Geometry and entropy of generalized rotation sets

TL;DR

This paper investigates the geometry, entropy, and statistical properties of generalized rotation sets for continuous maps, constructing specific potentials to realize any convex set as a rotation set and analyzing entropy behavior.

Contribution

It constructs potentials for subshifts of finite type to realize any convex set as a rotation set and establishes a variational principle for the entropy function.

Findings

Every compact convex set in R^m can be realized as a rotation set for some potential in subshifts of finite type.

The sets of rotation vectors and statistical limits generally differ, but criteria for equality are provided.

The entropy function is real-analytic in the interior of the rotation set for systems with strong thermodynamic properties.

Abstract

For a continuous map $f$ on a compact metric space we study the geometry and entropy of the generalized rotation set $\R(\Phi)$. Here $\Phi=(\phi_1,...,\phi_m)$ is a $m$-dimensional continuous potential and $\R(\Phi)$ is the set of all $\mu$-integrals of $\Phi$ and $\mu$ runs over all $f$-invariant probability measures. It is easy to see that the rotation set is a compact and convex subset of $\bR^m$. We study the question if every compact and convex set is attained as a rotation set of a particular set of potentials within a particular class of dynamical systems. We give a positive answer in the case of subshifts of finite type by constructing for every compact and convex set $K$ in $\bR^m$ a potential $\Phi=\Phi(K)$ with $\R(\Phi)=K$. Next, we study the relation between $\R(\Phi)$ and the set of all statistical limits $\R_{Pt}(\Phi)$. We show that in general these sets differ but also provide criteria that guarantee $\R(\Phi)= \R_{Pt}(\Phi)$. Finally, we study the entropy function $w\mapsto H(w), w\in \R(\Phi)$. We establish a variational principle for the entropy function and show that for certain non-uniformly hyperbolic systems $H(w)$ is determined by the growth rate of those hyperbolic periodic orbits whose $\Phi$-integrals are close to $w$. We also show that for systems with strong thermodynamic properties (subshifts of finite type, hyperbolic systems and expansive homeomorphisms with specification, etc.) the entropy function $w\mapsto H(w)$ is real-analytic in the interior of the rotation set.