Generic Rotation Sets

TL;DR

This paper investigates the geometric properties of rotation sets in topological dynamical systems, showing that generically these sets are strictly convex with smooth boundaries and that the rotation map is open and surjective.

Contribution

It establishes the generic convexity and smoothness of rotation sets and proves the openness and surjectivity of the rotation map in broad classes of systems.

Findings

Rotation sets of generic potentials are strictly convex.

Rotation sets have $C^{1}$ boundaries.

The rotation map is open and surjective.

Abstract

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.