Cocycle Stability of Linear Automorphisms and their Jointly Integrable Perturbations

For general partially hyperbolic systems $E^{s}$ and $E^{u}$ are integrable into invariant foliations $\mathcal{W}^{ss}$ and $\mathcal{W}^{uu}$. In our case, where $f$ is a perturbation of a linear map, also $E^{c}$, $E^{s}\oplus E^{c}$, and $E^{c}\oplus E^{u}$ are integrable into invariant foliations $\mathcal{W}^{c}$, $\mathcal{W}^{cs}$, and $\mathcal{W}^{cu}$ [HPS70].

Of course linear maps $F\in SL(d,\mathbb{Z})$ (thought as maps of the torus $\mathbb{T}^{d}$) are jointly integrable as the invariant splitting should be a splitting $\mathbb{R}^{d}=S\oplus C\oplus U$ into vector subspaces.

In our theorems we ask for hypothesis on $F$ with respect to an implicitly given partial hyperbolic splitting. Whenever we ask for $f$ to be jointly integrable, we mean with respect to the splitting that comes from perturbing the given splitting.

Asking for stronger conditions (called bunching conditions) in the domination of center it is known [Wil13] that relevant holonomy maps have certain amount of regularity.

Note that the bounds for partial hyperbolicity and for bunching are all open in the $C^{1}$ topology, so if $f$ is close enough to a linear map $F$ satisfying this conditions so will $f$.

On the other hand, for a linear map $F$ is easy to show that the partial hyperbolicity condition and the bunching conditions are equivalent to analogous conditions when replacing norms and conorms with the biggest and smallest (absolute value of) eigenvalue of the respective subspace. In this case if $F|_{C}$ only has eigenvalues of modulus one then it is $R$-bunched and strongly $R$-bunched for any $R$.

In this notes partial hyperbolicity and jointly integrability are always assumed.

Solving ($\star$ ‣ 1) without any hypothesis on $\varphi$ usually requires a certain Diophantine property. In our case this property is hidden behind an auxiliary construction called the central translations.

Is easy to see that in our case (jointly integrable perturbations of linear maps) every pair of $su$-leaf and central leaf intersect exactly once in the universal cover. Fix a “favorite” central leaf $\mathcal{C}=\mathcal{W}^{c}(0)$ of a fix point $0$.

We define an action $T_{n}:\mathcal{C}\to\mathcal{C}$ of $\mathbb{Z}^{d}$ in $\mathcal{C}$ by

where, by abuse of notation, we are thinking of $\mathcal{C},\mathcal{W}^{su}(x+n)\subset\mathbb{R}^{d}$ as living in the universal cover.

We refer to these as central translations. In § 3.2 we will show that these are, in a certain sense, Diophantine and so the perturbations can be linearized (§ 3.3) and we will be able to use this Diophantine property to solve the cohomological equation along the center in § 4.1.

Since the intention is to linearize these translations we start here by projecting them to a vector space

where $C\subset\mathbb{R}^{d}$ is the central subspace for the nearby linear map $F$ and $\Pi:\mathcal{C}\to C$ is the projection to this subspace along the $su$-leaves of $f$. See [Rod05, Proposition B.1] to see that $\Pi$ is indeed well defined and invertible.

This $\Pi$ shouldn’t be confused with $\pi^{c}:\mathbb{R}^{d}\to\mathcal{C}$ the projection to our favorite central leaf for $f$

Notice that $\pi^{c}$ and $\Pi$ are both “reasonable” holonomy maps, and that $T_{n}(x)=\pi^{c}(x+n)$. So according to [PSW97] we have

We describe what uniformly $C^{R}$ means right after lemma 2.7.

Let us define the holonomy functionals. Consider $x$ a point and $x^{+}\in\mathcal{W}^{ss}(x)$, and $x^{-}\in\mathcal{W}^{uu}(x)$. In each case we define the functional $\operatorname{Hol}$ in the following ways

and if $\gamma=(x_{0},x_{1},\dots,x_{k})$ is a $su$-sequence, by that we mean $x_{j}\in\mathcal{W}^{ss}(x_{j-1})$ or $x_{j}\in\mathcal{W}^{uu}(x_{j-1})$, we extend the functional

In the case where $\gamma$ is periodic, that is $x_{0}=x_{k}$, this is the periodic cycle functional $PCF_{\gamma}\varphi=\operatorname{Hol}_{\gamma}\varphi$.

The condition $PCF_{\gamma}\varphi=0$ for all $su$-sequence $\gamma$ is a natural necessary condition for solving the cohomological equation ($\star$ ‣ 1). In such case we say that $\varphi$ has trivial periodic cycle functionals, and in this notes we will always assume that $\varphi$ satisfies this condition.

So, under this hypothesis, if $y\in\mathcal{W}^{su}(x)$ (recall that in this notes we consider $f$ being jointly integrable) we write

where $\gamma=(x_{0}=x,x_{1},\dots,x_{k}=y)$ is a $su$-sequence going from $x$ to $y$ (the trivial periodic cycle hypothesis makes this well defined, regardless of the choice of $\gamma$).

