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Instability for the rotation set of homeomorphisms of the torus homotopic to the identity

Published online by Cambridge University Press:  09 March 2004

SALVADOR ADDAS-ZANATA
SALVADOR ADDAS-ZANATA
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil (e-mail: sazanata@ime.usp.br)
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Abstract

In this paper we consider homeomorphisms $f:T^2 \rightarrow T^2$ homotopic to the identity and their rotation sets $\rho (\tilde{f})$, which are compact convex subsets of the plane. We show that if $\rho (\tilde{f})$ has an extremal point $(t,\omega )$ which is not a rational vector, then arbitrarily C0 close to f we can find a homeomorphism g such that $\rho (\tilde{g})\cap \rho (\tilde{f})^c\neq \emptyset$. So in this case, we have instability for the rotation set.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 24 , Issue 2 , April 2004 , pp. 319 - 328
Copyright
2004 Cambridge University Press

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