Free curves and periodic points for torus homeomorphisms
Dynamical Systems
2007-12-06 v1
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.
Cite
@article{arxiv.0712.0643,
title = {Free curves and periodic points for torus homeomorphisms},
author = {Alejandro Kocsard and Andres Koropecki},
journal= {arXiv preprint arXiv:0712.0643},
year = {2007}
}
Comments
to appear in Ergodic Theory and Dynamical Systems