Generic Rotation Sets

2019/12/10 by Pavez-Molina, Sebastián
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.04428

Abstract

Let (X,T) be a topological dynamical system. Given a continuous vector-valued function F ∈ C(X, 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 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(⋅) 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 C¹ boundary. Furthermore, we prove that the map R(⋅) is surjective, extending a result of Kucherenko and Wolf.

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