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 Rot(Φ). Here Φ = (φ1, ..., φm) is a m-dimensional continuous potential and Rot(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 R. 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 R a potential Φ = Φ(K) with Rot(Φ) = K. Next, we study the relation between Rot(Φ) and the set of all statistical limits RotPt(Φ). We show that in general these sets differ but also provide criteria that guarantee Rot(Φ) = RotPt(Φ). Finally, we study the entropy function w 7→ H(w), w ∈ Rot(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 7→ H(w) is real-analytic in the interior of the rotation set.