Formally $-\sum\varphi(f^{n}x)$ and $\sum\varphi(f^{-n}x)$ solve ($\star$ ‣ 1), so, naively, we can think of “$\operatorname{Hol}_{x}^{y}\varphi=u(y)-u(x)$”. This, in a sense, shows that this functional solves the equation along $\mathcal{W}^{su}$.

Below we establish some trivial properties of these functionals to reference in the future (one of which formalizes the idea of the previous paragraph).

We establish too the regularity that we will need for these functionals.

By uniformly $C^{R}$ we mean that all partial derivatives (and the Hölder constants for the highest order derivatives if $R$ is not an integer) are bounded both along $\mathcal{W}^{\sigma}(x_{0})$ and as we vary $x_{0}\in\mathbb{T}^{d}$.

We remark that this definition depends on the choice of the splitting, and in many situations the optimal splitting would be when $C$ is the generalized eigenspace of eigenvalues of modulus $1$. In the latter case Katznelson irreducibility is just ergodicity (see below).

Notice that by partial hyperbolicity $F|_{C}$ and $F_{S\oplus U}$ have no common eigenvalues, so this condition is enough to apply Katznelson’s lemma [Kat71, Lemma 3].

To compare with more algebraic definition of irreducibility we give the following equivalence:

From this lemma the following implications are obvious for $F\in SL(d,\mathbb{R})$

and none of the reverse implications are true, except in the case where $F|_{C}$ has only eigenvalues of modulus $1$, in which case

since a rational polynomial can have only roots of modulus $1$ if those roots are roots of unity.

We show now that this condition is actually necessary to solve the cohomological equation under our hypotheses.

For $v\in\mathbb{R}^{d}=S\oplus C\oplus U$ we call $v^{c}\in C$ its projection along $S\oplus U$. If $\dim C=c$ we have, after perhaps rearranging the basis, that $e_{1}^{c},\dots,e^{c}_{c}$ is a basis of $C$.

Using this basis to identify $C$ with $\mathbb{R}^{c}$ we can write the projection to $C$ as a linear map $P:\mathbb{R}^{d}\to\mathbb{R}^{c}$, with $P(e_{i})=e_{i}$ for $i=1,\dots,c$. With this notations we can write (in the linear case) the central translations defined in § 2.2 as $T_{n}a=a+Pn$, and we can show that they satisfy the following Diophantine condition:

After showing this we will show, using Moser in § 3.3, that the general central translations (of a map $f$ that is a jointly integrable perturbation of a linear map $F$ instead of a linear map itself) is conjugated to the Diophantine translations of the nearby linear map.

During the proof we will actually use Katznelson’s lemma for $F^{t}$.

This is essentially already done in [Rod05], we sketch the construction below and explain the regularities.

Recall than in § 2.2 we projected $T_{n}$ to maps $\tilde{T}_{n}:C\to C$ of a vector space. We also make the identification $C=\mathbb{R}^{c}$ exactly like we did in § 3.2.

By means of bump functions we can construct $\tilde{h}:\mathbb{R}^{c}\rightarrow\mathbb{R}^{c}$ so that

for $\ell\in\mathbb{Z}^{c}\times\{0\}\subset\mathbb{Z}^{d}$, where $R_{\alpha}=x\mapsto x+\alpha$, and for a general $n\in\mathbb{Z}^{d}$

(the proximity to the respective translation for the linear map comes from [Rod05, Corollary 2.4]).

To improve this $\tilde{h}$ into a simultaneous linearization we follow Moser [Mos90]. Moser shows how to simultaneously linearize commuting perturbations of rotations of the circle with a Diophantine condition analogous to ours by using some KAM type of arguments.

Essentially all the difficulties of generalizing this to general dimension are already taken care by F. Rodriguez-Hertz (the only minor detail is that in the proof of [Rod05, Lemma B.3] it is used the fact that $F|_{C}$ is an isometry, but actually the argument works just as well with the hypothesis of partial hyperbolicity).

In here we will follow the constants in Moser to see how much they can be optimized. Clearly in our case $\tau=c$. In lemma 3.1 from Moser there is a convergence that is over $\mathbb{Z}^{c}$ in our case, making it so that we need $\sigma>2\tau+c=3c$.

In page 119 we leave, for now, the variables $\kappa$ and $\ell$ free, and set $N_{s}=\varepsilon_{s}^{-\xi}$ with $\xi$ free. The conditions needed to make the rest of the page work would be

For the next page we leave the variable $m$ free and we need the argument to work for $k$ up to $\hat{k}$, so the condition for the argument to work would be

For the argument to work we need existence of derivatives up to order $\ell+\sigma$ and $m+\sigma$ and to control derivatives up order $\ell$. So we need to optimize those numbers.

Taking $\kappa$ close enough to $1$ and $\xi$ close enough to $\frac{\kappa-1}{\sigma}$ the conditions are satisfied for

Since $\sigma$, $\ell$ and $m$ need to be integers we can solve the problem with $\sigma=3c+1$, $\ell=6c+3$ and $m=3\hat{k}+1$.

So far we show that, in a sense, the central translations are Diophantine. We now show that Diophantine translations have no restrictions to solve certain cohomological equations.

We remark that the condition $P\ell=\ell$ can be replaced by $\Phi$ being defined on the torus $\mathbb{T}^{c}$.

Here is a small lemma that we will need later: