Localized topological pressure and equilibrium states

TL;DR

This paper introduces localized topological pressure for continuous maps, establishing a local variational principle, and explores new phenomena in localized equilibrium states, including non-uniqueness even for classical systems.

Contribution

It develops a localized version of topological pressure and equilibrium states, revealing new behaviors not seen in classical thermodynamic formalism.

Findings

Localized pressure defined via statistical sums close to a target value

A local variational principle established for various systems

Existence of non-unique localized equilibrium states in classical settings

Abstract

We introduce the notion of localized topological pressure for continuous maps on compact metric spaces. The localized pressure of a continuous potential $\varphi$ is computed by considering only those $(n,\epsilon)$-separated sets whose statistical sums with respect to an $m$-dimensional potential $\Phi$ are "close" to a given value $w\in \bR^m$. We then establish for several classes of systems and potentials $\varphi$ and $\Phi$ a local version of the variational principle. We also construct examples showing that the assumptions in the localized variational principle are fairly sharp. Next, we study localized equilibrium states and show that even in the case of subshifts of finite type and H\"older continuous potentials, there are several new phenomena that do not occur in the theory of classical equilibrium states. In particular, ergodic localized equilibrium states for H\"older continuous potentials are in general not unique.