Realization of rotation vectors for volume preserving homeomorphisms of the torus

Abstract

In this note we study the level sets of rotation vectors for $C^0$-generic homeomorphisms in the space $\text{Homeo}_{0,\lambda}(\mathbb T^m)$ $(m \geq 3)$ of volume preserving homeomorphisms isotopic to the identity, and contribute to the ergodic optimization of vector valued observables. It is known that such homeomorphisms satisfy the specification property and their rotation sets are polyhedrons with rational vertices and non-empty interior, and stable \cite{BLV}, \cite{GL}, \cite{LV}. For a $C^0$-generic homeomorphism we prove uniform bounded deviations for the displacement of points in $\mathbb T^m$ in the support of any ergodic probability that generates a rotation vector in the boundary of the rotation set. As consequences, we show: (i) the support of ergodic probabilities generating rotation vectors in the boundary of rotation sets has empty interior, and (ii) weak version of Boyland's conjecture: the rotation vector of the Lebesgue measure lies in the interior of the rotation sets for a $C^0$-open and dense subset of homeomorphisms in $\text{Homeo}_{0,\lambda}(\mathbb T^m)$.