Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T04:05:52.322Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinite sequence of fixed-point free pseudo-Anosov homeomorphisms

Published online by Cambridge University Press:  24 November 2009

JÉRÔME LOS
JÉRÔME LOS*
Affiliation:
Laboratoire d’Analyse, Topologie, Probabilité (LATP), UMR CNRS 6632, Université de Provence, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct an infinite sequence of pseudo-Anosov homeomorphisms without fixed points and leaving invariant a sequence of orientable measured foliations on the same topological surface and the same stratum of the space of Abelian differentials. The existence of such a sequence shows that all pseudo-Anosov homeomorphisms fixing orientable measured foliations cannot be obtained by the Rauzy–Veech induction strategy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 6 , December 2010 , pp. 1739 - 1755
Copyright
Copyright © Cambridge University Press 2009

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

[1]Alseda, L., Llibre, J. and Misiurewicz, M.. Combinatorial Dynamics and Entropy in Dimension One (Advanced Series in Nonlinear Dynamics, 5). World Scientific, Singapore, 1993.CrossRefGoogle Scholar
[2]Arnoux, P.. Un exemple de semi-conjugaison entre un échange d’intervalles et une translation du tore. Bull. Soc. Math. France 116(4) (1988), 489–500.CrossRefGoogle Scholar
[3]Arnoux, P. and Yoccoz, J.-C.. Construction de diffomorphismes pseudo-Anosov. C. R. Math. Acad. Sci. Paris 292(1) (1981), 75–78.Google Scholar
[4]Bauer, M.. Examples of pseudo-Anosov homeomorphisms. Trans. Amer. Math. Soc. 330(1) (1992), 333–359.CrossRefGoogle Scholar
[5]Bestvina, M. and Handel, M.. Train tracks for surface homeomorphisms. Topology 34 (1995), 109–140.CrossRefGoogle Scholar
[6]Boyland, P.. Topological methods in surface dynamics. Topology Appl. 58 (1994), 223–298.CrossRefGoogle Scholar
[7]Bufetov, A.. Logarithmic asymptotics for the number of periodic orbits of the Teichmüller flow on Veech’s space of zippered rectangles. Mosc. Math. J. to appear.Google Scholar
[8]Casson, A. and Bleiler, S.. Automorphisms of Surfaces After Nielsen and Thurston (London Mathematical Society Student Texts, 9). Cambridge University Press, Cambridge, 1988.CrossRefGoogle Scholar
[9]Eskin, A. and Mirzakhani, M.. Counting closed orbits in modular space. Preprint, 2008.Google Scholar
[10]Fathi, A., Laudenbach, F. and Poenaru, V.. Travaux de Thurston sur les surfaces. Astérisque 66–67 (1979).Google Scholar
[11]Fehrenbach, J. and Los, J.. Roots, symmetry and conjugacy of pseudo-Anosov homeomorphims. Preprint, 2007.Google Scholar
[12]Hamenstädt, U.. Train tracks and the Gromov boundary of the complex of curves. Spaces of Kleinian Groups (London Mathematical Society Lecture Notes, 329). Eds. Minsky, Y., Sakuma, M. and Series, C.. London Mathematical Society, London, 2005.Google Scholar
[13]Jiang, B.. Lectures on Nielsen Fixed Point Theory. American Mathematical Society, Providence, RI, 1983.CrossRefGoogle Scholar
[14]Ko, K. H., Los, J. and Song, W. T.. Entropy of braids. J. Knot. Theory Ramifications 11(4) (2002), 647–666.Google Scholar
[15]Los, J.. On the forcing relation for surface homeomorphisms. Publ. Math. Inst. Hautes Études Sci. 85 (1997), 5–61.CrossRefGoogle Scholar
[16]Mosher, L.. The classification of pseudo-Anosov’s. Low-dimensional Topology and Kleinian Groups (Coventry/Durham, 1984) (London Mathematical Society Lecture Notes Series, 112). London Mathematical Society, London, 1986, pp. 13–75.Google Scholar
[17]Mosher, L.. Train-track expansions of measured foliations. Preprint monograph, 2003.Google Scholar
[18]Papadopoulos, A. and Penner, R.. A characterization of pseudo-Anosov foliations. Pacific J. Math. 130(2) (1987), 359–377.CrossRefGoogle Scholar
[19]Penner, R. C. and Harer, J. L.. Combinatorics of train tracks. Ann. of Math. Stud. 125 (1992).Google Scholar
[20]Rauzy, G.. Echanges d’intervalles et transformations induites. Acta. Arith. 34 (1979), 315–328.CrossRefGoogle Scholar
[21]Thurston, W.. On the geometry and dynamics of diffeomorphisms on surfaces. Bull. Amer. Math. Soc. 19 (1988), 417–431.CrossRefGoogle Scholar
[22]Veech, W. A.. Teichmüller geodesic flow. Ann. of Math. (2) 124 (1986), 441–530.CrossRefGoogle Scholar
[23]Williams, R.. One dimensional non wandering sets. Topology 6 (1967), 473–487.CrossRefGoogle Scholar