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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Published in:arXiv.org 2009-04
Main Authors: Heber Enrich, Guelman, Nancy, Larcanché, Audrey, Liousse, Isabelle
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Language:English
Subjects:
Entropy
Hypotheses
Isomorphism
Rotation
Topology
Toruses
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Guelman, Nancy
Larcanché, Audrey
Liousse, Isabelle
description 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.
doi_str_mv 10.48550/arxiv.0711.4728
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subjects Entropy
Hypotheses
Isomorphism
Rotation
Topology
Toruses
title Rotation set and Entropy
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