Restrictions on rotation sets for commuting torus homeomorphisms

Abstract

Let $K_1,\: K_2\subset \mathbb{R}^2$ be two convex, compact sets. We would like to know if there are commuting torus homeomorphisms $f$ and $h$ homotopic to the identity, with lifts $\tilde f$ and $\tilde h$ such that $K_1$ and $K_2$ are their rotation sets respectively. In this work, we prove some cases where it cannot happen, assuming some restrictions on rotation sets.