On Torus Homeomorphisms Semiconjugate to irrational Rotations
| Autor: | Tobias Jäger, Alejandro Passeggi |
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| Rok vydání: | 2013 |
| Předmět: |
Pure mathematics
Mathematics::Dynamical Systems Applied Mathematics General Mathematics Franks-Misiurewicz conjecture homeomorphisms invariant 'foliation' rotation vector Franks-Misiurewicz-Vermutung Homöomorphismen invarianten 'Foliierung' Rotationsvektor Mathematics::General Topology Torus Annulus (mathematics) Rotation matrix Dynamical Systems (math.DS) Foliation Irrational rotation Primary 54H20 Secondary 37E30 37E45 FOS: Mathematics Uniqueness ddc:510 Mathematics - Dynamical Systems Invariant (mathematics) Rotation (mathematics) Mathematics |
| DOI: | 10.48550/arxiv.1305.0723 |
| Popis: | 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 characterise 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. Generalising a well-known result of M. 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. Comment: 20 pages, 5 figures |
| Databáze: | OpenAIRE |
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