Pressure and equilibrium states

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Author: Dr. Jean-René Chazottes, Centre de Physique Théorique, CNRS-École Polytechnique, France
Author: Dr. Gerhard Keller, Mathematics Institute, University of Erlangen, Germany

Let $T:X\to X$ be a continuous transformation of a compact metric space $(X,d)$ . Let $C^0(X,\mathbb{R})$ denote the Banach algebra of real-valued continuous functions of $X$ equipped with the supremum norm. The topological pressure of $T$ will be a map $P(T,\cdot):C^0(X,\mathbb{R})\to\mathbb{R}\cup\{\infty\}$ . It contains topological entropy in the sense that $P(T,0)=h(T)$ where $0$ denotes the member of $C^0(X,\mathbb{R})$ identically equal to zero.

1 History
2 Equilibrium States in Finite Systems
3 Topological Pressure
4 Properties of Pressure
5 The Variational Principle
6 Equilibrium States
7 Equilibrium States on shift spaces
8 Application to Differentiable Dynamics
9 Connection with Statistical Mechanics
10 Recommended reading
11 Further reading
12 See also

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History

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Equilibrium States in Finite Systems

The following elementary setting contains germs of notions and results described hereafter.

Let $\Omega$ be a finite set, a configuration space without any specific structure.

A state is a probability vector $\mu=(\mu(\omega)| \omega\in\Omega)$ . The set of states is denoted by $\mathcal{M}$ .

The entropy of the state $\mu$ is defined as $H(\mu):=-\sum_{\omega\in \Omega} \mu(\omega) \log\mu(\omega)$ .

Each configuration $\omega\in\Omega$ is assigned an energy value $u(\omega)\in\mathbb{R}$ such that in state $\mu$ the system has mean energy $\mu(u):=\sum_{\omega\in\Omega} \mu(\omega)u(\omega)$ .

$Z(\beta):=\sum_{\omega\in\Omega} \exp(\beta u(\omega))$ is the partition function of $u$ where $\beta\in\mathbb{R}$ .

For each $\beta\in\mathbb{R}$ the Gibbs measure $\mu_\beta$ on $\Omega$ is defined by

$\mu_\beta:=\frac{1}{Z(\beta)} \exp(\beta u(\omega)).$

The following theorem is an elementary prototype of a much more general statement given later on.

Variational principle(elementary version):

Each Gibbs measure $\mu_\beta$ with $\beta\in\mathbb{R}$ satisfies

$H(\mu_\beta)+\mu_\beta(\beta u) = \log Z(\beta)=\sup\{H(\nu)+\nu(\beta u)\big|\ \nu \in\mathcal{M}\}.$

A measure $\nu$ for which this supremum is attained is called an equilibrium state for $\beta u$ . Thus Gibbs measures are equilibrium states. In fact, $\mu_\beta$ is the only equilibrium state for $\beta u$ .

Proof of the elementary version of the variational principle

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Topological Pressure

A set $E \subset X$ is said to be $(n,\varepsilon)$ -separated, if for every $x, y\in E$ with $x\neq y$ there is $i\in\{0,1,\dots,n-1\}$ such that $d(T^ix,T^iy)\ge\varepsilon$ . Let $s(n,\varepsilon)$ be the maximal cardinality of an $(n,\varepsilon)$ -separated set in $X$ . Again, by compactness, this number is always finite. For $f\in C^0(X,\mathbb{R})$ , $x\in X$ and $n\in \mathbb{N}_0$ define $S_n f(x):= \sum_{i=0}^{n-1} f(T^i(x))$ .

For $\varepsilon>0$ , $n\in\mathbb{N}$ , let

$Z(T,f,\varepsilon,n):=\sup\left\{\sum_{x\in E} e^{S_n f(x)}\ \big| \ E\subset X\;\mathrm{is}\;(n,\varepsilon)-\mathrm{separated}.\right\}$

Then

$P(T,f):=\lim_{\varepsilon\to 0}\limsup_{n\to\infty}\frac{1}{n}\log Z(T,f,\varepsilon,n)$

is called the topological pressure of $T$ with respect to $f$ .

