$ with $|Q(\psi(y)v)|=\delta>0$ and $|Q(\psi(t)v)|\leq\delta$ for all $t\in[x,y]$ . Then it follows that

\sup_{t\in[x,y]}|t^{\nu}Q(\psi(t)v)|=y^{\nu}\delta.

Now, by the $(C,\alpha)$ -good property of $|t^{\nu}Q(\psi(t)v)|$ , we get:

	$\displaystyle\left\|\{t\in[x,y]:\|Q(\psi(t)v)\|\leq\eta\}\right\|$	$\displaystyle=\left\|\{t\in[x,y]:\|t^{\nu}Q(\psi(t)v)\|\leq t^{\nu}\eta\}\right\|$
		$\displaystyle\leq\left\|\{t\in[x,y]:\|t^{\nu}Q(\psi(t)v)\|\leq y^{\nu}\eta\}\right\|$
(3.31)			$\displaystyle\leq C\left(\frac{y^{\nu}\eta}{y^{\nu}\delta}\right)^{\alpha}(y-x)=C\left(\frac{\eta}{\delta}\right)^{\alpha}(y-x).$

∎

Proof for Proposition 3.10.

Let $\psi:[0,\infty)\to\text{GL}(m,\mathbb{R})$ be a continuous unbounded curve definable in a polynomially bounded o-minimal structure such that $\lim_{t\to\infty}\psi(t)v\neq 0,\forall v\neq 0$ . Pick $T_{0},C,\alpha>0$ such that for all $v\in\mathbb{R}^{m}$ , the $(C,\alpha)$ -good property holds for the norm-map $t\mapsto\lVert\psi(t)v\rVert$ as in Proposition 1.15, and the right-max- $(C,\alpha)$ -property holds for the map $t\mapsto\lvert Q(\psi(t)v)\rvert$ as in Corollary 3.14. Let $\epsilon>0$ be given.

Given a compact set $\text{K}_{1}\subset\text{V}$ , let $R_{1}>0$ be such that $\text{K}_{1}\subset B_{R_{1}}(0)$ . Let $\text{K}_{2}:=\overline{\text{B}_{R_{2}}(0)}\cap\text{V}$ , where $R_{2}>R_{1}$ is such that $C(R_{1}/R_{2})^{\alpha}\leq\epsilon$ , see (3.34).

For any $\epsilon^{\prime}>0$ , define

\text{V}^{\epsilon^{\prime}}:=\{x\in\mathbb{R}^{m}:|Q(x)|<\epsilon^{\prime}\}.

Let $\Phi$ be a compact neighborhood of $\text{K}_{2}$ . Then, we can pick an $\epsilon_{1}>0$ such that

(3.32)

B_{R_{2}}(0)\cap\text{V}^{\epsilon_{1}}\subset\mathring{\Phi}.

We pick $\epsilon_{2}\in(0,\epsilon_{1})$ such that $K_{1}\subset B_{R_{1}}(0)\cap\text{V}^{\epsilon_{2}}$ and $C(\epsilon_{2}/\epsilon_{1})^{\alpha}\leq\epsilon$ , see (3.35). Let

(3.33)

\Psi:=\overline{B_{R_{1}}(0)\cap\text{V}^{\epsilon_{2}}}.

Let $v\in\mathbb{R}^{m}$ and an interval $[a,b]\subset[T_{0},\infty)$ such that:

•

$\psi(t)v\in\Phi,~{}\forall t\in[a,b]$ , and
•

$\psi(b)v\in\Phi\smallsetminus\mathring{\Phi}$ .

Then by (3.32), $\|\psi(b)v\|\geq R_{2}$ or $|Q(\psi(b)v)|\geq\epsilon_{1}.$ Now, if $\|\psi(b)v\|\geq R_{2}$ , then by (3.33) and Proposition 1.15,

	$\displaystyle\|\{t\in[a,b]\|\psi(t)v\in\Psi\}\|\leq$	$\displaystyle\|\{t\in[a,b]\|\\|\psi(t)v\\|\leq R_{1}\}\|$
(3.34)		$\displaystyle\leq$	$\displaystyle C\left(\frac{R_{1}}{R_{2}}\right)^{\alpha}(b-a)\leq\epsilon(b-a).$

Suppose now that $|Q(\psi(b)v)|\geq\epsilon_{1}$ . Since $t\mapsto|Q(\psi(t)v)|$ is continuous and definable, and since a definable map takes a given value only finitely many times, there is a decomposition

\{t\in[a,b]:|Q(\psi(t)v)|\leq\epsilon_{1}\}=[a_{1},b_{1}]\sqcup\cdots\sqcup[a_{k},b_{k}],

for some $k\in\mathbb{N}$ , where for all $1\leq i\leq k$ it holds

|Q(\psi(b_{i})v)|=\epsilon_{1}\text{ and }|Q(\psi(t)v)|\leq\epsilon_{1},\forall t\in[a_{i},b_{i}].

Now, since $\Psi=B_{R_{1}}(0)\cap\text{V}^{\epsilon_{2}}$ , we have

\displaystyle\{t\in[a,b]|\psi(t)v\in\Psi\}\subseteq\{t\in[a,b]:|Q(\psi(t)v)|\leq\epsilon_{2}\},

and since $\epsilon_{2}<\epsilon_{1}$ , we have

\displaystyle\{t\in[a,b]:|Q(\psi(t)v)|\leq\epsilon_{2}\}=\bigsqcup_{i=1}^{k}\{t\in[a_{i},b_{i}]:|Q(\psi(t)v)|\leq\epsilon_{2}\}.

By (3.30) of Corollary 3.14,

(3.35)

\displaystyle|\{t\in[a_{i},b_{i}]:|Q(\psi(t)v)|\leq\epsilon_{2}\}|\leq C\left(\frac{\epsilon_{2}}{\epsilon_{1}}\right)^{\alpha}(b_{i}-a_{i})\leq\epsilon(b_{i}-a_{i}),

for each $1\leq i\leq k$ . Thus we may conclude that

|\{t\in[a,b]|\psi(t)v\in\Psi\}|\leq\epsilon(b-a).

Thus, (3.29) holds in all cases. ∎

4. Non-escape of mass

Let $X\cup\{\infty\}$ be the one-point compactification of

X:=\text{SL}(n,\mathbb{R})/\text{SL}(n,\mathbb{Z}).

Let $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ be a continuous unbounded curve definable in a polynomially bounded o-minimal structure, and for $x\in X$ recall the measures $\mu_{T,\varphi,x}$ defined in (1.1).

By the Banach-Alaoglu theorem, any weak-star limit $\mu$ as $T\to\infty$ of the measures $\mu_{T,\varphi,x}$ is a probability measure on $X\cup\{\infty\}$ . First will show the following. Then we will extend the result to the case of finite volume quotient spaces of Lie groups.

Proposition 4.1.

If $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ is a continuous unbounded non-contracting curve definable in a polynomially bounded o-minimal structure. Then, there exists $T_{0}>0$ such that given a compact set $F\subset\text{SL}(n,\mathbb{R})$ and $\epsilon>0$ , there exists a compact set $K\subset\text{SL}(n,\mathbb{R})/\text{SL}(n,\mathbb{Z})$ such that for any $g\in F$ and $T\geq T_{0}$ , we have

\lvert\{t\in[T_{0},T]:\varphi(t)g\text{SL}(n,\mathbb{Z})/\text{SL}(n,\mathbb{Z})\in K\}\rvert\geq(1-\epsilon)(T-T_{0}).

We give some preliminaries before the proof. In the discussion that follows, we will refer to rank- $k$ discrete subgroups of $\mathbb{R}^{n}$ as $k$ -lattices. For a $k$ -lattice $\Lambda\subset\mathbb{Z}^{n}$ , let $\|\Lambda\|$ be the volume of a fundamental domain $F\leq\text{Span}_{\mathbb{R}}\{\Lambda\}$ of $\Lambda$ with respect to the usual measure on the subspace $\text{Span}_{\mathbb{R}}\{\Lambda\}\leq\mathbb{R}^{n}$ (which is obtained by restricting the usual Euclidean inner product). Recall that if $\{\textbf{v}_{1},...,\textbf{v}_{k}\}$ forms a $\mathbb{Z}$ -basis for $\Lambda$ , then

\|\Lambda\|=\|\textbf{v}_{1}\wedge\cdots\wedge\textbf{v}_{k}\|,

where the norm is the standard Euclidean norm defined through the inner product for which the pure wedges of $k$ -positively oriented tuples of the canonical basis vectors $e_{i},i=1,...,n$ are orthogonal and have norm equal to one. Consider the representation of $\text{SL}(n,\mathbb{R})$ on $\bigwedge^{k}\mathbb{R}^{n}$ defined by

(4.1)

g.(\textbf{v}_{1}\wedge\textbf{v}_{2}...\wedge\textbf{v}_{k}):=g\textbf{v}_{1}\wedge g\textbf{v}_{2}\wedge\cdots\wedge g\textbf{v}_{k},~{}g\in\text{SL}(n,\mathbb{R}).

For a fixed $k<n$ , the action of $\text{SL}(n,\mathbb{R})$ is transitive on the space of $k$ -lattices of rank $k<n$ . We denote $\mathbb{Z}^{k}:=\text{Span}_{\mathbb{Z}}\{e_{1},...,e_{k}\}$ , and observe that for a unimodular $n$ -lattice $L=g\mathbb{Z}^{n}\leq\mathbb{R}^{n}$ , where $g\in\text{SL}(n,\mathbb{R})$ , we have that

g\gamma\mathbb{Z}^{k},\gamma\in\text{SL}(n,\mathbb{Z}),

is the collection of primitive $k$ -sublattices of $L$ . The following powerful theorem on quantitative non-divergence due to Kleinbock [Kle07] will be needed.

Theorem 4.2.

Suppose an interval $B\subset\mathbb{R}$ , $C,\alpha>0$ , $0<\rho<1$ and a continuous map $h:B\to\text{SL}(n,\mathbb{R})$ are given. Assume that for any $\gamma\in\text{SL}(n,\mathbb{Z})$ , and $1\leq k<n$ we have

(1)

the function $x\to\|h(x)\gamma.e_{1}\wedge\cdots\wedge e_{k}\|$ is $(C,\alpha)$ -good on $B$ , and
(2)

$\sup_{x\in B}\|h(x)\gamma.e_{1}\wedge\cdots\wedge e_{k}\|\geq\rho^{k}$ .

Then, for any $\epsilon<\rho$ ,

(4.2)

|\{x\in B:\lambda_{1}(h(x)\mathbb{Z}^{n})\leq\epsilon\}|\leq Cn2^{n}\left(\frac{\epsilon}{\rho}\right)^{\alpha}|B|,

where $\lambda_{1}(\cdot)$ is the function that outputs the length of the shortest nonzero vector of an Euclidean lattice.

Proof of Proposition 4.1.

We identify $\text{SL}(n,\mathbb{R})/\text{SL}(n,\mathbb{Z})$ with the space of unimodular lattices

\mathcal{L}_{n}:=\{L:=\text{Span}_{\mathbb{Z}}\{v_{1},...,v_{n}\}:\det(v_{i,j})=1\}.

Consider

\mathcal{B}_{\delta}:=\{L\in\mathcal{L}_{n}:(L\smallsetminus 0)\cap B_{\delta}(0)\neq\emptyset\}.

Here $B_{\delta}(0)\subseteq\mathbb{R}^{n}$ is the ball of radius $\delta$ centered at the origin. By Mahler’s Criterion ([BM00], Theorem 3,2), $\mathcal{L}_{n}\smallsetminus\mathcal{B}_{\delta}$ is compact for all $\delta>0$ and thus $\mathcal{B}_{\delta}$ is a neighborhood of $\infty$ . Fix $g\mathbb{Z}^{n}\in\mathcal{L}_{n}$ for $g\in\text{SL}(n,\mathbb{R})$ , and consider the measures

(4.3)

\mu_{T}(f)=\frac{1}{T}\int_{0}^{T}f(\varphi(t)g\mathbb{Z}^{n})dt,~{}f\in C_{c}(\mathcal{L}_{n})

Let $\mu$ be a weak-* limit of the measures $\mu_{T}$ as $T\to\infty$ . In order to show that $\mu(\infty)=0$ , it is enough to prove that for every $\epsilon>0$ , there is $\delta$ such that

(4.4)

\displaystyle\limsup_{T\to\infty}\frac{|\{t\in[0,T]:\lambda_{1}(\varphi(t)g\mathbb{Z}^{n})\leq\delta\}|}{T}\leq\epsilon.

This will be concluded by Theorem 4.2 as follows. We first verify the conditions. For $k\in\{1,2,..,n-1\}$ , we denote by $\theta_{k}:\text{SL}(n,\mathbb{R})\to\bigwedge^{k}\mathbb{R}^{n}$ the representation (4.1). By Proposition 1.15, there exists $T_{0},C,\alpha>0$ such that for any fixed $k\in\{1,2,..,n-1\}$ , and $\gamma\in\text{SL}(n,\mathbb{Z})$ it holds that

(4.5)

\displaystyle\Theta(t)=\|\theta_{k}(\varphi(t)g\gamma)(e_{1}\wedge\cdots\wedge e_{k})\|=\|\varphi(t)g\gamma.(e_{1}\wedge\cdots\wedge e_{k})\|

is $(C,\alpha)$ -good in $[T_{0},\infty)$ . Let $F\subset\text{SL}(n,\mathbb{R})$ be a given compact subset. Then

\rho:=(\inf\{\|\varphi(T_{0})g\gamma.(e_{1}\wedge\cdots\wedge e_{k})\|:g\in F,\,\gamma\in\text{SL}(n,\mathbb{Z})\})^{\frac{1}{k}}>0.

Thus, by Theorem 4.2, for any $g\in F$ ,

(4.6)

\displaystyle|\{t\in[T_{0},T]:\lambda_{1}(\varphi(t)g\mathbb{Z}^{n})\leq\delta\}|\leq C\left(\frac{\delta}{\rho}\right)^{\alpha}(T-T_{0}),

∎

4.1. Non-divergence for homogeneous space of Lie groups

In the setting of Lie groups, one obtains the following result.

Proposition 4.3.

