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Transactions of the Moscow Mathematical Society

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Lyapunov exponents and other properties of $ N$-groups


Authors: D. A. Filimonov and V. A. Kleptsyn
Translated by: G. G. Gould
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva.
Journal: Trans. Moscow Math. Soc. 2012, 29-36
MSC (2010): Primary 37C85; Secondary 37E10, 37A35, 37D25, 37H15
DOI: https://doi.org/10.1090/S0077-1554-2013-00198-3
Published electronically: January 24, 2013
MathSciNet review: 3184966
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the class of minimally acting finitely generated groups of $ C^2$-diffeomorphisms of the circle which have the property that the nonexpandable points are fixed, where the set of nonexpandable points is nonempty. It turns out that the Lyapunov expansion exponent of any such action is zero. As a consequence, we have a singularity of the stationary measure for a random dynamical system given by any probability distribution whose support is a finite set of the generating elements of the group.


References [Enhancements On Off] (What's this?)

  • 1. R. Bowen, Invariant measures for Markov maps on the interval, Comm. Math. Phys. 69 (1979), 1-17. MR 547523 (81e:28010)
  • 2. A. B. Katok and B. Hasselblatt, Introduction to the theory of dynamical systems with a survey of latest achievements, Cambridge University Press, Cambridge, 2006.
  • 3. V. A. Kleptsyn and D. A. Filimonov, On actions on the circle with the fixed-point property for nonexpandable points, Funktsional. Anal. i Prilozhen, to appear.
  • 4. M. Herman, Sur la conjugaison différentiable des difféomorphismes du circle à des rotations, Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5-234. MR 538680 (81h:58039)
  • 5. B. Deroin, V. Kleptsyn and A. Navas, On the question of ergodicity for minimal group actions on the circle, Moscow Math. J. 9 (2009), no. 2, 263-303. MR 2568439 (2010m:37041)
  • 6. B. Deroin, V. Kleptsyn and A. Navas, Sur la dynamique unidimensionelle en régularité intermédiaire, Acta Math. 199 (2007), no. 2, 199-262. MR 2358052 (2010c:37059)
  • 7. É. Ghys and V. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv. 62 (1987), 185-239. MR 896095 (90c:57035)
  • 8. Y. Guivarc'h and Y. Le Jan, Asymptotic winding of the geodesic flow on modular surfaces and continuous fractions, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 1, 23-50. MR 1209912 (94a:58157)
  • 9. Y. Guivarc'h and C. R. E. Raja, Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces, Preprint arXiv:0908.0637.
  • 10. S. Hurder, Exceptional minimal sets and the Godbillon-Vey class, Ann. Inst. Fourier (Grenoble), to appear.
  • 11. T. Inoue, Ratio ergodic theorems for maps with indifferent fixed points, Ergodic Theory Dynam. Systems 17 (1997), 625-642. MR 1452184 (98e:58109)
  • 12. M. Mañé, Introducäo à teoria ergódica, Instituto de Matemática Pura e Aplicada (1983). MR 800092 (87d:58085)

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Additional Information

D. A. Filimonov
Affiliation: Moscow State University
Email: mityafil@gmail.com

V. A. Kleptsyn
Affiliation: CNRS, Institut de Recherche Mathématique de Rennes
Email: victor.kleptsyn@univ-rennes1.fr

DOI: https://doi.org/10.1090/S0077-1554-2013-00198-3
Keywords: Dynamical systems, group actions, diffeomorphisms of the circle, Lyapunov exponent, stationary measures.
Published electronically: January 24, 2013
Additional Notes: This work was carried out with the partial support of the Russo-French programme “Cooperation network in mathematics”, grant RFFI-10-01-00739-a and grant RFFI-CNRS-10-01-93115-NTsNIL-a.
Article copyright: © Copyright 2013 American Mathematical Society

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