Rigidity of pressures of Hölder potentials and the fitting of analytic functions via them

This work deals with traditional topics in thermodynamic formalism [Bow, Rue1], which originates from theoretical physics. We focus on shift spaces of finite type here, which model dynamics of some smooth systems such as Axiom-A Diffeomorphisms through Markov partitions. Given a symbolic set $\Lambda$ of finite symbols and a continuous potential (observable) $\phi$ on the shift space $\Lambda^{\mathbb{N}}$, a core concept in thermodynamic formalism is the pressure $P(\phi)$. People are particularly interested in the pressure function $P(t\phi)$ with the variable $t>0$ representing the inverse temperature. A sharp change in the pressure function (or other terms) is usually called a phase transition as $t$ varies, see for example [IRV, IT1, IT2, KQW, Lop1, Lop2, Lop3, Sar].

For Hölder continuous potentials, Ruelle [Rue2] proved that the pressure function $P(t\phi)$ is analytic for $t\in(0,\infty)$ (in fact he proved that $P(\psi)$ depends analytically on $\psi$ for $\psi$ in the Hölder space $C^{0,h}(X)$ with $X$ being a transitive subshift space of finite type and $0<h\leq 1$ being the exponent [GT]). A key ingredient in his proof is the use of Ruelle (transfer) operator [BDL, GLP] acting on functions in the Hölder space. Moreover, the equilibrium measure of $t\phi$ for any $t>0$ and Hölder potential $\phi$ is always unique, so there are in fact no phase transitions in this case. Let

$P^{(n)}(t)=P^{(n)}(t\phi)=\cfrac{d^{n}P(t\phi)}{dt^{n}}$

be the $n$-th differential of the pressure function $P(t\phi)$ with respect to $t\in(0,\infty)$ for some fixed Hölder potential $\phi$. We also write

$P^{(1)}(t)=P^{\prime}(t),P^{(2)}(t)=P^{\prime\prime}(t),P^{(3)}(t)=P^{\prime\prime\prime}(t),\cdots$

intermittently in the following. We discover that there is some rigid relationship between the differentials of the pressure function.

A potential $\phi$ is said to be generic (or we say it defines a non-lattice distribution, cf. [CP, Fel, PP]), if for any normalised potential $\psi$, the spectral radius of the complex Ruelle operator $\mathcal{L}_{\psi+it\phi}$ is less than $1$ for any $t\neq 0$. For pressure functions of generic potentials, Theorem 1.1 can be strengthened to the following result.

This means the second differential of the pressure function of a generic Hölder potential imposes some global subtle restriction on its third differential. It would be interesting to try to interpret the meaning of $P^{\prime\prime}(t)=\frac{1}{\sqrt[3]{2\pi}}$ for the pressure function at individual parameters. Let $\sigma:\Lambda^{\mathbb{N}}\rightarrow\Lambda^{\mathbb{N}}$ denote the shift map. Both the proofs of Theorem 1.1 and 1.2 require use of the Ruelle operator and the Central Limit Theorem (CLT) for the process $\{f\circ\sigma^{n}\}_{n\in\mathbb{N}}$, with the latter one depending on a finer CLT in the generic case. Recall that there are some expressions on the higher differentials of the pressure function by Kotani and Sunada in [KS1] for smooth systems, and we refer the readers to [KS2] for a CLT for random walks on crystal lattices.

It is well-known that $P(t\phi)$ is convex and Lipschitz for continuous $\phi$, moreover, the supporting lines of its graph must intersect the vertical axis in a closed bounded interval in $[0,\infty)$. Kucherenko and Quas have shown that any such function can be realised by the pressure function of some continuous potential on some shift space of finite type [KQ, Theorem 1], whose result fits into Katok’s flexibility programme [BKR]. However, the continuous potentials constructed in their work are not Hölder, so they ask the following question (their original problem is set in the multidimensional case).

Our following results are dedicated to an answer to their problem. We first point out that any convex, Lipschitz analytic function with its supporting lines intersecting the vertical axis in a closed bounded interval in $[0,\infty)$ can be approximated by sequences of pressure functions of locally constant potentials on some shift space of finite type.

