# User:Eugene M. Izhikevich/Proposed/Pressure and equilibrium states

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.

## 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 $$\beta u\ .$$

## 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 $$f\ .$$

## 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.

• $$f\leq g$$ 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 $$P(T_2,f)=P(T_1,f\circ\phi)\ .$$

## 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 $$\mu\ .$$

## 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.

## Equilibrium States on shift spaces

We consider a configuration space $$\Omega:=A^{\mathbb{N}}\ .$$

## Connection with Statistical Mechanics

[B] R. Bowen: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Second revised edition. Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008

[K] G. Keller: Equilibrium States in Ergodic Theory. London Mathematical Society Student Texts 42 Cambridge University Press, 1998

[R1] D. Ruelle: Thermodynamic Formalism: The Mathematical Structures of Equilibrium Statistical Mechanics. Second revised edition. Cambridge Mathematical Library. Cambridge University Press, 2004

[W] P. Walters: An introduction to ergodic theory. Graduate Texts in Mathematics. Springer, 2000.

[Z] M. Zinsmeister: Thermodynamic Formalism and Holomorphic Dynamical Systems. SMF/AMS Texts and Monographs 2 (2000) [Publié en français dans le numéro 4 (1996) de la série Panoramas et Synthèses]

[G] H.-O. Georgii: Gibbs measures and phase transitions. Studies in Mathematics 9. De Gruyter, Berlin, 1988

[I] R.B. Israel: Convexity in the theory of lattice gases. Princeton Series in Physics. Princeton University Press, 1979

[R2] D. Ruelle: Statistical Mechanics: Rigorous Results. World Scientific, 1999 [First edition: Benjamin, N.Y., 1969]

[S] Ya. Sinai: Gibbs measures in ergodic theory. Russian Mathematical Surveys (1972) Vol. 27 (4), 21-69.