Let $G$ be a Lie subgroup of $\text{SL}(n,\mathbb{R})$ and let $\Gamma$ be a lattice in $G$ . Suppose that $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ is a non-contracting curve definable in a polynomially bounded o-minimal structure $\mathscr{S}$ . Assume further that $\varphi([0,\infty))\subset G_{u}$ . Then, there exists a $T_{0}>0$ such that given $\epsilon>0$ and a compact set $F\subset G$ , there exists a compact set $K\subset G/\Gamma$ such that for all $g\in F$ and all $T>T_{0}$ ,

(4.7)

\lvert\{t\in[T_{0},T]:\varphi(t)g\Gamma\in K\}\rvert\geq(1-\epsilon)(T-T_{0}).

Remark 4.4.

We note that the assumption that $\varphi([0,\infty))\subset G_{u}$ is non restrictive for our purposes in view of the following. In Proposition 5.13, we show that for a curve contained in a Lie subgroup $G\subseteq\text{SL}(n,\mathbb{R})$ which is non-contracing and definable in a polynomially bounded o-minimal structure, it holds that its hull is contained in $G_{u}$ . Thus, by multiplying the original curve with a correcting curve, the assumption $\varphi([0,\infty))\subset G_{u}$ is satisfied.

Proof.

We follow the arguments as in [DM93, Theorem 6.1] to reduce the problem to the case when $G$ is a semisimple group and $\Gamma$ is an irreducible lattice in $G$ .

We note that if $G/\Gamma$ is compact, there is nothing to prove. So, we now assume that $G/\Gamma$ is not compact. Let $M$ be the smallest closed normal subgroup of $G$ such that $\bar{G}=G/M$ is semisimple with the trivial center and no compact factors. Then $M/(M\cap\Gamma)$ is compact, see [Rag72, Theorem 8.24] and [Sha96, Proof of Theorem 2.2]. Let $q:G\to\bar{G}$ denote the quotient homomorphism. Then $q(\Gamma)$ is a lattice in $\bar{G}$ , and the natural quotient map $\bar{q}:G/\Gamma\to\bar{G}/q(\Gamma)$ is proper. We note that if $\tilde{G}$ and $\tilde{M}$ denote the Zariski closures of $G$ and $M$ in $\text{SL}(n,\mathbb{R})$ , respectively, then $\bar{G}=G/M=\tilde{G}^{0}/\tilde{M}^{0}$ . Therefore, $q|_{G_{u}}:G_{u}\to\bar{G}$ is a rational homomorphism of real algebraic groups. Thus $q\circ\varphi$ is a curve definable in $\mathscr{S}$ . By Lemma 2.15, $q\circ\varphi:[0,\infty)\to\bar{G}$ is a non-contracting. Since $\bar{q}$ is proper, proving the theorem for $\bar{G}$ in place of $G$ is sufficient.

Therefore, we assume that $G$ is semisimple with the trivial center and no compact factors. Then, by [Rag72, Theorem 5.22], there exist closed normal subgroups $G_{1},\ldots,G_{k}$ of $G$ for some $k$ such that $G$ equals the direct product $G_{1}\times\cdots\times G_{k}$ , such that for $\Gamma_{i}=G_{i}\cap\Gamma$ is an irreducible lattice in $G_{i}$ . Moreover, $\Gamma_{1}\times\cdots\times\Gamma_{k}$ is a normal subgroup of finite index in $\Gamma$ . Hence, $G/\Gamma$ is finitely covered by $G_{1}/\Gamma_{1}\times\cdots\times G_{k}/\Gamma_{k}$ . Let $q_{i}:G\to G_{i}$ denote the natural factor map, and observe that $q_{i}\circ\varphi$ is a non-contracting curve in $G_{i}$ definable in $\mathscr{S}$ by Lemma 2.15. So, proving the theorem for $G_{i}/\Gamma_{i}$ separately for each $i$ will imply the theorem for $G/\Gamma$ . Therefore, without loss of generality, we may assume that $\Gamma$ is an irreducible lattice in $G$ , which is semisimple with the trivial center and no compact factors.

First, suppose that the real rank of $G$ is at least $2$ . Then, by the arithmeticity theorem due to G. A. Margulis, $\Gamma$ is arithmetic; that is, see [Zim84, page 3]: There exists an $m\in\mathbb{N}$ , an algebraic semisimple group $H\subset\text{SL}(m,\mathbb{R})$ defined over $\mathbb{Q}$ and a surjective algebraic homomorphism

\rho:H^{0}\to G,

such that $\ker\rho$ is compact and $\rho(H^{0}(\mathbb{Z}))$ is commensurable with $\Gamma$ , where $H^{0}(\mathbb{Z})=H^{0}\cap\text{SL}(m,\mathbb{Z})$ . Since $G$ has no compact factors, the map $\rho$ restricted to $H_{u}$ is a finite cover of $G$ . By Lemma 2.19, we can lift $\varphi$ to a $\mathscr{S}$ -definable curve $\psi:[0,\infty)\to H_{u}\subset H^{0}\subset\text{SL}(m,\mathbb{R})$ which is non contracting in $H_{u}$ , such that $\rho\circ\psi=\varphi$ . By Proposition 2.20, $\psi$ in non-contracting in $\text{SL}(m,\mathbb{R})$ . As in Proposition 4.1, let $T_{0}>0$ be such that for any compact set $F_{1}\subset\text{SL}(m,\mathbb{R})$ , there exists a compact set $K_{1}\subset Y$ such that for any $T\geq T_{0}$ and $h\in F_{1}$

(4.8)

\lvert\{t\in[T_{0},T]:\psi(t)h\text{SL}(m,\mathbb{Z})\in K_{1}\}\rvert\geq(1-\epsilon)(T-T_{0}).

Choose, $F_{1}=\rho^{-1}(F)\subset H^{0}$ . Since $H^{0}/H^{0}(\mathbb{Z})\hookrightarrow Y:=\text{SL}(m,\mathbb{R})/\text{SL}(m,\mathbb{Z})$ is a proper $H^{0}$ -equivariant continuous injection, we can pick a compact set $F_{2}\subset H^{0}$ such that $F_{2}\text{SL}(m,\mathbb{Z})\supset K_{1}\cap H^{0}\text{SL}(m,\mathbb{Z})$ . Since $\Gamma\cap\rho(H^{0}(\mathbb{Z}))$ contains a subgroup of finite index in $\rho(H^{0}(\mathbb{Z}))$ , we can pick a finite set $S\subset H^{0}(\mathbb{Z})$ such that $S\rho^{-1}(\Gamma)\supset H^{0}(\mathbb{Z})$ . Now $K:=\pi(\rho(F_{2}S))$ is a compact subset of $G/\Gamma$ . Then, for any $t\geq T_{0}$ and $g\in F$ , let $h\in\rho^{-1}(g)\subset F_{1}$ . Let $t\in[T_{0},T]$ such that $\psi(t)h\text{SL}(m,\mathbb{Z})\in K_{1}$ . Then $\psi(t)hH^{0}(\mathbb{Z})\in F_{2}H^{0}(\mathbb{Z})$ , and hence $\pi(\varphi(t)g)\in K$ . Therefore, for any $g\in F$ , and $h\in\rho^{-1}(g)\subset F_{1}$ , we have

\{t\in[T_{0},T]:\varphi(t)g\Gamma\in K\}\supset\{t\in[T_{0},T]:\psi(t)h\text{SL}(m,\mathbb{Z})\in K_{1}\}.

From this and by (4.8), we obtain (4.7). This completes the proof when the real rank of $G$ is at least $2$ .

Now suppose that the real rank of $G$ is $1$ . It is straightforward to adapt the proof of [Dan84, Proposition 1.2] to conclude (4.7). For this purpose, it is enough to use the following property of the map $\varphi$ in place of [Dan84, Lemmas 2.5 and 2.7]: Let $V=\wedge^{d}\mathfrak{g}$ , where $d$ is the dimension of the maximal unipotent subgroup of $G$ and $\mathfrak{g}$ the Lie algebra of $G$ . Now, $G$ acts on $V$ via $\wedge^{d}\text{Ad}$ . Then there exists $T>0$ , $C>0$ and $\alpha>0$ such that for any $g\in G$ , the map $t\mapsto\lVert\varphi(t)g.p\rVert$ has the $(C,\alpha)$ -good property on $[T_{0},\infty)$ by Proposition 1.15.

∎

As an immediate consequence of Proposition 4.3 we obtain the following.

Theorem 4.5.

If $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ be a continuous unbounded non-contracting curve definable in a polynomially bounded o-minimal structure. Let $G\leq\text{SL}(n,\mathbb{R})$ be a Lie subgroup, $\Gamma$ be a lattice in $G$ and suppose that $\varphi([0,\infty))\subset G_{u}$ . Then any weak-* limit $\mu$ of $\mu_{T,\varphi,x}$ has $\mu(\infty)=0$ for all $x\in G/\Gamma$ .

5. Unipotent invariance

We begin with the following general statement.

Proposition 5.1.

Let $G\leq\text{GL}(n,\mathbb{R})$ be a closed subgroup. Consider a continuous curve $\varphi:[0,\infty)\to G$ which is unbounded and definable in a polynomially bounded o-minimal structure such that $\det(\varphi(t))$ is constant in $t$ . Suppose further that for all $v\in\mathbb{R}^{n}\smallsetminus\{0\}$ , $\lim_{t\to\infty}\varphi(t)v\neq 0$ . Then there exists a nontrivial one-parameter unipotent group $P\leq G$ such that if $\Gamma\leq G$ is a discrete subgroup, $g\in G$ and $\nu$ is weak- $\ast$ limit of the measures

\nu_{T,\varphi,g\Gamma}(f):=\frac{1}{T}\int_{0}^{T}f(\varphi(t)g\Gamma)dt,f\in C_{c}(G/\Gamma),

then $\nu$ is invariant under $P$ .

The main idea behind the above statement is the fact that o-minimal curves exhibit “tangency at infinity”. More precisely, there are “change of speed” maps $h(t)$ with $\lim_{t\to\infty}h(t)=\infty$ , such that the limit of the differences $\lim_{t\to\infty}\varphi(h(t))\varphi(t)^{-1}=\rho$ exists. Importantly, under our polynomially boundedness and the non-contraction assumption, those matrices $\rho$ will generate a one-parameter unipotent group, and any limiting measure of averaging along $\varphi$ will be invariant under $\rho$ .

Such an idea was used in [Sha94] for averaging along polynomial curves. In the polynomial case, it is possible to use Taylor expansion in order to conclude such change of speed maps. For the o-minimal case, Peterzil and Steinhorn in [PS99] showed that the collection of all possible limits of the form $\lim_{t\to\infty}\varphi(h(t))\varphi(t)^{-1}$ , where $h$ is a definable function converging to $\infty$ forms a one-dimensional torsion free group. We will refer to this group as the Peterzil-Steinhorn subgroup, which we define below in Definition 5.3 in a simplified manner which suits our needs. Poulios in his thesis [Pou13] studied further the Peterzil-Steinhorn groups, and he proves that the P.S. group for an unbounded curve definable in polynomially bounded structure is either unipotent or $\mathbb{R}$ -diagonalizable. Importantly, Poulios establishes a convenient condition for the Peterzil-Steinhorn group to be unipotent.

We now recall the required notions and results from [Pou13]. For the following, for $r<1$ , consider

(5.1)

h_{r,s}(t):=\left(t^{1-r}+(1-r)s\right)^{\frac{1}{1-r}}=t+st^{r}+o(t^{r}),\text{ where }s\in\mathbb{R},

and let

(5.2)

h_{1,s}(t):=st,\text{ where }s>0.

Proposition 5.2.

[Pou13, Chapter 3] Let $G\leq\text{GL}(n,\mathbb{R})$ be a closed subgroup, and consider an unbounded curve $\varphi:[0,\infty)\to G$ definable in a polynomially bounded o-minimal structure. Then, there exists a unique $r\leq 1$ such that the limit

(5.3)

M_{\varphi}:=\lim_{t\to\infty}t^{r}\cdot\varphi^{\prime}(t)\cdot\varphi(t)^{-1}

exists and non-zero. Notice that $M_{\varphi}$ is naturally identified as a tangent vector in the Lie algebra $\mathfrak{g}$ of $G$ . We have that $M_{\varphi}$ is nilpotent $\iff$ $r<1$ , and $M_{\varphi}$ is $\mathbb{R}$ -diagonalizable $\iff r=1$ . We let $\mathfrak{p}=\text{Span}_{\mathbb{R}}\{M_{\varphi}\}\leq\mathfrak{g}$ , and denote as $P\leq G$ the connected subgroup with Lie-algebra $\mathfrak{p}$ . Then:

(1)

$r<1$ $\iff$ for each $s\in\mathbb{R}$ , the limit

(5.4) $\rho(s):=\lim_{t\to\infty}\varphi(h_{r,s}(t))\varphi(t)^{-1},$

exists. In this case $\rho$ defines an isomorphism $\mathbb{R}\to P$ . Here $\mathbb{R}$ denotes the additive group of real numbers.
(2)

$r=1\iff$ for each $s>0$ , the limit

(5.5) $\rho(s):=\lim_{t\to\infty}\varphi(h_{1,s}(t))\varphi(t)^{-1},$

exists. In this case $\rho$ defines an isomorphism $\rho:\mathbb{R}_{>0}\to P$ . Here $\mathbb{R}_{>0}$ denotes the multiplicative group of positive real numbers.

Definition 5.3.

We say that $r\leq 1$ is the P.S. order of an unbounded curve $\varphi:[0,\infty)\to\text{GL}(n,\mathbb{R})$ definable in a polynomially bounded o-minimal structure, if $r$ satisfies (5.3), and we call the one-parameter subgroup generated by $\exp(M_{\varphi})$ the P.S. group of $\varphi$ .

Remark 5.4.

We note that if $\varphi$ as in Proposition 5.2 is a restriction of one-parameter group to $[0,\infty)$ , then $r=0$ , $h_{r,s}(t)=t+s$ , and $\rho(s)=\varphi(s)$ for all $s\in\mathbb{R}$ ; that is, $\varphi$ is a restriction of a unipotent one-parameter subgroup.

5.1. Unipotent P.S. groups and limiting unipotent invariance

We now show that when the P.S. group is unipotent, any limiting measure of measures of the form (1.1) will be invariant under the P.S. group. This proof is found in [Sha94] and in [PS18], and we give it here also for completeness. We note the following elementary calculus lemma.