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Properties of Pressure

We give some properties of $P(T,\cdot):C^0(X,\mathbb{R})\to\mathbb{R}\cup\{\infty\}$ .

If $f,g \in C^0(X,\mathbb{R})$ , $\varepsilon>0$ and $c\in\mathbb{R}$ then the following are true.

$P(T,0)=h(T)$ where $h(T)$ is the topological entropy of $T$

$f\leq g$ implies $P(T,f)\leq P(T,g)$ . In particular $h(T)+\inf f \leq P(T,f)\leq h(T)+ \sup f$ .

$P(T,\cdot)$ is either finite valued or constantly $\infty$ .

$|P(T,f)-P(T,g)|\leq \|f-g\|$ .

$P(T,\cdot)$ is convex.

$P(T,f+c)=P(T,f)+c$ .

$P(T,f+g\circ T-g)=P(T,f)$

Now we look at how $P(T,\cdot)$ depends on $T$ .

If $k>0$ $P(T^k,S_k f)=k P(T,f)$ .

If $T$ is a homeomorphism $P(T^{-1},f)$ .

If $Y$ is a closed subset of $X$ with $TY\subset Y$ then $P(T|_{Y},f|_{Y})\leq P(T,f)$ .

If $T_i:X_i\to X_i$ ( $i=1,2$ ) is a continuous map of a compact of a compact metric space $(X_i,d_i)$ and if $\phi:X_1\to X_2$ is a surjective continuous map with $\phi\circ T_1=T_2\circ\phi$ then $P(T_2,f)\leq P(T_1,f\circ\phi)$ $\forall f \in C^{0}(X_2,\mathbb{R})$ . If $\phi$ is a homeomorphism then $P(T_2,f)=P(T_1,f\circ\phi)$ .

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The Variational Principle

Denote by $\mathcal{M}(X,T)$ the set of $T$ -invariant probability measures on $X$ (equipped with weak $^*$ or vague topology). We have the following theorem:

Let $T:X\to X$ be a continuous transformation of a compact metric space $(X,d)$ and let $f\in C^0(X,\mathbb{R})$ . Then

$P(T,f)=\sup\left\{h_\mu(T)+\int f d\mu\ \Big| \ \mu\in \mathcal{M}(X,T)\right\}$

where $h_\mu(T)$ is the Kolmogorov-Sinai entropy of $\mu$ .

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Equilibrium States

The variational principle gives a natural way of selecting members of $\mathcal{M}(X,T)$ . The concept extends the idea of measure with maximal entropy.

Let $T:X\to X$ be a continuous map of a compact metric space $X$ and let $f \in C^0(X,\mathbb{R})$ .

A member of $\mathcal{M}(X,T)$ is called an equilibrium state for $f$ if $P(T,f)=h_\mu(T)+\int f d\mu$ .

Let $\mathcal{E\!S}(f)$ denote the collection of all equilibrium states for $f$ . Notice that this set can be empty but if the entropy map is upper semi-continuous then $\mathcal{E\!S}(f)$ is a non-empty compact subset of $\mathcal{M}(X,T)$ .

Remarks.

$\mathcal{E\!S}(f)$ is a convex set.

If $f,g\in C^0(X,\mathbb{R})$ and if there exists $c\in\mathbb{R}$ such that $f-g-c$ belongs to the closure of the set $\{h\circ T-h| h\in C^0(X,\mathbb{R})$ in $C^0(X,\mathbb{R})$ , then $\mathcal{E\!S}(f)=\mathcal{E\!S}(g)$ .

The notion of equilibrium state is tied in with the notion of tangent functional to the convex function $P(T,\cdot):C^0(X,\mathbb{R})\to\mathbb{R}$ . See Tangent Functional to the Pressure.