Free curves and periodic points for torus homeomorphisms

TL;DR

This paper explores the connection between free curves and periodic points in torus homeomorphisms, establishing conditions under which rational rotation points are realized by periodic points and extending classical theorems.

Contribution

It provides a topological version of Franks' theorem and a Poincaré-Birkhoff type result for torus homeomorphisms without free curves.

Findings

Every rational point in the rotation set is realized by a periodic point if no free curve exists and the rotation set has empty interior.

In the absence of free curves, either a fixed point exists or the rotation set has nonempty interior.

The results extend classical theorems to a topological setting for torus homeomorphisms.

Abstract

We study the relationship between free curves and periodic points for torus homeomorphisms in the homotopy class of the identity. By free curve we mean a homotopically nontrivial simple closed curve that is disjoint from its image. We prove that every rational point in the rotation set is realized by a periodic point provided that there is no free curve and the rotation set has empty interior. This gives a topological version of a theorem of Franks. Using this result, and inspired by a theorem of Guillou, we prove a version of the Poincar\'e-Birkhoff Theorem for torus homeomorphisms: in the absence of free curves, either there is a fixed point or the rotation set has nonempty interior.