Restrictions on rotation sets for commuting torus homeomorphisms

TL;DR

This paper investigates the conditions under which two commuting torus homeomorphisms can have prescribed convex, compact rotation sets, establishing restrictions on possible rotation set configurations under certain assumptions.

Contribution

It provides new results identifying cases where specific convex, compact rotation sets cannot be realized simultaneously by commuting torus homeomorphisms homotopic to the identity.

Findings

Certain convex, compact sets cannot be rotation sets of commuting torus homeomorphisms.

Restrictions depend on properties of the rotation sets, such as convexity and compactness.

The work advances understanding of the relationship between rotation sets and commuting dynamics.

Abstract

Let $K_1$, $K_2$ $\subset$ $R^2$ 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 proof some cases where it cannot happen, assuming some restrictions on rotation sets.