Lemma 5.5.

Let $h:(0,\infty)\to\mathbb{R}$ be a differentiable function such that $\lim_{t\to\infty}h^{\prime}(t)=1$ . Then, for any bounded continuous function $F:[0,\infty)\to\mathbb{R}$ , we have

(5.6)

\frac{1}{T}\int_{0}^{T}[F(h(t))-F(t)]dt=0

Notice that for all $r<1$ , we have $h^{\prime}_{s,r}(t)=1+O(t^{r-1})$ .

Lemma 5.6.

Let $G\leq\text{GL}(n,\mathbb{R})$ be a closed subgroup and let $\Gamma\leq G$ be a discrete subgroup. Suppose that $\varphi:[0,\infty)\to G$ is a continuous, unbounded curve, definable in a polynomially bounded o-minimal structure such that the P.S. group is unipotent (namely, (5.3) holds for $r<1$ ). Let $\rho$ as in (5.4). Then

(5.7)

\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\{f(\rho(s)\varphi(t)x)-f(\varphi(t)x)\}dt=0,

for all $f\in C_{c}(G/\Gamma)$ , $x\in G/\Gamma$ and $s\in\mathbb{R}$ .

Proof.

For $f\in C_{c}(G/\Gamma)$ and an $\epsilon>0$ , by uniform continuity of $f$ , there is an $T_{\epsilon}$ such that

	$\displaystyle\|f(\rho(s)\varphi(t)x)-f(\varphi(h_{r,s}(t)x)\|$	$\displaystyle=\|f(\rho(s)\varphi(t)x)-f([\varphi(h_{r,s}(t))\varphi(t)^{-1}]\varphi(t)x)\|$
		$\displaystyle\leq\epsilon/2,$

for all $t\geq T_{\epsilon}$ . Now

		$\displaystyle\frac{1}{T}\int_{0}^{T}\|f(\rho(s)\varphi(t)x)-f(\varphi(t)x)\|dt\leq\frac{1}{T}\int_{0}^{T_{\epsilon}}\|f(\rho(s)\varphi(t)x)-f(\varphi(t)x)\|dt$
	$\displaystyle+$	$\displaystyle\frac{1}{T}\int_{T_{\epsilon}}^{T}\|f(\rho(s)\varphi(t)x)-f(\varphi(t)x)\|dt\leq\frac{1}{T}\int_{0}^{T_{\epsilon}}\|f(\rho(s)\varphi(t)x)-f(\varphi(t)x)\|dt$
	$\displaystyle+$	$\displaystyle\frac{1}{T}\int_{T_{\epsilon}}^{T}\|f(\varphi(h_{r,s}(t)x)-f(\varphi(t)x)\|dt+\frac{\epsilon}{2}.$

Now take $T\to\infty$ . The first term goes to zero by boundedness of the function $f$ , and the second term goes to zero by Lemma 5.5. ∎

Poulios’ observations were used to establish invariance for polynomially bounded o-minimal curves in nilmanifolds by Peterzil and Starchenko, see [PS18]. If a curve is contained in a unipotent group, as in [PS18], then the P.S. group is automatically unipotent. Here, in order to verify that the P.S. group is unipotent when the curve is non-contracting we give the following new observation.

Definition 5.7.

We call a curve $\{\varphi(t)\}\subset\text{GL}(n,\mathbb{R})$ essentially diagonal if there exists a decomposition

(5.8)

\varphi(t)=\sigma(t)b(t)C,

where $b(t)\in\text{GL}(n,\mathbb{R})$ is a diagonal matrix for all large $t$ , $C\in\text{GL}(n,\mathbb{R})$ is a constant matrix, and $\sigma(t)\in\text{SL}(n,\mathbb{R})$ is convergent as $t\to\infty$ .

The following is our key observation.

Proposition 5.8.

Let $\varphi:[0,\infty)\to\text{GL}(n,\mathbb{R})$ be an unbounded continuous curve definable in a polynomially bounded o-minimal structure. Suppose that $\det(\varphi(t))$ is constant for all large enough $t$ . Then, the P.S. group of $\varphi$ is unipotent if and only if the curve $\varphi$ is not essentially diagonal.

We will prove the above Proposition 5.8 in the subsection below. Before we proceed, we consider the following lemma which proves Proposition 5.1 by assuming Proposition 5.8 in combination with Lemma 5.6.

Lemma 5.9.

Let $\varphi:[0,\infty)\to\text{GL}(n,\mathbb{R})$ be an unbounded continuous curve definable in a polynomially bounded o-minimal structure. Suppose that the determinant $\det(\varphi(t))$ is constant in $t$ and suppose that $\lim_{t\to\infty}\varphi(t)v\neq 0,$ for all $v\in\mathbb{R}^{n}\smallsetminus\{0\}$ . Then, $\varphi$ is not essentially diagonal and in particular, the P.S. group of $\varphi$ is unipotent.

Proof.

Suppose for contradiction that the P.S. group is not unipotent. Then, by Proposition 5.8, $\varphi$ is essentially diagonal. Namely $\varphi(t)=\sigma(t)b(t)C$ , where $C$ , $\sigma(t)\in\text{SL}(n,\mathbb{R})$ are bounded and $b(t)$ is a diagonal matrix with $\det(b(t))=c_{0}$ for all $t$ . Since we assume that $\varphi$ is unbounded, it follows that $b(t)$ is unbounded, which in turn implies that there is an index $1\leq i\leq n$ such that the corresponding entry function $b_{ii}(t)$ on the diagonal of $b(t)$ decays to zero. In particular, for $v:=C^{-1}e_{i}$ it holds that $\lim_{t\to\infty}\varphi(t)v=0$ , which is a contradiction. ∎

5.1.1. Proving Proposition 5.8

We will call the $(i,j)$ index of an upper-triangular matrix $b\in\text{SL}(n,\mathbb{R})$ of the form

(5.9)

b=\begin{bmatrix}f_{1,1}&0&\cdots&0&0&\cdots&0&0&\cdots&0\\ 0&\ddots&\ddots&\vdots&\vdots&&\vdots&\vdots&&\vdots\\ \vdots&\ddots&\ddots&0&0&\cdots&0&0&\cdots&0\\ 0&\cdots&0&f_{i,i}&0&\cdots&0&\boldsymbol{f_{i,j}}&\cdots&f_{i,n}\\ \vdots&&\vdots&\ddots&\ddots&&&\vdots&&\vdots\\ \vdots&&\vdots&&\ddots&\ddots&&\vdots&&\vdots\\ \vdots&&\vdots&&&\ddots&\ddots&\vdots&&\vdots\\ \vdots&&\vdots&&&&0&f_{j,j}&\cdots&f_{j,n}\\ \vdots&&\vdots&&&&&\ddots&\ddots&\vdots\\ 0&\cdots&0&\cdots&\cdots&\cdots&\cdots&\cdots&0&f_{n,n}\end{bmatrix},

where $f_{i,j}\neq 0$ , the first non-zero off-diagonal entry. More precisely, $(i,j)$ is the first non-zero entry among the off-diagonal entries according to the following lexicographic order on $\mathbb{N}^{2}$ :

(5.10)

\displaystyle(i,j)\prec(k,l)\iff i<k\text{ or }(i=k\text{ and }j<l).

Let $B\leq\text{SL}(n,\mathbb{R})$ be the subgroup of upper-triangular matrices. Let $b:[0,\infty)\to B$ be a definable curve. Recall that each definable function $f(t)$ is either zero for all large enough $t$ or $|f(t)|>0$ for all large $t$ . Thus, for all large enough $t$ , there exists a unique first non-zero off-diagonal entry in $b(t)$ , or $b(t)$ is diagonal for all large $t$ . We will refer to this entry as the first non-zero off-diagonal entry of the curve $b(t)$ .

Lemma 5.10.

Let $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ be a curve definable in an polynomially bounded o-minimal structure. Then there is a definable curve $b:[0,\infty)\to B$ , such that $b(t)$ is either diagonal for all large $t$ , or the first non-zero off-diagonal entry $f_{i,j}$ satisfies

(1)

$\deg f_{i,i}\neq\deg f_{i,j}$ , and
(2)

$\deg f_{i,j}>\deg f_{j,j}$ ,

and importantly,

(5.11)

\varphi(t)=\sigma(t)b(t)C,

where $C\in\text{SL}(n,\mathbb{R})$ is constant, and $\sigma(t)\in\text{SL}(n,\mathbb{R})$ is convergent as $t\to\infty$ .

Proof.

Using the KAN decomposition, we first write $\varphi(t)\in\text{SL}(n,\mathbb{R})$ , as $\varphi(t)=k(t)p(t)$ , where $k(t)\in\text{SO}(n,\mathbb{R})$ and $p(t)\in B$ an upper-triangular matrix. Since $k(t)$ is obtained by performing the Gram-Schimdt process on the columns of $\varphi$ , we conclude that $k(t)$ is definable. As a consequence, $p(t)$ is definable. Since $k(t)$ is bounded and definable, $\lim_{t\to\infty}k(t)$ exists. For all $t$ large, $p(t)$ is either diagonal or $p(t)$ takes the form of (5.9).

We employ the following algorithm to achieve the outcome described in the lemma. At each step, we perform either a column or a row operation on $p(t)$ :

(1)

All off-diagonal entries are eventually $0$ . Then we are done.
(2)

We pick the first index $(i,j)$ (with $i<j$ ) such that the $(i,j)$ -th entry is not eventually $0$ , and proceed to the next step.
(3)
Suppose that $\deg(f_{i,j})=\deg(f_{i,i})$ . Then there is an $c\in\mathbb{R}$ such that $\deg(f_{i,j}-cf_{i,i})<\deg(f_{i,i})$ . For this step, subtract from the $j$ -th column the $i$ -th column multiplied by $c$ . This amounts to multiplying $p(t)$ by a constant unipotent matrix from the right. The obtained matrix is the same besides the $i,j$ -th entry which is replaced with $f_{i,j}-cf_{i,i}$ . There are two possibilities now:
1. (a)
  
  $f_{i,i}-cf_{i,j}$ is eventually zero. If all off-diagonal entries are eventually $0$ , then we are done. Otherwise, the first eventually non-zero off-diagonal entry in the resulting matrix has a strictly larger index (according to the lexicographic order), and we go back to step 2.
2. (b)
  
  $f_{i,i}-cf_{i,j}$ is not eventually zero: Then the first requirement of Lemma 5.10 is satisfied. We continue then with the following step.
(4)

If $\deg(f_{i,j})>\deg(f_{j,j})$ , then the second requirement of Lemma 5.10 is satisfied, and we are done, otherwise we proceed to the next step.
(5)

Now suppose $\deg(f_{i,j})\leq\deg(f_{j,j})$ . Then subtract from the $i$ -th row the $j$ -th row multiplied by $\frac{f_{i,j}}{f_{j,j}}$ . This amounts to multiplying from the left by a unipotent matrix, which converges as $t\to\infty$ . This is so because

$\deg\left(\frac{f_{i,j}}{f_{j,j}}\right)=\deg(f_{i,j})-\deg(f_{j,j})\leq 0.$

Then, either all the off-diagonal entries in the resulting matrix are eventually $0$ , and we are done; or the first eventually non-zero off-diagonal entry in the resulting matrix has a strictly larger index (according to the lexicographic order), and we go back to step 2.

The algorithm ends with finitely many steps with either a diagonal matrix or a definable curve $b(t)$ satisfying the requirements of the lemma. ∎

Proof for Proposition 5.8.

First, note that it is enough to prove the statement for $\varphi\subset\text{SL}(n,\mathbb{R})$ since the P.S. group is unchanged if we multiply the curve by a constant invertible matrix from the right. In the notations of Lemma 5.10:

\varphi(t)=\sigma(t)b(t)C,

where $\lim_{t\to\infty}\sigma(t)=g\in\text{SL}(n,\mathbb{R}).$ Let $r\leq 1$ be such that (5.3) holds for $\{b(t)\}$ . Let $h_{s,r}(t)$ be is as in (5.1)–(5.2). We note that

(5.12)

\lim_{t\to\infty}\varphi(h_{s,r}(t))\varphi(t)^{-1}=g\left[\lim_{t\to\infty}b(h_{s,r}(t))b(t)^{-1}\right]g^{-1}.

Therefore, by Proposition 5.2, the $r$ corresponding to (5.3) for $\varphi(t)$ and $b(t)$ are the same.

Now, $b(t)$ is either eventually diagonal or upper-triangular and satisfies the conditions of the Lemma 5.10 for the first (eventually) non-zero off-diagonal entry.

If $b(t)$ is eventually diagonal, then for all $s>0$ , $\lim_{t\to\infty}b(h_{1,s}(t))b^{-1}(t)$ converges to a diagonal matrix. Namely, the P.S. group is diagonal in this case.

Otherwise, let $(i,j)$ be the first non-zero off-diagonal entry in $b(t)$ (for all large enough $t$ ). We observe that $(i,j)$ -th entry in the matrix $b^{\prime}(t)b(t)^{-1}$ is

(5.13)

\displaystyle\frac{-f_{i,i}^{\prime}f_{i,j}+f_{i,j}^{\prime}f_{i,i}}{f_{i,i}f_{j,j}}=\left(\frac{f_{i,j}}{f_{i,i}}\right)^{\prime}\cdot\frac{f_{i,i}}{f_{j,j}}.

Since $r_{i,i}:=\deg f_{i,i}\neq r_{i,j}:=\deg f_{i,j}$ , we have that

\deg\left(\frac{f_{i,j}}{f_{i,i}}\right)^{\prime}={r_{i,j}-r_{i,i}-1}.

Thus,

\deg\left(\left(\frac{f_{i,j}}{f_{i,i}}\right)^{\prime}\cdot\frac{f_{i,i}}{f_{j,j}}\right)=r_{i,j}-r_{j,j}-1.

Recall by Lemma 5.10 that $r_{i,j}-r_{j,j}>0$ . Thus the $(i,j)$ -th entry in $tb^{\prime}(t)b(t)^{-1}$ is unbounded. So (5.5) is not satisfied. Hence $r\neq 1$ . ∎

5.2. Unipotent invariance in quotients

For our purposes, it will not suffice only to establish unipotent invariance for our limiting measures, but we will also require unipotent invariance in certain quotients. We will now describe more precisely our motivation for the results in this section and how they play a role in the proof of Theorem 1.8 in Section 6.

