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Dynamics of homeomorphisms of the torus homotopic to Dehn twists

Published online by Cambridge University Press:  15 January 2013

SALVADOR ADDAS-ZANATA ,
FÁBIO A. TAL  and
BRÁULIO A. GARCIA
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 (email: [email protected], [email protected], [email protected])
FÁBIO A. TAL
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil (email: [email protected], [email protected], [email protected])
BRÁULIO A. GARCIA
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil (email: [email protected], [email protected], [email protected])
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Abstract

In this paper, we consider torus homeomorphisms $f$ homotopic to Dehn twists. We prove that if the vertical rotation set of $f$ is reduced to zero, then there exists a compact connected essential ‘horizontal’ set $K$, invariant under $f$. In other words, if we consider the lift $\hat {f}$ of $f$ to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of $\hat {f}$. Also, we give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. As a corollary of the above results, we prove a version of Boyland’s conjecture to this setting: if $f$ is area preserving and has a lift $\hat {f}$ to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under $\hat {f}$, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 2 , April 2014 , pp. 409 - 422
Copyright
©2013 Cambridge University Press 

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