Free curves and periodic points for torus homeomorphisms

Details Free curves and periodic points for torus homeomorphisms Free curves and periodic points for torus homeomorphisms

2007-12-05

arxiv.org/abs/0712.0643v1

Mathematics

Dynamical Systems

to appear in Ergodic Theory and Dynamical Systems

Scientific paper

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.

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