Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T13:40:04.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Unremovable periodic orbits of homeomorphisms

Published online by Cambridge University Press:  24 October 2008

Toby Hall
Toby Hall
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW
Get access Rights & Permissions [Opens in a new window]

Extract

In [1], Asimov and Franks give conditions under which a collection of periodic orbits of a diffeomorphism f:M→M of a compact manifold persists under arbitrary isotopy of f. Together with the Nielsen–Thurston theory, their result has been of pivotal importance in recent work on the periodic orbit structure of surface automorphisms (for example [3, 4, 7, 8, 9, 12, 13]). However, their proof uses bifurcation theory and as such depends crucially upon the differentiability of f. The periodic orbit results which make use of the Asimov–Franks theorem are therefore applicable only in the differentiable case, a limitation which belies their topological character. In this paper we shall use classical Nielsen-theoretic methods to prove the analogue of the Asimov–Franks result for homeomorphisms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 3 , November 1991 , pp. 523 - 531
Copyright
Copyright © Cambridge Philosophical Society 1991

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]Asimov, D. and Franks, J.. Unremovable closed orbits. Preprint (1989). (This is a revised version of a paper which appeared in Geometric dynamics, Lecture Notes in Math. vol. 1007 (Springer-Verlag, 1981).)Google Scholar
[2]Bowen, R.. Entropy and the fundamental group. In The Structure of Attractors in Dynamical Systems, Lecture Notes in Math. vol. 668 (Springer-Verlag, 1977), pp. 21–29.Google Scholar
[3]Boyland, P.. An analog of Sharkovski's theorem for twist maps. In Hamiltonian Dynamical Systems, Contemp. Math. vol. 81 (American Mathematical Society, 1988), pp. 119–133.Google Scholar
[4]Boyland, P.. Rotation sets and monotone periodic orbits for annulus homeomorphisms. Preprint (1990).Google Scholar
[5]Boyland, P.. Braid types and a topological method of proving positive entropy. Preprint (1984).Google Scholar
[6]Poénaru, V. and others. Travaux de Thurston sur les Surfaces. Astérisque 66–67 (1979).Google Scholar
[7]Gambaudo, J-M., Van Strien, S. and Tresser, C.. The periodic orbit structure of orientation- preserving diffeomorphisms of D2 with topological entropy zero. Preprint (1988).Google Scholar
[8]Guaschi, J., Llibre, J. and Mackay, R.. A classification of braid types for periodic orbits of diffeomorphisms of surfaces of genus one with topological entropy zero. Preprint (1989).Google Scholar
[9]Hall, T.. The bifurcations creating horseshoes. Preprint (1990).Google Scholar
[10]Handel, M.. The rotation set of a homeomorphism of the annulus is closed. Comm. Math. Phys. 127 (1990), 339–349.Google Scholar
[11]Jiang, B.. Nielsen Fixed Point Theory. Contemp. Math. vol. 14 (American Mathematical Society, 1983).Google Scholar
[12]Llibre, J. and Mackay, R.. A classification of braid types for diffeomorphisms of surfaces of genus zero with topological entropy zero. Preprint (1988).Google Scholar
[13]Matsuoka, T.. Braids of periodic points and a 2-dimensional analogue of Sharkovskii's ordering. In Dynamical Systems and Nonlinear Oscillations (World Science Publishing, 1986), pp. 58–72.Google Scholar