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On non-contractible periodic orbits for surface homeomorphisms

Published online by Cambridge University Press:  19 March 2015

FÁBIO ARMANDO TAL
FÁBIO ARMANDO TAL*
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil email fabiotal@ime.usp.br
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Abstract

In this work we study homeomorphisms of closed orientable surfaces homotopic to the identity, focusing on the existence of non-contractible periodic orbits. We show that, if $g$ is such a homeomorphism, and if ${\hat{g}}$ is its lift to the universal covering of $S$ that commutes with the deck transformations, then one of the following three conditions must be satisfied: (1) the set of fixed points for ${\hat{g}}$ projects to a closed subset $F$ which contains an essential continuum; (2) $g$ has non-contractible periodic points of every sufficiently large period; or (3) there exists a uniform bound $M>0$ such that, if $\hat{x}$ projects to a contractible periodic point, then the ${\hat{g}}$ orbit of $\hat{x}$ has diameter less than or equal to $M$. Some consequences for homeomorphisms of surfaces whose rotation set is a singleton are derived.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 5 , August 2016 , pp. 1644 - 1655
Copyright
© Cambridge University Press, 2015 

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