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On torus homeomorphisms semiconjugate to irrational rotations

Published online by Cambridge University Press:  30 June 2014

T. JÄGER  and
A. PASSEGGI
T. JÄGER
Affiliation:
Department of Mathematics, TU-Dresden, 01062 Dresden, Germany email Tobias.Oertel-Jaeger@tu-dresden.de, alepasseggi@gmail.com
A. PASSEGGI
Affiliation:
Department of Mathematics, TU-Dresden, 01062 Dresden, Germany email Tobias.Oertel-Jaeger@tu-dresden.de, alepasseggi@gmail.com
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Abstract

In the context of the Franks–Misiurewicz conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterize these maps by the existence of an invariant ‘foliation’ by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalizing a well-known result of Herman on forced circle homeomorphisms, we provide a criterion, in terms of topological properties of the annular continua, for the uniqueness of the rotation vector. As a byproduct, we obtain a simple proof for the uniqueness of the rotation vector on decomposable invariant annular continua with empty interior. In addition, we collect a number of observations on the topology and rotation intervals of invariant annular continua with empty interior.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 7 , October 2015 , pp. 2114 - 2137
Copyright
© Cambridge University Press, 2014 

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