One can see that in the above proof the increasing sequence of pressures $\{P(t\phi_{n,-})\}_{n\in\mathbb{N}}$ satisfies

$P(t\phi_{n,-})\nearrow F(t)$

as $n\rightarrow\infty$ since $\{\phi_{n,-}\}_{n\in\mathbb{N}}$ is an increasing sequence tending to $\phi_{F}$ (see [Wal1, Theorem 9.7(ii)]). Alternatively, one can take

$\phi_{n,+}(x)=\sup\{\phi_{F}(x):x\in[x_{-n}x_{-n+1}\cdots x_{n}]\}$,

which results in a decreasing sequence of locally constant potentials approximating $\phi_{F}(x)$, or

$\phi_{n,\pm}(x)=\cfrac{\phi_{n,-}(x)+\phi_{n,+}(x)}{2}$,

which also results in a sequence of locally constant potentials approximating $\phi_{F}(x)$, while their pressure functions both approximate $F(t)$. See Corollary 5.5 for an interpretation of the result from another point of view.

The following result confirms that some convex analytic functions cannot be fitted by the pressure of any Hölder potential on any shift space of finite type, which gives a negative answer to Problem 1.3.

For an explicit example of convex analytic functions in Theorem 1.7, one can simply take

$F_{2,3,1}(t)=\cfrac{2t^{2}+3t+te^{-t^{2}}+e^{-t^{2}}}{t}$

on $(\alpha,\infty)$ for any $\alpha>0$. See Proposition 4.2 for a family of such examples. Thus one can see that there are in fact elementary functions which cannot be fitted by pressures of Hölder potentials on shift spaces of finite type.

In the following we consider fitting convex analytic functions locally instead of globally, only by pressures of locally constant potentials on shift spaces of finite type. Let

$\Lambda_{n}=\{1,2,\cdots,n\}$

be the symbolic set of $n$ symbols.

This means we can fit some germs at $t_{*}$ up to level $2$ by pressures of some locally constant potentials when the number of symbols of the shift space is large enough. The values $\delta_{n},\{c_{i,n}\}_{i=1}^{n}$ all depend on $t_{*},a_{0},a_{1},n$ and $a_{2}$ in fact, while we only indicate the dependence of $m_{t_{*},a_{0},a_{1},n}$ and $M_{t_{*},a_{0},a_{1},n}$ as we are particularly interested in their values in the context of Theorem 1.8. There are some results on the values of

$\{m_{t_{*},a_{0},a_{1},n},M_{t_{*},a_{0},a_{1},n}\}_{n\in\mathbb{N}}$

subject to $t_{*}>0$ and $(a_{0},a_{1})\in\mathbb{R}^{2}$ satisfying (1.5) at the end of Section 5.

We choose to present all our results in the one dimensional case, while many of these results can in fact be extended to convex Lipschitz or analytic functions $F(t_{1},t_{2},\cdots,t_{m})$ of $m$ variables naturally. Most of our results also hold on transitive subshift spaces of finite type, with some technical adjustments in their proofs involving the transition matrix.

The organization of the work is as following. In Section 2 we introduce some basics in thermodynamic formalism and the Central Limit Theorem for the process generated by a potential and the shift map on the symbolic space of finite type. We give an explicit bound on the tail term in the CLT. Section 3 is devoted to the proof of Theorem 1.1 and 1.2. We formulate some expression of the derivatives of the pressure (Corollary 3.11) linking directly to the CLT, which allows us to unveil the relationship between derivatives of the pressure function of various orders. Section 4 is devoted to the proof of Theorem 1.7. In Section 5 we consider fitting 1- and 2-level candidate analytic germs locally by pressure functions of locally constant potentials (Problem 5.2) on symbolic spaces of finite type. We conjecture that any reasonable analytic germ of finite level can be fitted by the pressure function of some locally constant potential locally, as long as the number of the symbols is large enough.

In this section we collect some basic notions and results in thermodynamic formalism for later use. We start from the pressure. Let $\Lambda$ be some symbolic set of finite symbols, $\Lambda^{\mathbb{N}}$ be the shift space equipped with the visual metric

$d(x,y)=\cfrac{1}{2^{l(x,y)}}$

for distinct $x=x_{0}x_{1}x_{2}\cdots,y=y_{0}y_{1}y_{2}\cdots\in\Lambda^{\mathbb{N}}$, in which

$l(x,y)=\min\{i\in\mathbb{N}:x_{i}\neq y_{i}\}$.

For a continuous potential $\phi:\Lambda^{\mathbb{N}}\rightarrow\mathbb{R}$ on the compact metric space $\Lambda^{\mathbb{N}}$, Let

$S_{m,\phi}(x)=\sum_{i=0}^{m-1}\phi\circ\sigma^{i}(x)$

for $m\in\mathbb{N}$, in which $\sigma$ is the shift map.

One can refer to [Wal1, p208] for a definition for continuous potentials on general compact metric spaces. It satisfies the well-known variational formula

$P(\phi)=\sup\{h(\mu)+\int\phi d\mu:\mu\mbox{ is a }\sigma-\mbox{invariant measure on }\Lambda^{\mathbb{N}}\}$.

