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.
Beppo Levi's theorem for the vector-valued McShane integral and applications
