Generic Rotation Sets

Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb{R}^{d})$ called a potential we define its rotation set $R(F)$ as the set of integrals of $F$ with respect to all $T$-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 if $T$ is 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.