Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T14:05:00.780Z Has data issue: false hasContentIssue false
  • English
  • Français

A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds

Published online by Cambridge University Press:  19 September 2008

Stephan Pelikan  and
Edward E. Slaminka
Stephan Pelikan
Affiliation:
Department of Mathematics, University of Cincinatti, Cincinnati, Ohio 45221, USA;
Edward E. Slaminka
Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The study of area preserving maps of manifolds has an extensive history in the theory of dynamical systems. One interest has been in the behaviour of such maps near an isolated fixed point. In 1974 Carl Simon proved the existence of an upper bound for the index of an isolated fixed point for Ck area preserving diffeomorphisms of a surface. We extend his result to homeomorphisms of an orientable two manifold. The proof utilizes the notion of free modification, developed by Morton Brown, and enlarges the scope of the problem to the consideration of ‘nice’ measures, i.e. uniformly equivalent to Lebesgue measure on compact sets. By suitably modifying the homeomorphism and the measure, we obtain the following theorem.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 3 , September 1987 , pp. 463 - 479
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[1]Boucher, K.. Private communication.Google Scholar
[2]Brown, M.. A new proof of Brouwer's Lemma on translation arcs. Houston J. Math. 10 (1984), 35–41.Google Scholar
[3]Brown, M.. Private communication.Google Scholar
[4]Brown, M.. Homeomorphisms of two-dimensional manifolds. To appear in Houston J. Math.Google Scholar
[5]Schmitt, B.. Sur les plongements, admettant zero ou un point fixe, du disque dans le plan. Topology 14(1975), 357–365.CrossRefGoogle Scholar
[6]Simon, C.. A bound for the fixed-point index of an area-preserving map with applications to mechanics. Inventiones Math. 26 (1974), 187–200.CrossRefGoogle Scholar