Consider a non-contracting curve $\varphi:[0,\infty)\to G:=\mathbf{G}(\mathbb{R})$ definable in a polynomially bounded o-minimal structure. By Theorem 4.5, there exists a subsequence of the measures $\mu_{T_{i},\varphi,x}$ as in (1.1) that converges to a probability measure $\mu$ as $i\to\infty$ . By Proposition 5.1, $\mu$ is invariant under a nontrivial unipotent one-parameter subgroup. We will let $W$ be the subgroup generated by all unipotent one-parameter subgroups that preserve $\mu$ . Then, using Ratner’s theorem describing finite ergodic invariant measures for $W$ , and the linearization technique (which is applicable due to the $(C,\alpha)$ -good property for the norm of trajectories of $\varphi(t)$ on representations of $\text{SL}(n,\mathbb{R})$ ), we will prove that $\varphi(t)$ is contained in a certain algebraic subgroup $N$ such that $W\trianglelefteq N$ and that $H_{\varphi}\subseteq N$ . Our goal will be to prove that $H_{\varphi}\subset W$ . To achieve this goal, we would like to show that if $\varphi(t)$ is not bounded modulo $W$ , then the image of $\varphi$ in $N/W$ , which is definable, is, in fact, non-contracting, and hence its corresponding P.S. group in $N/W$ is unipotent. Consequently, we will show that $\mu$ is invariant under a unipotent one-parameter subgroup of $N$ that is not contained in $W$ . This will contradict our choice of $W$ .

Remark 5.11.

As the following example demonstrates, one cannot expect a non-contracting curve modulo a subgroup to have a unipotent P.S. group. Consider $\varphi(t):=\begin{bmatrix}t&t^{2}\\ 0&1/t\end{bmatrix}$ for $t\geq 1$ . Then $\varphi$ is non-contracting in $\text{SL}(2,\mathbb{R})$ , and its P.S. group is $U:=\{u(s):=\begin{bmatrix}1&s\\ 0&1\end{bmatrix}:s\in\mathbb{R}\}$ . Now consider the group $P\leq\text{SL}(2,\mathbb{R})$ of upper triangular matrices. Then, $P/U$ is naturally identified with the group $A\leq\text{SL}(2,\mathbb{R})$ of diagonal matrices. In particular, $\varphi(t)$ modulo $H$ is a diagonal curve. In view of our result below, this is explained by the fact that the hull of the curve is not contained in $P$ .

We will prove the following.

Proposition 5.12.

Let $\psi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ be an unbounded, non-contracting curve definable in a polynomially bounded o-minimal structure. Suppose that $\psi\subseteq H_{\psi}$ , where $H_{\psi}$ is the hull of $\psi$ . Let $N\leq\text{SL}(n,\mathbb{R})$ be a Zariski closed subgroup such that $H_{\psi}\leq N$ , and let $H\trianglelefteq N$ be a closed normal subgroup. Let $q:N\to N/H$ be the natural map.

Assume that the image of $\psi$ in $N/H_{u}$ is unbounded. Then, there exists $r<1$ such that

(5.14)

\lim_{t\to\infty}q(\psi(h_{s,r}(t)))q(\psi(t))^{-1}=p(s),s\in\mathbb{R},

where $p(s),s\in\mathbb{R}$ is a (non-trivial) one-parameter subgroup of $N/H$ . Here, $h_{s,r}(t)$ is given by (5.1). Importantly, there is a one-parameter unipotent subgroup $U\leq N$ such that

q(U)=\{p(s):s\in\mathbb{R}\}.

Proof of Proposition 5.12.

Let $\tilde{H}_{u}:=\text{zcl}(H_{u})$ with $H_{u}$ being the subgroup generated by all one-parameter unipotent subgroups contained in $H$ . Since $H\trianglelefteq N$ , it follows that $H_{u}\trianglelefteq N$ , and by Zariski density we get that $\tilde{H}_{u}\trianglelefteq N$ . Now, $H_{u}$ is of finite index in $\tilde{H}_{u}$ , therefore, since $\psi$ is unbounded in $N/H_{u}$ it follows that $\psi$ is unbounded in $N/\tilde{H}_{u}$ .

By [Spr98, Theorem 5.5.3] and since $\tilde{H}_{u}$ has no non-trivial real character, there is a finite dimensional rational representation $\rho:\text{SL}(n,\mathbb{R})\to\text{GL}(V)$ with an $v\in V$ such that

\tilde{H}_{u}=\{g\in\text{SL}(n,\mathbb{R}):\rho(g)v=v\}.

Now let

W:=\{x\in V:\rho(h)x=x,\forall h\in\tilde{H}_{u}\}.

We have that $v\in W$ so that $W$ is not the trivial subspace. Because $\tilde{H}_{u}$ is a normal subgroup of $N$ , we get that the action of $N$ keeps $W$ invariant. Now recall that one-parameter unipotent subgroups generate $H_{\psi}$ . Therefore, $\det(\rho(H_{\psi})|_{W})=\{1\}$ . In particular, $\det(\rho(\psi)|_{W})=\{1\}$ .

Since $\psi$ is not contracting, we have in particular that $\lim_{t\to\infty}\rho(\psi(t))w\neq 0,~{}\forall w\in W\smallsetminus\{0\}.$ Thus, by Lemma 5.9, there is an $r<1$ such that

\lim_{t\to\infty}\rho(\psi(h_{s,r}(t))\psi(t)^{-1})|_{W}=:\bar{p}(s),s\in\mathbb{R},

where $\overline{p}(s)$ is a one-parameter unipotent subgroup. It is a general fact if $f:G_{1}\to G_{2}$ is a homomorphism of algebraic groups, and $U_{2}\leq G_{2}$ is a unipotent one-parameter subgroup, then there is a unipotent one-parameter subgroup $U_{1}\leq G_{1}$ such that $f(U_{1})=U_{2}$ . This follows by Jordan decomposition. Thus, there is a unipotent one-parameter subgroup $U\leq N$ such that $\rho(U)|_{W}=\{\bar{p}(s):s\in\mathbb{R}\}$ . Now consider the diagram:

(5.15)

Since images of algebraic groups under algebraic homomorhpisms are closed (see e.g. [Spr98, Proposition 2.2.5]), we conclude that $\tilde{\tau}$ is an injective proper map. Since $H_{u}$ is of finite index in $\tilde{H}_{u}$ , the map $\tau$ is a covering map with finite fibers, and it follows that $\tilde{\tau}\circ\tau$ is a proper map with finite fibers. Then, we obtain for each $s\in\mathbb{R}$ ,

(5.16)

\lim_{t\to\infty}\vartheta(\psi(h_{s,r}(t))\psi(t)^{-1})=\lim_{t\to\infty}\vartheta(\psi(h_{s,r}(t)))\vartheta(\psi(t))^{-1}=:\tilde{p}(s).

Now we show that $\tilde{p}(s),s\in\mathbb{R}$ is a one-parameter group, where $s\mapsto\tilde{p}(s)$ is an homomorphism from the additive group of real numbers. This argument appears in [Pou13] and given here for completeness. Note that for $r<1$ and for all $s,l\in\mathbb{R}$ that

h_{s,r}\circ h_{l,r}=h_{s+l,r}.

Denote $\xi(t):=\vartheta(\psi(t))$ . Now if (5.14) holds for $r<1$ , then for $s,l\in\mathbb{R}$ we have

	$\displaystyle\tilde{p}(s)\tilde{p}(l)$	$\displaystyle=(\lim_{t\to\infty}\xi(h_{s,r}(t))\xi(t)^{-1})(\lim_{t\to\infty}\xi(h_{l,r}(t))\xi(t)^{-1})$
		$\displaystyle=(\lim_{t\to\infty}\xi(h_{s,r}(h_{l,r}(t)))\xi(h_{l,r}(t))^{-1})(\lim_{t\to\infty}\xi(h_{l,r}(t))\xi(t)^{-1})$
		$\displaystyle=\lim_{t\to\infty}\xi(h_{s,r}(h_{l,r}(t)))\xi(h_{l,r}(t))^{-1}\xi(h_{l,r}(t))\xi(t)^{-1}$
		$\displaystyle=\lim_{t\to\infty}\xi(h_{s+l,r}(t))\xi(t)^{-1}=\tilde{p}(s+l).$

Since the above diagram commutes, we get that $\vartheta(U)=\{\tilde{p}(s):s\in\mathbb{R}\}$ is a non-trivial subgroup of $N/H_{u}$ . Finally, consider the natural quotient map $\eta:N/H_{u}\to N/H$ , and note that $q=\eta\circ\vartheta$ . Let $p(s):=\eta(\tilde{p}(s))$ . Then, due to (5.16), we get by continuity that for all $s\in\mathbb{R}$ ,

p(s)=\lim_{t\to\infty}\eta\circ\vartheta(\psi(h_{s,r}(t))\psi(t)^{-1})=\lim_{t\to\infty}q(\psi(h_{s,r}(t)))q(\psi(t))^{-1}.

Finally, since $U\not\subset H_{u}$ and since $U$ is unipotent, it follows that $q(U)\leq N/H$ is not trivial, and $q(U)=\eta\circ\vartheta(U)=\{p(s):s\in\mathbb{R}\}.$

∎

5.3. Inclusion of the hull in a Lie group

Here, we will prove the following.

Proposition 5.13.

Let $G\leq\text{SL}(n,\mathbb{R})$ be a connected unimodular closed subgroup, and let $\varphi:[0,\infty)\to G$ be a non-contracting, continuous curve, definable in a polynomially bounded o-minimal structure. Then, $H_{\varphi}\leq G_{u}$ where $H_{\varphi}$ is the hull of $\varphi$ and $G_{u}\leq G$ is the closed subgroup generated by the one-parameter unipotent subgroups contained in $G$ . In particular, both the hull and the range of the correcting curve of $\varphi$ are in $G$ .

The proof will be based on Proposition 5.12 and the following key lemma.

Lemma 5.14.

Let $\mathfrak{g}$ be a Lie subalgebra of $\mathfrak{gl}(n,R)$ . Let $\mathfrak{g}_{u}$ be the subalgebra generated by nilpotent matrices in $\mathfrak{g}$ . Then for any $X\in\mathfrak{g}$ ,

\text{Trace}(\text{ad}X)=\text{Trace}(\text{ad}X|_{\mathfrak{g}_{u}}).

Proof.

We will be using repeatedly the following fact – If $T$ is a linear map on a vector space $V$ preserving a subspace $W$ , and $T_{V/W}$ is the factored map on $V/W$ , then

\text{Trace}(T|_{V})=\text{Trace}(T|_{W})+\text{Trace}(T_{V/W}),

where $T|_{W}$ is the restriction of $T$ to $W$ .

Let $X\in\mathfrak{g}$ . Then $\text{ad}X$ acts trivially on $\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$ . Hence,

\text{Trace}(\text{ad}X|_{\mathfrak{g}})=\text{Trace}(\text{ad}X|_{[\mathfrak{g},\mathfrak{g}]}).

Let $\tilde{\mathfrak{g}}$ denote the Lie algebra of the Zariski closure of the Lie group associated with $\mathfrak{g}$ in $\text{GL}(n,\mathbb{R})$ . By [Che51, Chapter II, Theorem 13], $[\tilde{\mathfrak{g}},\tilde{\mathfrak{g}}]=[\mathfrak{g},\mathfrak{g}]$ . Thus,

\text{Trace}(\text{ad}X|_{\mathfrak{g}})=\text{Trace}(\text{ad}X|_{[\tilde{\mathfrak{g}},\tilde{\mathfrak{g}}]}).

Let $\mathfrak{u}$ denote the radical of the $[\tilde{\mathfrak{g}},\tilde{\mathfrak{g}}]$ . Then $\mathfrak{u}$ consists of nilpotent matrices; we can derive this from [Rag72, 2.5 A structure theorem]. We also note that $[\tilde{\mathfrak{g}},\tilde{\mathfrak{g}}]$ and $\mathfrak{u}$ are ideals in $\tilde{\mathfrak{g}}$ . Since $[\tilde{\mathfrak{g}},\tilde{\mathfrak{g}}]/\mathfrak{u}$ is semisimple, the trace of the derivation on it corresponding to $\text{ad}X$ is zero. Therefore,

\text{Trace}(\text{ad}X|_{[\tilde{\mathfrak{g}},\tilde{\mathfrak{g}}]})=\text{Trace}(\text{ad}X|_{\mathfrak{u}}).

Now let $\mathfrak{v}$ denote the radical of $\mathfrak{g}_{u}$ . Since $\mathfrak{g}_{u}$ is a Lie subalgebra generated by nilpotent matrices, $\mathfrak{v}$ consists of nilpotent matrices. Also, $\mathfrak{g}_{u}$ , and hence $\mathfrak{v}$ are ideals of $\mathfrak{g}$ . As above, since $\mathfrak{g}_{u}/\mathfrak{v}$ is semisimple, we conclude that

\text{Trace}(\text{ad}X|_{\mathfrak{g}_{u}})=\text{Trace}(\text{ad}X|_{\mathfrak{v}}).

It remains to prove:

\text{Trace}(\text{ad}X|_{\mathfrak{v}})=\text{Trace}(\text{ad}X|_{\mathfrak{u}}).

Now $\mathfrak{u}\subset[\tilde{\mathfrak{g}},\tilde{\mathfrak{g}}]=[\mathfrak{g},\mathfrak{g}]\subset\mathfrak{g}$ . Hence, $\mathfrak{u}\subset\mathfrak{g}_{u}$ , and so $\mathfrak{u}\subset\mathfrak{v}$ . Moreover, $\mathfrak{v}\cap[\tilde{\mathfrak{g}},\tilde{\mathfrak{g}}]$ is an ideal in $\tilde{\mathfrak{g}}$ , so it is contained in $\mathfrak{u}$ . Thus,

\mathfrak{v}\cap[\tilde{\mathfrak{g}},\tilde{\mathfrak{g}}]=\mathfrak{u}.

