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Pseudo-Anosov homeomorphisms on a sphere with four punctures have all periods

Published online by Cambridge University Press:  24 October 2008

Jaume Llibre  and
Robert S. Mackay
Jaume Llibre
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Spain
Robert S. Mackay
Affiliation:
Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Coventry CV4 7AL, England
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Abstract

It is proved that if ƒ is a homeomorphism of the two-sphere with an invariant set V of cardinality N = 4, then either ƒ has periodic orbits of all periods or it belongs to one of a small number of algebraically finite isotopy classes relative to V. For N ≤ 4, the second case always holds. On the other hand, for each N ≥ 7 we give examples of pseudo-Anosov homeomorphisms of the sphere, relative to a set of N points, for which not all periods occur.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 3 , November 1992 , pp. 539 - 549
Copyright
Copyright © Cambridge Philosophical Society 1992

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