Let $C^{0}(\Lambda^{\mathbb{N}})$ be the collection of all the continuous potentials on $\Lambda^{\mathbb{N}}$. Two potentials $\psi,\phi\in C^{0}(\Lambda^{\mathbb{N}})$ are said to be cohomologous [Wal2] in case there exists a continuous map $\varphi:\Lambda^{\mathbb{N}}\rightarrow\mathbb{R}$ such that

$\psi(x)-\phi(x)=\varphi(x)-\varphi\circ\sigma(x)$.

We write $\psi\sim\phi$ to denote the equivalence relationship between two potentials cohomologous to each other. The maps in

$\{\varphi(x)-\varphi\circ\sigma(x):\varphi\in C^{0}(\Lambda^{\mathbb{N}})\}$

are called coboundaries. The importance of the cohomologous relationship is revealed in the following result.

One can find a proof in [Rue1] or [PP]. Another important tool in thermodynamic formalism is the Ruelle operator.

One can see easily that its compositions satisfy

for any $m\in\mathbb{N}$. In case of $\psi$ being Hölder, it admits a simple maximum isolated eigenvalue $\lambda=e^{P(\psi)}$ such that,

for some eigenfunction $w_{\psi}(x)\in C^{0,h}(\Lambda^{\mathbb{N}})$, refer to [Rue1]. The unique equilibrium measure for the Hölder potential $\psi$ is denoted by $\mu_{\psi}$. It then follows that

for $w_{\psi}(x)\in C^{0,h}(\Lambda^{\mathbb{N}})$. A potential $\psi$ is said to be normalized if

$P(\psi)=0$ and $w_{\psi}=1_{\Lambda^{\mathbb{N}}}$,

in which $1_{\Lambda^{\mathbb{N}}}$ is the identity map on $\Lambda^{\mathbb{N}}$. In case of $\psi$ being not normalized, we call

$\bar{\psi}=\psi+\log w_{\psi}-\log w_{\psi}\circ\sigma-P(\psi)$

the normalization of $\psi$. It is easy to check that $\bar{\psi}$ is a normalized potential. Moreover, $\bar{\psi}$ and $\psi$ share the same equilibrium state.

Now we turn to the Central Limit Theorem for the random process $\{\phi\circ\sigma^{j}(x)\}_{j=0}^{\infty}$ with the equilibrium measure $\mu_{\psi}$ defined by some Hölder potential $\psi$, while $\phi$ is also assumed to be Hölder. It deals with the asymptotic behaviour of the distribution of $\cfrac{S_{m,\phi}}{\sqrt{m}}$ with respect to $\mu_{\psi}$ as $m\rightarrow\infty$. The Ruelle operator comes in here, see [CP, Lal, Rou]. Let

$G_{m}(y)=\mu_{\psi}\Big{\{}x\in\Lambda^{\mathbb{N}}:\cfrac{S_{m,\phi}(x)}{\sqrt{m}}<y\Big{\}}$

for $y\in\mathbb{R}$. For $a,b\in\mathbb{R}$ and $b>0$, Let $N_{a,b}(y)$ be the normal distribution with expectation $a$ and standard deviation $\sqrt{b}$ on $\mathbb{R}$, that is,

$\cfrac{dN_{a,b}(y)}{dy}=\cfrac{1}{\sqrt{2\pi b}}e^{-(y-a)^{2}/2b}$

for $y\in\mathbb{R}$. For Hölder potentials $\psi,\phi$ on a shift space of finite type, since the pressure $P(\psi+t\phi)$ is analytic in a small neighbourhood around $0$, denote by

$\Delta_{m}=P^{(m)}(\psi+t\phi)|_{t=0}$

for $m\in\mathbb{N}$ for convenience, while the readers can understand its dependence on $\psi,\phi$ easily from the contexts in the following. Let

$P(\psi+t\phi)=\sum_{m=0}^{\infty}\cfrac{\Delta_{m}}{m!}t^{m}=\sum_{m=0}^{3}\cfrac{\Delta_{m}}{m!}t^{m}+t^{4}\kappa(t)$,

in which $\kappa(t)=\sum_{m=0}^{\infty}\cfrac{\Delta_{m+4}}{(m+4)!}t^{m}$.