Therefore $[X,\mathfrak{v}]\subset\mathfrak{v}\cap[\tilde{\mathfrak{g}},\tilde{\mathfrak{g}}]=\mathfrak{u}$ . Therefore, the action of $\text{ad}X$ on $\mathfrak{v}/\mathfrak{u}$ is zero. So,

\text{Trace}(\text{ad}X|_{\mathfrak{v}})=\text{Trace}(\text{ad}X|_{\mathfrak{u}}).

∎

Corollary 5.15.

Let $G$ be a closed connected Lie subgroup of $\text{GL}(n,\mathbb{R})$ . Suppose that $G$ is unimodular. Let $G_{u}$ be the closed subgroup generated by unipotent one-parameter subgroups contained in $G$ and let $\mathfrak{g}_{u}$ denote the Lie algebra of $G_{u}$ . Then for any $g\in G$ ,

\det(\text{Ad}(g)|_{\mathfrak{g}_{u}})=1.

Proof.

Suppose that $g=\exp(X)$ . Then,

\det(\text{Ad}(g)|_{\mathfrak{g}_{u}})=\exp(\text{Trace}(\text{ad}X|_{\mathfrak{g}_{u}})).

Then, Lemma 5.14 shows that $\det(\text{Ad}(g)|_{\mathfrak{g}_{u}})=\det(\text{Ad}(g))$ . Since $G$ is connected and unimodular, we have that $\det(\text{Ad}(g))=1$ . Then, $\det(\text{Ad}(g)|_{\mathfrak{g}_{u}})=1$ for all $g$ in some identity neighborhood in $G$ . Since the identity neighborhood generates $G$ , the corollary follows. ∎

Proof of Proposition 5.13.

Let $\varphi:[0,\infty)\to G$ be a non-contracting, continuous curve, definable in a polynomially bounded o-minimal structure. Let $\beta$ be a correcting curve for $\varphi$ , and denote $\psi(t):=\beta(t)\varphi(t)$ .

Consider the exterior product of the Adjoint representation of $\text{SL}(n,\mathbb{R})$ on $\wedge^{\dim\mathfrak{g}_{u}}\mathfrak{sl}(n,\mathbb{R})$ . Let $p\in\wedge^{\dim\mathfrak{g}_{u}}\mathfrak{g}_{u}\setminus\{0\}$ , and let $N$ be the stabilizer of $p$ in $\text{SL}(n,\mathbb{R})$ . Note that $N$ is Zariski closed, $N$ is observable, and $G_{u}\trianglelefteq N$ . By Corollary 5.15, $G\subset N$ . Since $N$ is observable, and $\varphi\subset N$ , we conclude that $H_{\varphi}\subseteq N$ by the minimality of the hull, Theorem 2.4,(1). Now, by Remark 2.6, we have that $\psi\subset H_{\psi}=H_{\varphi}$ . Assume for contradiction that $\psi$ is not bounded modulo $G_{u}$ . Then, by Proposition 5.12, there exists a one-parameter unipotent subgroup of $\{u(s)\}_{s\in\mathbb{R}}\leq N$ such that

q(u(s))=\lim_{t\to\infty}q(\psi(h_{s,r}(t)))q(\psi(t))^{-1},s\in\mathbb{R}.

Also $q\circ u$ is a nontrivial unipotent one-parameter subgroup of $N/G_{u}$ . Let $\beta_{\infty}=\lim_{t\to\infty}\beta(t)$ . Then, for all $s\in\mathbb{R}$ ,

q(\beta_{\infty}^{-1}u(s)\beta_{\infty})=\lim_{t\to\infty}q(\varphi(h_{s,r}(t)))q(\varphi(t))^{-1},

and since $\varphi\subset G\subset N$ , we get that $\{q(\beta_{\infty}^{-1}u(s)\beta_{\infty}):s\in\mathbb{R}\}$ is a nontrivial one-parameter subgroup of $G/G_{u}$ . It then follows that $\{\beta_{\infty}^{-1}u(s)\beta_{\infty}:s\in\mathbb{R}\}$ is a nontrivial one-parameter unipotent subgroup contained in $G$ not contained in $G_{u}$ , a contradiction. Finally, since $\varphi$ is bounded modulu $G_{u}$ , and since $\text{zcl}(G_{u})$ is observable, we get that $H_{\varphi}\subseteq G_{u}$ by the minimality of the hull, Theorem 2.4,(1). ∎

6. Linearization

We begin with preliminaries needed for the linearization technique. Suppose that $\Gamma\leq\text{SL}(n,\mathbb{R})$ is a discrete subgroup, and let $\mu$ be a probability measure on $X:=\text{SL}(n,\mathbb{R})/\Gamma$ . Suppose that there exists a subgroup $W\leq G$ of positive dimension which is generated by one-parameter unipotent subgroups contained in $W$ such that $\mu$ is invariant under the $W$ -left translates. By the celebrated Ratner’s theorem (see [Rat91]) every $W$ -ergodic component of $\mu$ is homogeneous.

Definition 6.1.

Let $\mathcal{H}$ be the class of all closed connected subgroups $H$ of $\text{SL}(n,\mathbb{R})$ such that $H/H\cap\Gamma$ admits an $H$ -invariant probability measure and the closed subgroup $H_{u}\leq H$ which is generated by all unipotent one-parameter subgroups of $H$ acts ergodically on $H/H\cap\Gamma$ with respect to the $H$ -invariant probability measure.

We note the following results:

•

[Rat91, Theorem 1.1]: The collection $\mathcal{H}$ is countable.

For $H\in\mathcal{H}$ , define

	$\displaystyle N(H,W)=$	$\displaystyle\{g\in\text{SL}(n,\mathbb{R}):W\subset gHg^{-1}\},\text{ and}$
	$\displaystyle S(H,W)=$	$\displaystyle\bigcup_{H^{\prime}\in\mathcal{H},H^{\prime}\subsetneq H}N(H^{\prime},W).$

In what follows, we denote by $\pi:\text{SL}(n,\mathbb{R})\to X$ the canonical projection. Consider

(6.1)

T_{H}(W)=\pi(N(H,W)\smallsetminus S(H,W))=\pi(N(H,W))\smallsetminus\pi(S(H,W)),

for the second equality see [MS95, Lemma 2.4].

Theorem 6.2.

[MS95, Theorem 2.2] Let $\nu$ be a $W$ -invariant Borel probability measure on $X$ . For $H\in\mathcal{H}$ , let $\nu_{H}$ denote the restriction of $\nu$ on $T_{H}(W)$ . Then $\nu$ decomposes as:

\nu=\sum_{H\in\mathcal{H}^{*}}\nu_{H},

where $\mathcal{H}^{*}\subset\mathcal{H}$ is a countable set of subgroups, each is a representative of a distinct $\Gamma$ -conjugacy class.

Moreover, for $H\in\mathcal{H}$ it holds that $\nu_{H}$ is $W$ -invariant, and any $W$ -ergodic component of $\nu_{H}$ is the unique $gHg^{-1}$ -invariant probability measure on the closed orbit $gH\Gamma/\Gamma\subset X$ , for some $g\in N(H,W)\smallsetminus S(H,W)$ .

6.1. o-minimal curves in representations – a dichotomy theorem

For any $d\in\mathbb{N}$ , consider the action of $\text{SL}(n,\mathbb{R})$ on

(6.2)

V_{d}=\bigwedge^{d}\mathfrak{sl}(n,\mathbb{R}),

induced by the Adjoint representation of $\text{SL}(n,\mathbb{R})$ on its Lie algebra $\mathfrak{sl}(n,\mathbb{R})$ ; that is, $g\cdot(\bigwedge_{i=1}^{d}x_{i}):=\bigwedge_{i=1}^{d}\text{Ad}_{g}(x_{i})$ and extended linearly. Let $H\in\mathcal{H}$ , and let $\mathfrak{h}$ be the Lie algebra of $H$ and $d_{H}=\dim\mathfrak{h}$ . Fix a vector $p_{H}\in\bigwedge^{d_{H}}\mathfrak{h}\setminus\{0\}$ .

Theorem 6.3.

Let $H\in\mathcal{H}$ . Then:

(1)

[DM93, Proof of Proposition 3.2] Let $\mathfrak{w}$ be the Lie algebra of $W$ , and consider the variety

(6.3) $A_{H}:=\{v\in V_{d_{H}}:v\wedge w=0\in V_{d_{H}+1},\forall w\in\mathfrak{w}\}.$

Then, $g.p_{H}\in A_{H}$ if and only if $g\in N(H,W)$ . In other words, let $\eta_{H}:\text{SL}(n,\mathbb{R})\to V_{d_{H}}$ denote the orbit map

(6.4) $\eta_{H}(g)=g.p_{H},\,\forall g\in\text{SL}(n,\mathbb{R}),$

then $\eta_{H}^{-1}(A_{H})=N(H,W).$
(2)

[DM93, Theorem 3.4] $\Gamma.p_{H}$ is discrete in $V_{d_{H}}$ .

We present the following variant of [MS95, Proposition 3.4].

Theorem 6.4.

Let $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ be a continuous, unbounded, non-contracting curve that is definable in a polynomially bounded o-minimal structure. Let $\beta:[0,\infty)\to\text{SL}(n,\mathbb{R})$ be a correcting curve of $\varphi$ and let $H_{\varphi}$ be the hull of $\varphi$ such that $\beta(t)\varphi(t)\subset H_{\varphi}$ for all $t$ , see Definition 2.5. Fix $H\in\mathcal{H}$ . Let $\emph{\text{K}}_{1}\subset A_{H}$ be a compact set, and let $0<\epsilon<1$ . Then there exists a closed set $\mathcal{S}\subset\pi(S(H,W))$ such that the following holds.

For a given compact set $K\subset X\smallsetminus\mathcal{S}$ , there exists a neighborhood $\Psi$ of $\emph{\text{K}}_{1}$ in $V_{d_{H}}$ and such that for any $x\in X$ , at least one of the following is satisfied:

(1)

There exists an $w\in\pi^{-1}(x).p_{H}\cap\Psi$ such that

(6.5) $H_{\varphi}.w=w.$
(2)

For all sufficiently large $T>0$ (depending on $x$ ),

(6.6) $|\{t\in[0,T]:\beta(t)\varphi(t).x\in K\cap\pi(\eta_{H}^{-1}(\overline{\Psi}))\}|\leq\epsilon T.$

Here, $\eta_{H}$ is the orbit map (6.4).

First, recall some observations and results that will be used in the proof. We let $N(H)$ be the normalizer of $H$ in $\text{SL}(n,\mathbb{R})$ , and denote

N_{\Gamma}:=N(H)\cap\Gamma.

We recall that for $\gamma\in N_{\Gamma}$ it holds that

(6.7)

\det(\text{Ad}_{\gamma}|_{\mathfrak{h}})\in\{\pm 1\},

see [DM93, Lemma 3.1].

In view of (6.7), either $N_{\Gamma}.p_{H}=\{p_{H}\}$ or $N_{\Gamma}.p_{H}=\{\pm p_{H}\}$ . If the latter case is true, we will denote by $\tilde{V}_{H}:=V_{d_{H}}/\{\pm 1\}$ , the space $V_{d_{H}}$ modulo the equivalence relation defined by identifying $v\in V_{d_{H}}$ with $-v$ . Otherwise, namely if $N_{\Gamma}.p_{H}=\{p_{H}\}$ , then we denote $\tilde{V}_{d_{H}}:=V_{d_{H}}$ .

Proposition 6.5.

[MS95, Proposition 3.2] Let $D$ be a compact subset of $A_{H}$ , and consider

(6.8)

S(D)=\{g\in\eta_{H}^{-1}(D):g\gamma\in\eta_{H}^{-1}(D)\text{ for some }\gamma\in\Gamma\smallsetminus N_{\Gamma}\}\subset\text{SL}(n,\mathbb{R}).

Then:

(1)

$S(D)\subset S(H,W)$ .
(2)

$\pi(S(D))$ is closed in $X$ .
(3)

Suppose that $K\subset X\smallsetminus\pi(S(D))$ is compact. Then, there is a compact neighborhood $\Phi\subseteq V_{d_{H}}$ of $D$ such that for every $y\in\pi(\eta_{H}^{-1}({\Phi}))\cap K$ , the set $(\pi^{-1}(y).\tilde{p}_{H})\cap\tilde{{\Phi}}$ consists of a single element. Here $\tilde{{\Phi}}$ are the images of $p_{H}$ and ${\Phi}$ in $\tilde{V}_{d_{H}}$ .

Proof of Theorem 6.4.

We assume for simplicity that $N_{\Gamma}.p_{H}=\{p_{H}\}$ , namely $\tilde{V}_{d_{H}}=V_{d_{H}}$ . The case for which $N_{\Gamma}.p_{H}=\{\pm p_{H}\}$ is treated in essentially the same way and is left for the readers. Let $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ be an unbounded, continuous non-contracting curve definable in a polynomially bounded o-minimal structure. Fix $H\in\mathcal{H}$ . Let $\text{K}_{1}\subset A_{H}$ be a compact subset and $0<\epsilon<1$ be given. We will apply Proposition 3.10 with $\psi(t)$ being the image of $\beta(t)\varphi(t)$ in the representation ${V}_{d_{H}}$ . Let $\text{K}_{2}\subset A_{H}$ be a compact set given by Proposition 3.10 with $\text{K}_{1}\subset\text{K}_{2}$ .

In the notations of Proposition 6.5, we take the compact set $D:=\text{K}_{2}$ and let $\mathcal{S}:=\pi(S(\text{K}_{2})).$ Let $K\subset X\smallsetminus\mathcal{S}$ be a compact set given in the hypothesis of the proposition. Let $\Phi\subset{V}_{d_{H}}$ be a compact neighborhood of $\text{K}_{2}$ which satisfies the outcome of Proposition 6.5,(3). As a consequence, for any $t\geq 0$ , $w_{1},w_{2}\in\pi^{-1}(x).p_{H}$ , and $g\in\text{SL}(n,\mathbb{R})$ , if $gx\in K$ and $\beta(t)\varphi(t).w_{i}\in\Phi$ for $i=1,2$ , then $w_{1}=w_{2}$ ; let us call this the injective property of $K$ in $\Phi$ .

We now choose an open neighborhood $\Psi$ of $\text{K}_{1}$ in $V_{d_{H}}$ , with $\overline{\Psi}\subset\mathring{\Phi}$ , as in Proposition 3.10, where $\mathring{\Phi}$ denotes the interior of $\Phi$ .

Case 1 – There exists $w\in\pi^{-1}(x).p_{H}\cap\overline{\Psi}$ such that $H_{\varphi}.w=w$ for all $t\geq 0$ .

Case 2 – For all $w\in\pi^{-1}(x).p_{H}\cap\overline{\Psi}$ it holds that $\lim_{t\to\infty}\beta(t)\varphi(t).w=\infty$ . By Theorem 2.4,(3), the two mutually exclusive cases exhaust all possibilities. To complete the proof, we now assume that Case 1 does not occur and Case 2 occurs. We want to show that (6.6) holds for all sufficiently large $T$ .

We take $T^{\prime}_{0}>0$ to satisfy the outcomes of Propositions 1.15 and 3.10, and consider

(6.9)

J=\{t\geq T^{\prime}_{0}:\beta(t)\varphi(t)x\in\pi(\eta^{-1}_{H}(\overline{\Psi}))\cap K\}.

Then, to prove (6.6), we only need to show that

(6.10)

\lvert J\cap[T_{0}^{\prime},T]\rvert\leq\epsilon T\text{, for all sufficiently large $T\geq T_{0}^{\prime}$.}

For every $t\in J$ , there exists $w\in\pi^{-1}(x).p_{H}$ such that $\beta(t)\varphi(t).w\in\overline{\Psi}\subset\Phi$ and we have $\beta(t)\varphi(t)x\in K$ . By the injective property of $K$ in $\Phi$ , we get that such a $w$ is unique, and denote it by $w(t)$ . We note that for any $t\in J$ , $\lim_{s\to\infty}\beta(s)\varphi(s).w(t)=\infty$ , because otherwise $w(t)$ is fixed by $H_{\varphi}$ due to (3) of Theorem 2.4, and hence $w(t)\in\overline{\Psi}\cap\pi^{-1}(x).p_{H}$ , contradicting our assumption that Case 1 does not occur.

Therefore, given $t\in J$ and its corresponding $w(t)\in\pi^{-1}(x).p_{H}$ , there exists the largest closed interval $I(w(t)):=[t^{-},t^{+}]\subset[T^{\prime}_{0},T]$ containing $t$ such that

(1)

$\beta(s)\varphi(s).w(t)\in\Phi$ for all $s\in I(w(t))$ ,
(2)

$t^{-}\in J$ , and
(3)

$\beta(t^{+})\varphi(t^{+}).w(t)\in\Phi\smallsetminus\mathring{\Phi}$ or $t^{+}=T$ .

We note that $I(w(t))$ is uniquely determined by the $t\in J$ . For any $s\in J\cap I(w(t))$ , we have $\beta(s)\varphi(s)x\in K$ and $\beta(s)\varphi(s)\cdot w(t)\in\Phi$ , so by the injectivity of $K$ in $\Phi$ , we have $w(s)=w(t)$ . Therefore, by (6.9),

	$\displaystyle J\cap I(w(t))$	$\displaystyle=\{s\in I(w(t))\cap J:\beta(s)\varphi(s)\cdot w(s)\in\Psi\}$
		$\displaystyle=\{s\in I(w(t))\cap J:\beta(s)\varphi(s)\cdot w(t)\in\Psi\}$
(6.11)			$\displaystyle\subseteq\{s\in I(w(t)):\beta(s)\varphi(s)\cdot w(t)\in\Psi\}.$

We claim that for any $t_{1},t_{2}\in J$ and their corresponding $w(t_{1}),w(t_{2})\in\pi^{-1}(x).p_{H}$ , we have

(6.12)

\text{either }I(w(t_{1}))=I(w(t_{2}))\text{ or }I(w(t_{1}))\cap I(w(t_{2}))=\varnothing.

To verify this claim, suppose that $I(w(t_{1}))\cap I(w(t_{2}))\neq\varnothing$ . Then either $t_{1}^{-}\in I(w(t_{2}))$ or $t_{2}^{-}\in I(w(t_{1}))$ . Since $t_{1}^{-},t_{2}^{-}\in J$ , we have $J\cap I(w(t_{1}))\cap I(w(t_{2}))\neq\varnothing$ . So, by our earlier observation, $w(t_{1})=w(t_{2})$ . This proves the claim.

To prove (6.10), we only need to consider the case of $J$ being unbounded, and $J\cap[T_{0}^{\prime},T]\neq\varnothing$ . Let $t_{1}^{-}=\min(J\cap[T_{0}^{\prime},T])$ and $w_{1}=w(t_{1}^{-})=[t_{1}^{-},t_{1}^{+}]$ . We inductively define $w_{1},\ldots,w_{l(T)}$ as follows: For any $i\geq 1$ such that $w_{i}$ is defined, if $J\cap(t_{i}^{+},T]\neq\varnothing$ , let $t_{i+1}^{-}=\inf(J\cap(t_{i}^{+},T])$ , and $w_{i+1}:=w(t_{i+1}^{-})$ , and $I(w_{i+1})=[t_{i+1}^{-},t_{i+1}^{+}]$ . Otherwise, we let $l(T)=i$ and stop. Thus,

(6.13)

J\cap[T_{0}^{\prime},T]\subseteq I(w_{1})\sqcup...\sqcup I(w_{l(T)}).

Here among $\{w_{1},\ldots,w_{l(T)}\}\subset\pi^{-1}(x).p_{H}$ , each $w_{i}$ appears at most finitely many times, because $\beta(s)\varphi(s)w_{i}\to\infty$ as $s\to\infty$ . Since $\pi^{-1}(x).p_{H}$ is a discrete subset of $V_{d_{H}}$ , and since and $J$ is unbounded, $\lVert w_{l(T)}\rVert\to\infty$ as $T\to\infty$ .

So, by (6.11) and (6.13),

(6.14)

\lvert J\cap[T_{0}^{\prime},T]\rvert\leq\sum_{k=1}^{l(T)}\left|\{t\in I(w_{k}):\beta(t)\varphi(t).w_{k}\in\Psi\}\right|.

Let $1\leq i\leq l(T)-1$ . Then $t_{i}^{+}<t_{l_{T}}^{-}<T$ . So, by the defining property (3) of $I(w_{i})=[t_{i}^{-},t_{i}^{+}]$ , we have $\beta(t_{i}^{+})\varphi(t_{i}^{+})\cdot w_{i}\in\Phi\setminus\mathring{\Phi}$ . Now, by Proposition 3.10 we have that

\left|\{t\in I(w_{k}):\beta(t)\varphi(t).w_{k}\in\Psi\}\right|\leq\epsilon|I(w_{k})|.

Therefore,

(6.15)

\sum_{k=1}^{l(T)-1}\left|\{t\in I(w_{k}):\beta(t)\varphi(t).w_{k}\in\Psi\}\right|\leq\epsilon(T-T_{0}).

Now, it is possible that $\beta(t_{l(T)}^{+})\varphi(t_{l(T)}^{+})\cdot w_{l(T)}\in\mathring{\Phi}$ and $t_{l(T)}^{+}=T$ . So, we will estimate the $l(T)$ -th term differently. By Proposition 1.15, and recalling (1.6), given $\epsilon>0$ we can pick $M_{0}>0$ such for any $w\in V_{d_{H}}$ , if $\lVert\beta(T^{\prime}_{0})\varphi(T^{\prime}_{0}).w\rVert\geq M_{0}$ , then

\left|\{t\in[T_{0}^{\prime},T]:\beta(t)\varphi(t).w\in\Psi\}\right|\leq\epsilon(T-T_{0}^{\prime}).

Since, $\lVert w_{l(T)}\rVert\to\infty$ as $T\to\infty$ , we choose $T_{x}\geq T_{0}^{\prime}$ , such that for all $T\geq T_{x}$ , $\lVert\beta(T^{\prime}_{0})\varphi(T^{\prime}_{0}).w_{l(T)}\rVert\geq M_{0}$ . Then, for any $T\geq T_{x}$ , we have

(6.16)

\left|\{t\in I(w_{l(T)}):\beta(t)\varphi(t).w_{l(T)}\in\Psi\}\right|\leq\epsilon(T-T_{0}^{\prime}).

Thus, by combining (6.14), (6.15) and (6.16), for all $T\geq T_{x}$ we get

\lvert J\cap[T_{0}^{\prime},T]\rvert\leq 2\epsilon(T-T_{0}^{\prime}).

Therefore, (6.10) follows for $3\epsilon$ in place of $\epsilon$ . ∎

6.2. Proving Theorem 1.8

Let $G:=\mathbf{G}(\mathbb{R})$ . Suppose that $\mathcal{G}\leq G$ is a connected, closed Lie subgroup with a lattice $\Gamma\leq\mathcal{G}$ . Suppose that $\varphi:[0,\infty)\to G$ be a continuous, unbounded, non-contracting curve for $G$ definable in a polynomially bounded o-minimal structure. Suppose that $\varphi\left([0,\infty)\right)\subset\mathcal{G}$ . Let $H_{\varphi}$ be the hull of $\varphi$ in $G$ and let $\beta$ be a definable correcting curve in $G$ , so that $\beta\varphi\subset H_{\varphi}$ , see Definition 2.5. Then, by Proposition 5.13, $H_{\varphi}\subset\mathcal{G}$ and $\beta([0,\infty))\subset\mathcal{G}$ . We choose an injective algebraic map $\iota:G\hookrightarrow\text{SL}(n,\mathbb{R})$ for some $n\geq 2$ . Note that $\iota(\Gamma)$ is a discrete subgroup of $\text{SL}(n,\mathbb{R})$ . Importantly, $\iota\circ\varphi$ is non-contracting in $\text{SL}(n,\mathbb{R})$ by Lemma 2.15, and $\iota(H_{\varphi})=H_{\iota\circ\varphi}$ by Lemma 2.8. By abuse of notations, in the following we replace $\varphi$ with $\iota\circ\varphi$ , $\Gamma$ with $\iota(\Gamma)$ and $G$ with $\iota(G)$ .

Let $x_{0}\in\mathcal{G}/\Gamma\hookrightarrow\text{SL}(n,\mathbb{R})/\Gamma$ , and fix a limiting measure $\mu$ of $\mu_{T,\beta\varphi,x_{0}}$ (as in (1.1)), with respect to the weak- $\ast$ topology on the space of probability measures on $\text{SL}(n,\mathbb{R})/\Gamma$ , $T\to\infty$ . We fix a sequence $\{T_{i}\}$ such that $T_{i}\to\infty$ and

(6.17)

\lim_{i\to\infty}\mu_{T_{i},\beta\varphi,x_{0}}=\mu.

By Theorem 4.5, $\mu$ is a probability measure. We let $W$ be the subgroup of $\text{SL}(n,\mathbb{R})$ generated by all unipotent one-parameter subgroups under which the limiting measure $\mu$ is invariant. By Proposition 5.1, $W$ has a positive dimension. Using Theorem 6.2, there exists $H\in\mathcal{H}$ (of smallest dimension) such that

(6.18)

\displaystyle\mu(\pi(N(H,W)))>0\text{ and }\mu(\pi(S(H,W)))=0.

Define

(6.19)

\displaystyle N^{1}(H):=\eta_{H}^{-1}(p_{H})=\{g\in\text{SL}(n,\mathbb{R}):g.p_{H}=p_{H}\}.

and recall

•

[DM93, Theorem 3.4]: $N^{1}(H)\Gamma/\Gamma\subset X$ is closed.

Lemma 6.6.

Fix $H\in\mathcal{H}$ such that (6.18) holds. Then there exists an $g_{0}\in\text{SL}(n,\mathbb{R})$ such that $x_{0}=\pi(g_{0})$ and

(6.20)

g_{0}^{-1}\beta(t)\varphi(t)g_{0}\subset N^{1}(H),~{}\forall t\geq 0,

Importantly,

(6.21)

\nu:=g_{0}^{-1}\mu,

is supported on $\pi(N^{1}(H))$ , $\nu$ is $H$ -invariant, and the largest group generated by unipotent one-parameter subgroups preserving $\nu$ is $H_{u}$ .

Proof.

In view of (6.1), due to (6.18), there is a compact subset $C\subset N(H,W)\smallsetminus S(H,W)$ such that $\mu(\pi(C))=\alpha$ for some $\alpha>0$ , and $\pi(C)\cap\pi(S(H,W))=\varnothing$ . We will now apply Theorem 6.4. We consider the compact subset $\text{K}_{1}\subset A_{H}$ given by $\text{K}_{1}:=C.p_{H}$ and fix $\epsilon:=\frac{\alpha}{2}$ . Let $\mathcal{S}\subset\pi(S(H,W))$ be the closed subset given in Theorem 6.4. Now let $\Psi$ be an arbitrary open neighborhood of $\text{K}_{1}$ , and let $K\subseteq X\smallsetminus\mathcal{S}$ be an arbitrary open neighborhood of $\pi(C)$ , which exists as $\pi(C)$ is compact and $\mathcal{S}$ is closed. Then, $K\cap\pi(\eta_{H}^{-1}(\Psi))$ is an open neighborhood of $\pi(C)$ . Since we assume (6.17), for all $i$ large enough we have

|\{t\in[0,T_{i}]:\beta(t)\varphi(t).x\in K\cap\pi(\eta_{H}^{-1}(\Psi))\}|\geq\epsilon T_{i}.

Namely, condition 6.4,(2) fails, and so the first outcome of Theorem 6.4,(1) must hold. That is, there is a $g_{0}\in\pi^{-1}(x)$ such that $g_{0}.p_{H}\in\Psi$ and

(6.22)

\beta(t)\varphi(t)g_{0}.p_{H}=g_{0}.p_{H},\text{ for }t\geq 0.

Namely, $\beta(t)\varphi(t)g_{0}\in g_{0}N^{1}(H)$ for all $t\geq T_{0}$ . Since $\pi(N^{1}(H))$ is closed, we obtain that $\mu$ is supported in $\pi(g_{0}N^{1}(H))$ .

Since $\pi^{-1}(x).p_{H}$ is discrete, and since we may choose the $\text{K}_{1}$ -neighborhood $\Psi$ arbitrarily, we obtain that $g_{0}.p_{H}\in A_{H}$ . Then, by Theorem 6.3,(1), we conclude that $g_{0}\in N(H,W)$ . Then $g_{0}N^{1}(H)\subseteq N(H,W)$ .

