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Generic rotation sets
Published online by Cambridge University Press: 29 December 2020
Abstract
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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.
MSC classification
Secondary:
37E45: Rotation numbers and vectors
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