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Invariant rigid geometric structures and expanding maps

Published online by Cambridge University Press:  06 May 2011

YONG FANG
YONG FANG*
Affiliation:
Département de Mathématiques, Université de Cergy-Pontoise, avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France (email: yfang@math.u-cergy.fr)
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Abstract

In the first part of this paper, we consider several natural problems about locally homogeneous rigid geometric structures. In particular, we formulate a notion of topological completeness which is adapted to the study of global rigidity of chaotic dynamical systems. In the second part of the paper, we prove the following result: let φ be a C∞ expanding map of a closed manifold. If φ preserves a topologically complete C∞ rigid geometric structure, then φ is C∞ conjugate to an expanding infra-nilendomorphism.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 3 , June 2012 , pp. 941 - 959
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Amore, A. M.. Vector fields of a finite type G-structure. J. Differential Geom. 14 (1979), 1–6.Google Scholar
[2]Benoist, Y.. Orbites des structures rigides (d’après M. Gromov) (Progress in Mathematics, 145). Birkhäuser, Boston, 1997, pp. 1–17.Google Scholar
[3]Benoist, Y.. Réseaux des groupes de Lie, Cours de M2, Université de Paris VI, 2008.Google Scholar
[4]Benoist, Y., Foulon, P. and Labourie, F.. Fots d’Anosov à distributions stable et instable différentiables. J. Amer. Math. Soc. 5 (1992), 33–74.Google Scholar
[5]Benoist, Y. and Labourie, F.. Sur les difféomorphismes d’Anosov affines à feuilletages stable et instable différentiables. Invent. Math. 111 (1993), 285–308.Google Scholar
[6]Candel, A. and Quiroga-Barranco, R.. Gromov’s centralizer theorem. Geom. Dedicata 100 (2003), 123–155.Google Scholar
[7]Candel, A. and Quiroga-Barranco, R.. Parallelisms, prolongations of Lie algebras and rigid goemetric structures. Manuscripta Math. 114 (2004), 335–350.CrossRefGoogle Scholar
[8]Chevalley, C.. Théorie des groupes de Lie. Hermann, Paris, 1968.Google Scholar
[9]Fang, Y.. Structures géométriques rigides et systèmes dynamiques hyperboliques. PhD Thesis, Université de Paris-Sud, 2004, Web address: http://www.u-cergy.fr/rech/pages/yfang/index.html.Google Scholar
[10]Farrell, F. T. and Jones, L. E.. Examples of expanding endomorphisms on exotic tori. Invent. Math. 45 (1978), 175–179.CrossRefGoogle Scholar
[11]Feres, R.. Hyperbolic dynamical systems, invariant geometric structures, and rigidity. Math. Res. Lett. 1 (1994), 11–26.CrossRefGoogle Scholar
[12]Feres, R.. A differential-geometric view of normal forms of contractions. Modern Dynamical Systems and Applications. Cambridge University Press, Cambridge, 2004, pp. 103–121.Google Scholar
[13]Feres, R.. Rigid geometric structures and actions of semisimple Lie groups. Rigidité, groupe fondamental et dynamique (Panoramas et Synthèses, 13). Ed. Foulon, P.. Société Mathématiques de France, Paris, 2002.Google Scholar
[14]Gromov, M. and D’ambra, G.. Lectures on transformation groups: geometry and dynamics. Surveys in Differential Geometry. Lehigh University, Bethlehem, 1991, pp. 19–111.Google Scholar
[15]Gromov, M.. Rigid transformation groups. Géométrie différentielle (Travaux en Cours, 33). Hermann, Paris, 1988.Google Scholar
[16]Gromov, M.. Groups of polynomial growth and expanding maps. Publ. Math. Inst. Hautes Études Sci. 53 (1981), 53–73.Google Scholar
[17]Guysinski, M. and Katok, A.. Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations. Math. Res. Lett. 5 (1998), 149–163.Google Scholar
[18]Hasselblatt, B.. Problems in dynamical systems and related topics raised in connection with the Clay Mathematics Institute workshop on ‘Recent Progress in Dynamics’.Google Scholar
[19]Kobayashi, S. and Nomizu, K.. Foundations of Differential Geometry, Vol. I (Interscience Tracts in Pure and Applied Mathematics, 15). Wiley-Interscience, New York, 1996.Google Scholar
[20]Shub, M.. Endomorphisms of compact differentiable manifolds. Amer. J. Math. 91 (1969), 175–199.Google Scholar
[21]Springer, T. A.. Linear Algebraic Groups, 2nd edn(Progress in Mathematics, 9). Birkhäuser, Boston, 1998.Google Scholar
[22]Witte-Morris, D.. Ratner’s Theorems on Unipotent Flows (Chicago Lecture Notes in Mathematics). University of Chicago Press, Chicago, 2005.Google Scholar
[23]Zeghib, A.. Sur les groupes de transformations rigides: théorème de l’orbite dense-ouverte. Rigidité, groupe fondamental et dynamique (Panoramas et Synthèses, 13). Ed. Foulon, P.. Société Mathématiques de France, Paris, 2002.Google Scholar