2012/09/29 by Kucherenko, Tamara, Wolf, Christian · 2 citations
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1210.0135
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=(ϕ1,...,ϕm) 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 \bRm. 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 \bRm a potential Φ=Φ(K) with \R(Φ)=K. Next, we study the relation between \R(Φ) and the set of all statistical limits \RPt(Φ). We show that in general these sets differ but also provide criteria that guarantee \R(Φ)= \RPt(Φ). Finally, we study the entropy function w↦ H(w), w∈ \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↦ H(w) is real-analytic in the interior of the rotation set.