In particular, we conclude that $\mu(\pi(g_{0}N^{1}(H)\smallsetminus S(H,W))=1$ . By [MS95, Lemma 2.4], for each $g_{0}n\in g_{0}N^{1}(H)\smallsetminus S(H,W)$ , the group

H^{\prime}=(g_{0}n)H(g_{0}n)^{-1}=g_{0}Hg_{0}^{-1}

is the smallest group containing $W$ such that the orbit $H^{\prime}\pi(g_{0}n)$ is closed. Then, by [Sha91, Theorem 2.3], $W$ acts ergodically on $H^{\prime}\pi(g_{0}n)$ with respect to $g_{0}\mu_{H}$ , where $\mu_{H}$ is the $H$ invariant measure on $H/H\cap\Gamma\cong H\Gamma/\Gamma$ . Namely, $\mu$ -almost every $W$ -ergodic component of $\mu$ is $g_{0}Hg_{0}^{-1}$ -invariant. By performing ergodic decomposition on $\mu$ , we obtain that $\mu$ is $g_{0}Hg_{0}^{-1}$ -invariant. Then, $\nu=g_{0}^{-1}\mu$ is $H$ -invariant. So $H_{u}$ preserves $\nu$ . Since $g_{0}^{-1}Wg_{0}\leq H$ , we get $H_{u}=g_{0}^{-1}Wg_{0}$ by the maximality of $W$ .

∎

We will use the following notations: $\psi(t):=g_{0}^{-1}\beta(t)\varphi(t)g_{0}$ , for $t\geq 0$ , $N:=N^{1}(H)$ , $\Delta:=N^{1}(H)\cap\Gamma$ , and $q:N\to N/H$ denotes the natural quotient map.

Consider $\bar{N}:=q(N)$ , $\bar{\Delta}:=q(\Delta)$ . Since $H\Delta=\Delta H$ is closed, we have that $\bar{\Delta}\leq\bar{N}$ is discrete (since $\Delta H$ is closed, then $\Delta H/H\cong\Delta/\Delta\cap H$ ). We let $\bar{q}:N/\Delta\to\bar{N}/\bar{\Delta}$ be the natural map. Notice that the fibers of $\bar{q}$ are $H$ -orbits. Namely,

\bar{q}^{-1}(\bar{n}\bar{\Delta}))=nH\Delta/\Delta=Hn\Delta/\Delta.

Since $\mu$ is a probability measure on $N/\Delta$ , it follows that $\bar{\nu}:=\bar{q}_{*}\nu$ is a probability measure on $\bar{N}/\bar{\Delta}$ .

Lemma 6.7.

Suppose that $\psi(t)$ is unbounded in $N/H_{u}$ . Then there exists an unipotent one-parameter subgroup $U\leq N$ such that $U$ is not a subgroup of $H$ and $\bar{\nu}$ is invariant under $q(U)$ and $q(U)$ is a (non-trivial) one-parameter subgroup.

Proof.

We note that the measure push-forward map $q_{*}$ is continuous, and in particular,

\bar{\nu}(f)=\lim_{i\to\infty}\frac{1}{T_{i}}\int_{0}^{T_{i}}f(q(\psi(t))\bar{\Delta})dt,\forall f\in C_{c}(\bar{N}/\bar{\Delta}).

Thus, we may conclude the statement by applying Proposition 5.12, and Lemma 5.5 (see the proof of Lemma 5.6. The only difference here is that $N/H$ is not a subgroup of $\text{GL}(n,\mathbb{R})$ ). ∎

We now note the following, which is obtained by the ergodic decomposition for the $H$ -action on $N/\Delta$ (see [EW11, Theorem 8.20])

Lemma 6.8.

Let $d(h\Delta)$ denote the volume element of the $H$ -invariant probability measure on $H\Delta/\Delta$ . For $f\in C_{c}(N/\Delta)$ let:

(6.23)

P(f)(q(n)\bar{\Delta}):=\int_{H\Delta/\Delta}f(nh\Delta)d(h\Delta).

Then,

\displaystyle\int_{N/\Delta}f(n\Delta)d\nu(n\Delta)=\int_{\bar{N}/\bar{\Delta}}P(f)(q(n)\bar{\Delta})d\bar{\nu}(q(n)\bar{\Delta}).

We now have the following key observation.

Corollary 6.9.

It holds that $\psi(t)\in H_{u}$ for all $t\geq T_{0}$ , and $\nu$ is the $H$ -invariant probability on $H\pi(e)$ , where $\pi(e)$ is the identity coset in $X=\text{SL}(n,\mathbb{R})/\Gamma$ . Moreover, $H$ is the smallest subgroup such that $g_{0}^{-1}H_{\varphi}g_{0}\leq H$ and such that the orbit $H\pi(e)$ is closed.

Proof.

We first show that $\psi(t)$ is bounded in $N/H_{u}$ . In fact, if $\psi(t)$ is unbounded in $N/H_{u}$ , then by Lemma 6.7, $\bar{\nu}$ is invariant under $q(U)$ , where $U$ is a one-parameter unipotent subgroup and $q(U)$ is a non-trivial one-parameter subgroup of $\bar{N}=N/H$ . It then follows from Lemma 6.8 than $\nu$ is invariant by $U$ and $U$ is not a subgroup of $H$ . Since $\nu$ was invariant by $H$ as well, we get that $\nu$ is invariant by a group generated by one-parameter subgroups which strictly includes $H_{u}$ . This is a contradiction since $H_{u}$ is a maximal group with this property.

Since the image of $\psi(t)=g_{0}^{-1}\beta(t)\varphi(t)g_{0}$ is boundned in $\text{SL}(n,\mathbb{R})/H_{u}$ , we get that the image of $\beta(t)\varphi(t)$ is bounded in $\text{SL}(n,\mathbb{R})/g_{0}H_{u}g_{0}^{-1}$ . Thus, since $\text{zcl}(H_{u})$ is observable (see [Gro97, Corollary 2.8]), we conclude by Theorem 2.4 that $H_{\varphi}\leq g_{0}H_{u}g_{0}^{-1}$ , and in particular $\beta(t)\varphi(t)\in g_{0}H_{u}g_{0}^{-1}$ for all $t$ . Namely, $\psi(t)\in g_{0}^{-1}H_{\varphi}g_{0}\leq H_{u}$ for all $t\geq T_{0}$ . In particular, $\bar{\nu}$ is the Dirac measure at the identity coset, and so $\nu=\mu_{H}$ by Lemma 6.8. In particular $\{\psi(t)\pi(e)\}_{t\geq 0}$ is dense in $H\pi(e)$ . In particular $g_{0}^{-1}H_{\varphi}g_{0}\pi(e)$ is dense in $H\pi(e)$ , which finishes the proof.

∎

The proof of the main Theorem 1.8 follows directly from the above corollary.

7. Curves with a largest possible hull

The following result gives a condition for the hull to be semi-simple.

Lemma 7.1.

Suppose that $\varphi:[0,\infty)\to\text{SL}(n,\mathbb{R})$ is a continuous curve definable in an o-minimal structure such that $\varphi$ has the following divergence property:

(7.1)

\lim_{t\to\infty}\varphi(t).(v_{1}\wedge\dots\wedge v_{k})=\infty,

for all $1\leq k\leq n-1$ and all linearly independent vectors $v_{1},...,v_{k}\in\mathbb{R}^{n}$ . Then, the observable hull $H_{\varphi}$ of $\varphi$ is a connected semi-simple with no compact factors, and the action of $H_{\varphi}$ on $\mathbb{R}^{n}$ is irreducible.

Proof.

Since $\text{zcl}(H_{\varphi})$ is observable, we have a finite dimensional rational representation $V$ of $\text{SL}(n,\mathbb{R})$ and a $v\in V$ such that $\text{zcl}(H_{\varphi})$ is the isotropy group of $v$ . Note that $\varphi(t).v,t\geq 0$ is bounded since $\varphi$ is bounded in $\text{SL}(n,\mathbb{R})/H_{\varphi}$ . Since we assume (7.1), it follows by Lemma 2.1 that the orbit $\text{SL}(n,\mathbb{R}).v$ is Zariski closed. Then, Matsushima criterion [Mat60] implies that $\text{zcl}(H_{\varphi})$ is reductive. Since $H_{\varphi}$ is connected and generated by one-parameter unipotent subgroups, it follows that $H_{\varphi}$ is a connected semi-simple group with no compact factors. Finally, we prove that $H_{\varphi}$ acts on $\mathbb{R}^{n}$ irreducibly. Suppose for contradiction that $\mathbb{R}^{n}=V_{1}\oplus V_{2}$ where both $V_{1}$ and $V_{2}$ are non-trivial sub-spaces invariant under $H_{\varphi}$ . Let $v_{1},...,v_{k}$ by a basis for $V_{1}$ . Here $1\leq k<n$ because $V_{2}$ is not trivial. Since semi-simple groups have no non-trivial characters, we obtain that $H_{\varphi}.(v_{1}\wedge\cdots\wedge v_{k})=v_{1}\wedge\cdots\wedge v_{k}$ . Since $\varphi([0,\infty))$ is bounded in $\text{SL}(n,\mathbb{R})/H_{\varphi}$ , we get $\varphi(t).(v_{1}\wedge\cdots\wedge v_{k})$ is bounded as $t\to\infty$ . This contradicts (7.1). ∎

Let $\varphi$ be the curve given by (1.5). First, we will verify that $\varphi$ satisfies the following stronger divergence property.

Lemma 7.2.

For any $1\leq k\leq n$ , and $0\neq v\in\wedge^{k}\mathbb{R}^{n+1}$ , $\lim_{t\to\infty}\varphi(t).v\to\infty$ as $t\to\infty$ .

Proof.

We denote by $e_{0},e_{1},...,e_{n}$ the canonical basis vectors of $\mathbb{R}^{n+1}$ , where $e_{i}$ denotes the vectors whose coordinates equal $0$ besides the $(i+1)$ th position in which $1$ appears. Fix an integer $1\leq k\leq n$ and let $\mathcal{I}$ be the collection of all subsets of $\{0,\ldots,n\}$ of cardinality $k$ . Let $\varphi$ be the curve given by (1.5). For any $I:=\{i_{1},\ldots,i_{k}\}\in\mathcal{I}$ , where $i_{1}<i_{2}<\cdots<i_{k}$ , let $e_{I}=e_{i_{1}}\wedge\cdots\wedge e_{i_{k}}$ . There are two cases for $\varphi(t).e_{I}$ :

If $i_{1}=0$ , then

	$\displaystyle\varphi(t).e_{I}=$	$\displaystyle f_{0}(t)e_{0}\wedge[f_{i_{2}}(t)e_{0}+h_{i_{2}}(t)^{-1}e_{i_{2}}]\wedge\cdots\wedge[f_{i_{k}}(t)e_{0}+h_{i_{k}}(t)^{-1}e_{i_{k}}]$
(7.2)		$\displaystyle=$	$\displaystyle f_{0}(t)\left(\prod_{l=2}^{k}h_{i_{l}}(t)^{-1}\right)e_{I};$

and if $i_{1}\geq 1$ , then

	$\displaystyle\varphi(t).e_{I}=$	$\displaystyle[f_{i_{1}}(t)e_{0}+h_{i_{1}}(t)^{-1}e_{i_{1}}]\wedge\cdots\wedge[f_{i_{k}}(t)e_{0}+h_{i_{k}}(t)^{-1}e_{i_{k}}]$
	$\displaystyle=$	$\displaystyle\sum_{l=1}^{k}(-1)^{l-1}f_{i_{l}}(t)\left(\prod_{j\in\{1,\ldots,k\}\setminus\{l\}}h_{i_{j}}(t)^{-1}\right)e_{\{0\}\cup I\setminus\{i_{l}\}}$
(7.3)			$\displaystyle+\prod_{j\in\{1,\ldots,k\}}h_{i_{j}}(t)^{-1}e_{I}.$

Let $\mathbf{v}\in\wedge^{k}\mathbb{R}^{n+1}\setminus\{0\}$ . Then $\mathbf{v}=\sum_{I\in\mathcal{I}}c_{I}e_{I}$ , where $c_{I}\in\mathbb{R}$ . Then,

(7.4)

\varphi(t).\mathbf{v}=\sum_{I\in\mathcal{I}}C_{I}(t)e_{I},

where $C_{I}(t)\in\mathbb{R}$ . The coefficients of our interest are $C_{I}(t)$ such that $0\in I$ . For any $1\leq i_{2}<\cdots<i_{k}\leq n$ , and $J=\{i_{2},\ldots,i_{k}\}$ ,

(7.5)

\displaystyle C_{\{0\}\cup J}(t)=\left(\prod_{j\in J}h_{j}(t)^{-1}\right)\sum_{l\in\{0,1,\ldots,k\}\setminus J}f_{l}(t)c_{J\cup\{l\}}(-1)^{j_{l}},

where $j_{l}=\min\{j\in\{2,\ldots,k\}:i_{j}>l\}$ .

Let $\mathcal{I}_{\mathbf{v}}=\{I\in\mathcal{I}:c_{I}\neq 0\}\neq\varnothing$ , as $\mathbf{v}\neq 0$ . Let $i_{1}=\min\cup\mathcal{I}_{\mathbf{v}}$ . We pick $I\subset\mathcal{I}_{\mathbf{v}}$ such that $i_{i}\in I$ . Let $J=I\setminus\{i_{1}\}$ . Then, $c_{J\cup\{l\}}=0$ for every $l<i_{1}$ . Therefore, by (7.5)

(7.6)

C_{\{0\}\cup J}(t)=\left(\prod_{j\in J}h_{j}(t)\right)^{-1}\left(f_{i_{1}}c_{I}+\sum_{l\in\{i_{1}+1,\ldots,k\}\setminus J}f_{l}(t)c_{J\cup\{l\}}(-1)^{j_{l}}\right).

