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Lefschetz numbers of periodic orbits of pseudo-Anosov homeomorphisms

Published online by Cambridge University Press:  24 October 2008

John Guaschi
John Guaschi
Affiliation:
Centre de Recerca Matemàtica, Institut d'estudis Catalans, Apartat 50, 08193 Bellaterra (Barcelona), Spain
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Abstract

Given a surface homeomorphism isotopic to the identity which is pseudo-Anosov relative to a finite set, we show that the sum of the Lefschetz numbers of periodic points of any period greater than one is non-negative. If this period is odd and greater than a number which depends only on the surface, the sum is zero. If we consider sequences of periods such that each element is twice that of its predecessor, then this sum is increasing beyond a certain point also depending on the surface. As a corollary, for each periodic orbit contained within the boundary of the surface there exists one of the same period contained in the interior.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 1 , January 1994 , pp. 121 - 132
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Asimov, D. and Franks, J.. Unremovable closed orbits, preprint. (This is a revised version of a paper which appeared in Springer Lecture Notes in Mathematics 1007, Springer-Verlag, Berlin, 1983.)Google Scholar
[2]Alsedà, Ll., Llibre, J. and Misiurewicz, M.. Combinatorial dynamics and entropy in dimension one (World Scientific Publications, to appear).Google Scholar
[3]Bestvina, M. and Handel, M.. Train tracks for surface homeomorphisms, preprint.Google Scholar
[4]Birman, J. S.. Mapping class groups and their relationship to braid groups. Comm. Pure Appl. Math. 22 (1969), 213–238.CrossRefGoogle Scholar
[5]Birman, J. S.. Braids, links and mapping class groups. Ann. Math. Stud. 82 (Princeton University Press, 1974).Google Scholar
[6]Boyland, P. L.. Rotation sets and monotone periodic orbits for annulus homeomorphisms. Comment. Math. Helv. 67 (1992), 203–213.CrossRefGoogle Scholar
[7]Boyland, P. L. and Franks, J.. Notes on dynamics of surface homeomorphisms, lecture notes (University of Warwick 1989).Google Scholar
[8]Boyland, P. L. and Hall, T.. Personal communication.Google Scholar
[9]Brown, R. F.. The Lefschetz Fixed Point Theorem (Scott, Foresman and Co. 1971).Google Scholar
[10]Fadell, E. and Husseini, S.. The Nielsen number on surfaces. Cont. Math. 21 (1983), 59–98.CrossRefGoogle Scholar
[11]Fathi, A., Laudenbach, F. and Poénaru, V.. Travaux de Thurston sur les surfaces. Astérisque 66–67 (1979).Google Scholar
[12]Franks, J.. Generalizations of the Poincaré–Birkhoff theorem. Ann. Math. 128 (1988), 139–151.CrossRefGoogle Scholar
[13]Gambaudo, J.-M., Guaschi, J. and Hall, T.. Cascades in two-dimensional dynamics. Preprint submitted to Erg. Th. and Dyn. Sys.Google Scholar
[14]Gambaudo, J.-M. and Llibre, J.. A note on periods of surface homeomorphisms, preprint.Google Scholar
[15]Gambaudo, J.-M., Sullivan, D. and Tresser, C.. Infinite sequences of braids and smooth dynamical systems. Submitted to Topology.Google Scholar
[16]Guaschi, J.. Pseudo-Anosov braid types of the disc or sphere of low cardinality imply all periods. To appear in J. London Math. Soc.Google Scholar
[17]Handel, M.. The rotation set of a homeomorphism of the annulus is closed. Comm. Math. Phys. 127 (1990), 339–349.CrossRefGoogle Scholar
[18]Jiang, B.. Nielsen fixed point theory. Cont. Math. 14 (1983).CrossRefGoogle Scholar
[19]Kolev, B.. Periodic points of period 3 in the disc. To appear in Nonlinearity.Google Scholar
[20]Llibre, J. and MacKay, R. S.. Pseudo-Anosov homeomorphisms on a sphere with four punctures have all periods. Math. Proc. Cambridge Phil. Soc. 112 (1992), 539–550.CrossRefGoogle Scholar
[21]Papadopoulos, A. and Penner, R. C.. Enumerating pseudo-Anosov foliations. Pacific J. Math. 142 (1990), 159–173.CrossRefGoogle Scholar
[22]Sharkovskii, A. N.. Coexistence of cycles of a continuous map of a line into itself. Ukr. Math. Z. 16 (1964), 61–71.Google Scholar
[23]Shub, M.. Global Stability of Dynamical Systems (Springer-Verlag, 1987).CrossRefGoogle Scholar
[24]Thurston, W. P.. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417–431.CrossRefGoogle Scholar