Rotation set and Entropy

In 1991 Llibre and MacKay proved that if \(f\) is a 2-torus homeomorphism isotopic to identity and the rotation set of \(f\) has a non empty interior then \(f\) has positive topological entropy. Here, we give a converselike theorem. We show that the interior of the rotation set of a 2-torus \(C^{1+...

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Bibliographic Details
Published in:arXiv.org 2009-04
Main Authors: Heber Enrich, Guelman, Nancy, Larcanché, Audrey, Liousse, Isabelle
Format: Article
Language:English
Subjects:
Entropy
Hypotheses
Isomorphism
Rotation
Topology
Toruses
Online Access:Get full text
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Summary:In 1991 Llibre and MacKay proved that if \(f\) is a 2-torus homeomorphism isotopic to identity and the rotation set of \(f\) has a non empty interior then \(f\) has positive topological entropy. Here, we give a converselike theorem. We show that the interior of the rotation set of a 2-torus \(C^{1+ \alpha}\) diffeomorphism isotopic to identity of positive topological entropy is not empty, under the additional hypotheses that \(f\) is topologically transitive and irreducible. We also give examples that show that these hypotheses are necessary.
ISSN:2331-8422
DOI:10.48550/arxiv.0711.4728