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Free curves and periodic points for torus homeomorphisms

Published online by Cambridge University Press:  15 September 2008

ALEJANDRO KOCSARD  and
ANDRES KOROPECKI
ALEJANDRO KOCSARD
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, RJ, Brasil (email: alejo@impa.br)
ANDRES KOROPECKI
Affiliation:
Universidade Federal Fluminense, Instituto de Matemática, Rua Mário Santos Braga S/N, 24020-140 Niterói, RJ, Brasil (email: koro@mat.uff.br)
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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 non-trivial 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 Poincaré–Birkhoff theorem for torus homeomorphisms: in the absence of free curves, either there is a fixed point or the rotation set has non-empty interior.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 6 , December 2008 , pp. 1895 - 1915
Copyright
Copyright © 2008 Cambridge University Press

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