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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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References

[1]Addas-Zanata, S.. On the existence of a new type of periodic and quasi-periodic orbits for twist maps of the torus. Nonlinearity 15 (2002), 1399–1416.CrossRefGoogle Scholar
[2]Addas-Zanata, S.. On properties of the vertical rotation interval for twist mappings. Ergod. Th. & Dynam. Sys. 25 (2005), 641–660.Google Scholar
[3]Addas-Zanata, S.. Some extensions of the Poincaré–Birkhoff theorem to the cylinder and a remark on mappings of the torus homotopic to Dehn twists. Nonlinearity 18 (2005), 2243–2260.Google Scholar
[4]Addas-Zanata, S. and Tal, F. A.. Homeomorphisms of the annulus with a transitive lift. Math. Z. 267 (2009), 971–980.Google Scholar
[5]Birkhoff, G.. Nouvelles recherches sur les systèmes dynamiques. Collec. Math. Pap. 2 (1935), 530–661.Google Scholar
[6]Doeff, E.. Rotation measures for homeomorphisms of the torus homotopic to a Dehn twist. Ergod. Th. & Dynam. Sys. 17 (1997), 1–17.Google Scholar
[7]Doeff, E. and Misiurewicz, M.. Shear rotation numbers. Nonlinearity 10 (1997), 1755–1762.CrossRefGoogle Scholar
[8]Franks, J.. Generalizations of the Poincaré–Birkhoff theorem. Ann. of Math. (2) 128 (1988), 139–151.Google Scholar
[9]Le Roux, F.. Homéomorphismes de surfaces: théorèmes de la fleur de Leau-Fatou et de la variété stable. Astérisque 292 (2004), 120 pp.Google Scholar
[10]Le Calvez, P.. Propriétés dynamiques des régions d’instabilité. Ann. Sci. Éc. Norm. Supér. 20 (1987), 443–464.Google Scholar
[11]Llibre, J. and Mackay, R.. Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity. Ergod. Th. & Dynam. Sys. 11 (1991), 115–128.CrossRefGoogle Scholar
[12]Misiurewicz, M. and Ziemian, K.. Rotation sets for maps of tori. J. Lond. Math. Soc. 40 (1989), 490–506.Google Scholar