This fits into special cases of the Berry-Esseen Theorem [Fel]. There is nothing new in the version here comparing with [CP, Theorem 2, Theorem 3] or [PP, Theorem 4.13], except the explicit bound on the tail term $O(1/\sqrt{m})$ in (2.4). In the following we justify this explicit bound. To do this, let

$\chi_{m}(z)=\displaystyle\int e^{iz\frac{S_{m,\phi}}{\sqrt{m}}}d\mu_{\psi}$

be the Fourier transformation of $G_{m}(y)$. Note that the Fourier transformation of $N_{0,\Delta_{2}}(y)$ is $e^{-\frac{z^{2}\Delta_{2}}{2}}$.

Equipped with Lemma 2.4 we can justify the explicit bound on the tail term in the Central Limit Theorem in (2.4).

Proof of the tail term in CLT:

We will deal with the pressure function $P(\psi+t\phi)$ for $t\geq 0$ and $\psi,\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ for some $0<h\leq 1$ in the following sections. By [Rue2], $P(\psi+t\phi)$ depends analytically on $t$ in case that $\psi,\phi$ are Hölder. We will often assume that

$\int\phi d\mu_{\psi}=0$

in the following when dealing with the higher differentials of $P(\psi+t\phi)$ because if $\int\phi d\mu_{\psi}=c\neq 0$, we have

$P(\psi+t(\phi-c))=P(\psi+t\phi)-ct$,

then

for any $n\geq 2$ while $\int(\phi-c)d\mu_{\psi}=0$. We can also assume that $\psi$ is normalized when dealing with the differentials of $P(\psi+t\phi)$. If this is not the case we can simply change $\psi$ to its normalization $\bar{\psi}$ while

for $n\geq 1$ because

$P(\bar{\psi}+t\phi)=P(\psi+t\phi)-P(\psi)$

for any $t\in\mathbb{R}$.

In this section we formulate some explicit expressions for the derivatives of the pressure $P(t\phi)=P(t)$ in terms of the derivatives of the eigenfunction of $\mathcal{L}_{t\phi}$ for $\phi\in C^{0,h}(\Lambda^{\mathbb{N}})$ with respect to $t$. We give basically two expressions of the derivatives, one of which allows the introduction of the random stochastic process $\{\phi\circ\sigma^{j}(x)\}_{j=0}^{m}$ for $m\in\mathbb{N}$. Upon the expression we prove Theorem 1.1 and 1.2 in virtue of the CLT for the random process $\{\phi\circ\sigma^{j}(x)\}_{j=0}^{\infty}$.

First we define some basics to deal with the higher derivatives of compositional functions by the Faà di Bruno’s formula. For an integer $j\in\mathbb{N}$, we say

$\tau=\tau_{1}\tau_{2}\cdots\tau_{q}$

with $q\in\mathbb{N}$ is a partition of $j$ if the non-increasing sequence of positive integers $j\geq\tau_{1}\geq\tau_{2}\geq\cdots\geq\tau_{q}\geq 1$ satisfies $\sum_{i=1}^{q}\tau_{i}=j$. Denote the collection of all the possible partitions of $j$ by $\mathfrak{P}(j)$. For example, Table 1 lists all the partitions in $\mathfrak{P}(5)$.

We sometimes simply write $\tau$ to denote the set $\{\tau_{1},\tau_{2},\cdots,\tau_{q}\}$ for convenience in the following, so $\#\tau=q$. Now for $\tau$ being a partition of $j\geq 1$, let $\{B_{j}^{\tau}\}$ be the number of different choices of dividing a set of $j$ different elements into $\#\tau=q$ sets of sizes $\{\tau_{i}\}_{i=1}^{q}$ respectively (with no order on the sets of partitions). Set $B_{0}^{0}=1$ for convenience. For example, consider the cases $j=5$ and $\tau=3,1,1$, the number of different choices of dividing a set of $5$ different elements into $q=3$ sets of sizes $3,1,1$ respectively is

$C_{5}^{3}=10=B_{5}^{3,1,1}$.

Table 2 lists all the numbers $\{B_{5}^{\tau}\}_{\tau\in\mathfrak{P}(5)}$.

For a smooth map $f:X\rightarrow Y$ between two metric spaces $X,Y$ and some partition $\tau=\tau_{1},\tau_{2},\cdots,\tau_{q}\in\mathfrak{P}(j)$ with $j\geq 1$, let

$f^{(\tau)}(x)=f^{(\tau_{1})}(x)f^{(\tau_{2})}(x)\cdots f^{(\tau_{q})}(x)$

be the product of the derivatives. For $j=0$ and $\tau=0\in\mathfrak{P}(0)$, set $f^{(0)}(x)=1$. Then for two smooth functions $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ between metric spaces $X,Y,Z$, we have

in virtue of Faà di Bruno’s formula.

Now we turn to the higher differentials of the pressure function. We start by considering some standard case, then extend the result to the general case.