Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T02:32:53.485Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Boyland, P. L., Rotation sets and monotone periodic orbits for annulus homeomorphisms. Preprint (1990).Google Scholar
[2]Fathi, A., Laudenbach, F. and Poenaru, V.. Travaux de Thurston sur les Surfaces. Astérisque, vol. 66–67, 1979.Google Scholar
[3]Franks, J., Homology and Dynamics. CBMS Regional Conf. Ser. in Math. no. 49 (American Mathematical Society, 1982).CrossRefGoogle Scholar
[4]Gambaudo, J. M. and Llibre, J.. A note on the periods of surface homeomorphisms. Preprint (1991).Google Scholar
[5]Gambaudo, J. M., van Strien, S. and Tresser, C., Vers un ordre de Sharkovskii pour les plongements du disque préservant l'orientation. C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 294–297.Google Scholar
[6]Guaschi, J., Dynamics of surface homeomorphisms: braid types and coexistence of periodic orbits. Ph.D. thesis, Warwick University (1991).Google Scholar
[7]Handel, M.. Global shadowing of pseudo-Anosov homeomorphisms. Ergodic Theory Dynamical Systems 5 (1985), 373–377.CrossRefGoogle Scholar
[8]Handel, M. and Thurston, W. P.. New proofs of some results of Nielsen. Adv. in Math. 56 (1985), 173–191.CrossRefGoogle Scholar
[9]Katok, A.. Bernoulli diffeomorphisms on the disk. Ann. of Math. 110 (1979), 529–547.CrossRefGoogle Scholar
[10]Kolev, B.. Periodic orbits of period 3 in the disk. Nonlinearity, to appear.Google Scholar
[11]Li, T. and Yorke, J.. Period three implies chaos. Amer. Math. Monthly 82 (1975), 985–992.CrossRefGoogle Scholar
[12]Llibre, J. and MacKay, R. S.. A classification of braid types for diffeomorphisms of surfaces of genus zero with topological entropy zero. J. London Math. Soc. (2) 42 (1990), 562–576.CrossRefGoogle Scholar
[13]Manning, A. K.. There are no new Anosov diffeomorphisms of tori. Amer. J. Math. 96 (1974), 422–429.CrossRefGoogle Scholar
[14]Manning, A. K.. A Markov partition which displays the homology of a toral automorphism. In Papers presented to Christopher Zeeman, Warwick (1988).Google Scholar
[15]Niven, I. and Zuckerman, H. S.. An Introduction to the Theory of Numbers, fourth edition (John Wiley and Sons, 1980).Google Scholar
[16]Percival, I. C. and Vivaldi, F.. Arithmetical properties of strongly chaotic motions. Phys. D 25 (1987), 105–130.CrossRefGoogle Scholar
[17]Sharkovskii, A. N.. Co-existence of the cycles of a continuous mapping of the line into itself. Ukrain. Math. Zh. 16 (1964), 61–71.Google Scholar