Generic rotation sets

Abstract

AbstractLet$(X,T)$be a topological dynamical system. Given a continuous vector-valued function$F \in C(X, \mathbb {R}^{d})$called apotential, we define its rotation set$R(F)$as the set of integrals ofFwith respect to allT-invariant probability measures, which is a convex body of$\mathbb {R}^{d}$. In this paper we study the geometry of rotation sets. We prove that ifTis a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map$R(\cdot )$is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has$C^{1}$boundary. Furthermore, we prove that the map$R(\cdot )$is surjective, extending a result of Kucherenko and Wolf.