Geometry and entropy of generalized rotation sets

For a continuous map $f$ on a compact metric space we study the geometry and entropy of the generalized rotation set $\R(Φ)$. Here $Φ=(ϕ_1,...,ϕ_m)$ is a $m$-dimensional continuous potential and $\R(Φ)$ is the set of all $μ$-integrals of $Φ$ and $μ$ 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 $Φ=Φ(K)$ with $\R(Φ)=K$. Next, we study the relation between $\R(Φ)$ and the set of all statistical limits $\R_{Pt}(Φ)$. We show that in general these sets differ but also provide criteria that guarantee $\R(Φ)= \R_{Pt}(Φ)$. Finally, we study the entropy function $w\mapsto H(w), w\in \R(Φ)$. 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 $Φ$-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.

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