By condition (1), $\sum_{i=1}^{n}\deg h_{i}=n$ and $\deg h_{1}\geq\cdots\geq\deg h_{n}$ . Therefore

(7.7)

\sum_{i=1}^{j}\deg h_{i}\geq j

for each $1\leq j\leq n$ . Since $1\leq k\leq n$ and $\deg h_{n}>0$ , we get

(7.8)

\deg h_{i_{2}}+\cdots+\deg h_{i_{k}}\leq\deg h_{1}+\cdots+\deg h_{k-1}\leq n-\deg h_{n}<n,

and if $i_{1}\geq 1$ , then by (7.7),

(7.9)

\deg h_{i_{2}}+\cdots+\deg h_{i_{k}}\leq n-(\deg h_{1}+\cdots+\deg h_{i_{1}})\leq n-i_{1}.

Since $c_{I}\neq 0$ , by our assumptions (2) and (3),

(7.10)

\deg\left(c_{I}f_{i_{1}}+\sum_{l\in\{i_{1}+1,\ldots,k\}\setminus J}(-1)^{j_{l}}c_{J\cup\{l\}}f_{l}(t)\right)\begin{array}[]{ll}\geq n&\text{if $i_{1}=0$}\\ >n-{i_{1}}&\text{if $i_{1}\geq 1$}.\end{array}

By combining (7.6), (7.8), (7.9) and (7.10), we conclude that $\deg C_{\{0\}\cup J}(t)>0$ . Hence, the conclusion of the lemma follows from (7.4). ∎

Lemma 7.3.

Let $\{\rho(s):s\in\mathbb{R}\}$ be the P.S. group of $\varphi$ (Definition 5.3). Then, there exists $v=(v_{1},\ldots,v_{n})\in\mathbb{R}^{n}\setminus\{0\}$ , such that

(7.11)

\rho(s):=\begin{bmatrix}1&sv\\ 0&1\end{bmatrix}\in\text{SL}(n+1,\mathbb{R}),\,\forall s\in\mathbb{R}.

Proof.

Due to Lemma 7.2, by Lemma 5.9, $\rho(s)$ is unipotent. Moreover, there exists $r<1$ such that for any $s\in\mathbb{R}$ ,

(7.12)

\varphi(h_{r,s}(t))\varphi(t)^{-1}\to\rho(s),

where $h_{r,s}$ is as in (5.1). We note that $\varphi(s)$ is contained in the subgroup $F$ of $\text{SL}(n+1,\mathbb{R})$ whose nonzero entries are only on the diagonal and in the top row. Therefore, by (7.12), $\rho(s)$ is a one-parameter unipotent subgroup of $F$ . Hence, $\rho(s)$ is of the form given by (7.11). ∎

Proof of Proposition 1.10.

Let $\{e_{0},\ldots,e_{n}\}$ denote the standard basis of $\mathbb{R}^{n+1}$ . Let $(v_{1},\ldots,v_{n})\in\mathbb{R}^{n}$ be as in Lemma lem:ps group of our main example. Let $W_{1}$ denote the ortho-complement of $v=\sum_{i=1}^{n}v_{i}e_{i}$ in the span of $\{e_{1},\ldots,e_{n}\}$ . Then the fixed points space of $\rho(s)$ in $\mathbb{R}^{n+1}$ is $W:=\mathbb{R}e_{0}+W_{1}$ .

Now let $\beta$ denote the correcting definable curve such that $\beta(t)\varphi(t)\subset H_{\varphi}$ . Since $\beta(t)\to\beta(\infty)\in\text{SL}(n+1,\mathbb{R})$ as $t\to\infty$ , by (7.12), we conclude that $\rho_{1}(s):=\beta(\infty)\rho(s)\beta(\infty)^{-1}\in H_{\varphi}$ for all $s\in\mathbb{R}$ . By Lemma 7.2 and Lemma 7.1, $H_{\varphi}$ is a semisimple group acting irreducibly on $\mathbb{R}^{n+1}$ . Therefore, by Jacobson-Morosov theorem, there exists a homomorphism $\eta:\text{SL}(2,\mathbb{R})\to H_{\varphi}$ such that $\eta(u(s))=\rho_{1}(s)$ , where $u(s):=\begin{bmatrix}1&s\\ 0&1\end{bmatrix}\in\text{SL}(2,\mathbb{R})$ . As noted earlier, the fixed points space of $\rho_{1}$ on $\mathbb{R}^{n+1}$ is $\beta(\infty)W$ , and it has dimension $n$ . Expressing the representation of $\text{SL}(2,\mathbb{R})$ on $\mathbb{R}^{n+1}$ via $\eta$ as a direct sum of irreducible representations of $\text{SL}_{2}(\mathbb{R})$ , and noting that the dimension of the $u(s)$ -fixed subspace is $n$ , we conclude that $\mathbb{R}^{n+1}=V_{1}\oplus V_{2}$ , where $V_{1}$ is isomorphic to the standard representation of $\text{SL}(2,\mathbb{R})$ on $\mathbb{R}^{2}$ , and $\text{SL}(2,\mathbb{R})$ acts trivially on $V_{2}$ . Therefore, by [SY24, Theorem A.7], either $H_{\varphi}=\text{SL}(n+1,\mathbb{R})$ or $n+1=2d$ for some $d\geq 2$ and $H_{\varphi}$ is conjugate to the standard symplectic group $\text{SP}(\mathbb{R}^{2d})$ .

Consider the case when $H_{\varphi}\neq\text{SL}(n+1,\mathbb{R})$ . Then, there exists a basis $w_{1},w^{\prime}_{1},w_{2},w^{\prime}_{2},\ldots,w_{d},w^{\prime}_{d}$ of $\mathbb{R}^{2d}$ such that $H_{\varphi}$ stabilizes

\mathbf{v}=w_{1}\wedge w^{\prime}_{1}+w_{2}\wedge w^{\prime}_{2}+\cdots+w_{n}\wedge w^{\prime}_{n}\in\wedge^{2}\mathbb{R}^{2d}.

Now, by Lemma 7.2, $\varphi(t)$ cannot be bounded in $\text{SL}(2d,\mathbb{R})/H_{\varphi}$ , contradictions the assumption that $H_{\varphi}$ is the hull of $\varphi$ . Therefore, $H_{\varphi}=\text{SL}(n+1,\mathbb{R})$ . This completes the proof of Proposition 1.10. ∎

Proof of Corollary 1.11.

If $G$ is a linear Lie group, the result follows immediately from Proposition 1.10 and Theorem 1.8. If $G$ is not a linear Lie group, for our proof to work, it is sufficient that after the composition with the Adjoint representation of $G$ on its Lie algebra $\mathfrak{g}$ , the curve is definable and non-contracting in $\text{GL}(\mathfrak{g})$ . This property is valid in our situation. ∎

Appendix A Kempf’s Lemma for real algebraic groups

We note that [Kem78, Lemma 1.1(b)] is stated for reductive algebraic groups over an arbitrary field. We need the same result for any algebraic group defined over $\mathbb{R}$ . Here, we verify that the proof given by Kempf is valid for real algebraic groups $G$ using the most naive language.

For finite-dimensional real vector spaces $V$ and $W$ , we say that $P:V\to W$ is a polynomial map if, after choosing bases for $V$ and $W$ , the map $P$ in those coordinates is a multi-variable polynomial in each coordinate. We say that the map is $G$ -equvairant if $g.P(x)=P(g.x)$ for all $g\in G$ and $x\in V$ .

Lemma A.1 (Kempf).

Let $G$ be the set of $\mathbb{R}$ -points of an algebraic group defined over $\mathbb{R}$ . Let $V$ be a rational representation of $G$ . Suppose that for $v\in V$ , it holds that $S:=\text{zcl}(G.v)\smallsetminus G.v$ is non-empty. Then there exists a rational representation $W$ and an equivariant polynomial map $P:V\to W$ such that $P(S)=\{0\}$ and $P(v)\neq 0$ .

Proof.

First, note that since $\text{zcl}(G.v)$ and $G.v$ are $G$ -invariant, we get that $S$ is $G$ -invariant. Let $\mathbb{R}[V]$ be the coordinate ring on the vector space $V$ and consider $I\subseteq\mathbb{R}[V]$ the polynomial ideal of the polynomials vanishing on $S$ . For $d\in\mathbb{N}$ , let $I(d)\subseteq I$ be the subset of polynomials in $I$ with degree bounded by $d$ . We may choose $d$ such that the ideal generated by $I(d)$ is $I$ . Notice that $I(d)$ is a finite dimensional vector space. Consider the right action of $G$ on $\mathbb{R}[V]$ defined by

(A.1)

(f.g)(x):=f(g.x),~{}g\in G,~{}x\in V,~{}f\in\mathbb{R}[V].

The action preserves the degree of the polynomial. So, it preserves $I(d)$ because $S$ is $G$ -invariant. Let $I(d)^{*}$ be the space of linear functionals on $I(d)$ . Using the representation (A.1), we get a left action on $I(d)^{*}$ defined by

(A.2)

g.l(f):=l(f.g),~{}g\in G,~{}l\in I(d)^{*},~{}f\in I(d).

Now consider the map $P:V\to I(d)^{*}$ defined by

(A.3)

P(x):=\text{ev}_{x},

where $\text{ev}_{x}$ is the functional mapping $f\in I(d)$ to $f(x)$ . It is straightforward to verify that $P$ is an equivariant polynomial map. Finally, we claim that $P(v)=\text{ev}_{v}$ is a non-zero functional. By the definition of $S$ and $P$ , $P(x)=0$ for all $x\in S$ . It remains to show that $P(v)\neq 0$ . For that, recall the closed orbit lemma [Bor91, Section 1.8], which states that $S$ is Zariski closed. This implies that there exists a $p\in I(d)$ such that $p(S)=0$ and $p(v)\neq 0$ . Since $P(v)(p)=p(v)$ , $P(v)\neq 0$ . ∎

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	$\displaystyle 1$	$\displaystyle=\\|(w_{1}^{-},...,w_{n}^{-})\\|_{1}$
		$\displaystyle=\lim_{s\to\infty}\frac{\\|(c_{1}^{-}(s),...,c_{n}^{-}(s))\\|_{1}}{\\|(c^{-}_{1}(s),...,c^{-}_{n}(s),c^{+}_{0}(s)l(s)^{r_{0}},...,c^{+}_{m}(s)l(s)^{r_{m}})\\|_{1}}$
		$\displaystyle\leq\lim_{s\to\infty}\frac{\\|(c_{1}^{-}(s),...,c_{n}^{-}(s))\\|_{1}}{\\|(c^{-}_{1}(s),...,c^{-}_{n}(s),c^{+}_{0}(s),...,c^{+}_{m}(s))\\|_{1}}\text{, as $l(s)\to\infty$ and $r_{i}\geq 0$}$
		$\displaystyle=\\|(v_{1}^{-},...,v_{n}^{-})\\|_{1}\leq 1.$

	$\displaystyle\left\|\{t\in[x,y]:\|Q(\psi(t)v)\|\leq\eta\}\right\|$	$\displaystyle=\left\|\{t\in[x,y]:\|t^{\nu}Q(\psi(t)v)\|\leq t^{\nu}\eta\}\right\|$
		$\displaystyle\leq\left\|\{t\in[x,y]:\|t^{\nu}Q(\psi(t)v)\|\leq y^{\nu}\eta\}\right\|$
(3.31)			$\displaystyle\leq C\left(\frac{y^{\nu}\eta}{y^{\nu}\delta}\right)^{\alpha}(y-x)=C\left(\frac{\eta}{\delta}\right)^{\alpha}(y-x).$

		$\displaystyle\frac{1}{T}\int_{0}^{T}\|f(\rho(s)\varphi(t)x)-f(\varphi(t)x)\|dt\leq\frac{1}{T}\int_{0}^{T_{\epsilon}}\|f(\rho(s)\varphi(t)x)-f(\varphi(t)x)\|dt$
	$\displaystyle+$	$\displaystyle\frac{1}{T}\int_{T_{\epsilon}}^{T}\|f(\rho(s)\varphi(t)x)-f(\varphi(t)x)\|dt\leq\frac{1}{T}\int_{0}^{T_{\epsilon}}\|f(\rho(s)\varphi(t)x)-f(\varphi(t)x)\|dt$
	$\displaystyle+$	$\displaystyle\frac{1}{T}\int_{T_{\epsilon}}^{T}\|f(\varphi(h_{r,s}(t)x)-f(\varphi(t)x)\|dt+\frac{\epsilon}{2}.$

Equidistribution of polynomially bounded o-minimal curves in homogeneous spaces

Abstract.

1. Introduction

1.1. O-minimal curves

Definition 1.1.

Remark 1.2.

1.2. o-minimal curves in affine algebraic groups

Definition 1.3.

1.2.1. The non-contraction property

Definition 1.4.

Remark 1.5.

Proposition 1.6.

Proposition 1.7.

1.3. The main equidistribution result

Theorem 1.8.

Remark 1.9.

Proposition 1.10.

Corollary 1.11.

1.4. Growth of o-minimal functions

Definition 1.12.

Example 1.13.

Theorem 1.14.

Proposition 1.15.

1.5. Proof ideas and structure of the paper

1.6. Some conventions about notation

Acknowledgement.

2. Consequences of the non-contracting property

2.1. The criterion for the non-contracting property

Proof of Proposition 1.6.

Lemma 2.1.

Proof.

2.2. On the hull and the correcting curve

Definition 2.2.

Definition 2.3.

Theorem 2.4.

Definition 2.5.

Remark 2.6.

Remark 2.7.

Lemma 2.8.

Proof.

2.3. Proving Theorem 2.4

Lemma 2.9.

Proof.

Lemma 2.10.

Proof.

Lemma 2.11.

Proof.

Corollary 2.12.

Proof.

Corollary 2.13.

Proof.

Lemma 2.14.

Proof.

2.4. The non-contraction property under homomorphisms

Lemma 2.15.

Proof.

Lemma 2.16.

Proof.

Lemma 2.17.

Proof.

Lemma 2.18.

Proof.

Lemma 2.19.

Proof.

Proposition 2.20.

Proof.

3. (C,α)(C,\alpha)-good property

Definition 3.1.

Theorem 3.2.

3.1. Proving Theorem 1.14 via δ\delta-goodness

Corollary 3.3.

Proof.

Proof of Theorem 1.14.

3.2. Proving Theorem 3.2

Proposition 3.4 ([DM93, Lemma 4.1]).

Proof.

Lemma 3.5.

Proof.

Lemma 3.6.

Proof.

3. $(C,\alpha)$ -good property

3.1. Proving Theorem 1.14 via $\delta$ -goodness