Lyapunov exponents and other properties of $N$-groups
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D. A. Filimonov and V. A. Kleptsyn
Translated by: G. G. Gould - Trans. Moscow Math. Soc. 2012, 29-36
- DOI: https://doi.org/10.1090/S0077-1554-2013-00198-3
- Published electronically: January 24, 2013
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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
- Rufus Bowen, Invariant measures for Markov maps of the interval, Comm. Math. Phys. 69 (1979), no. 1, 1â17. MR 547523
- A. B. Katok and B. Hasselblatt, Introduction to the theory of dynamical systems with a survey of latest achievements, Cambridge University Press, Cambridge, 2006.
- 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.
- Michael-Robert Herman, Sur la conjugaison diffĂ©rentiable des diffĂ©omorphismes du cercle Ă des rotations, Inst. Hautes Ătudes Sci. Publ. Math. 49 (1979), 5â233 (French). MR 538680
- Bertrand Deroin, Victor Kleptsyn, and AndrĂ©s Navas, On the question of ergodicity for minimal group actions on the circle, Mosc. Math. J. 9 (2009), no. 2, 263â303, back matter (English, with English and Russian summaries). MR 2568439, DOI 10.17323/1609-4514-2009-9-2-263-303
- Bertrand Deroin, Victor Kleptsyn, and AndrĂ©s Navas, Sur la dynamique unidimensionnelle en rĂ©gularitĂ© intermĂ©diaire, Acta Math. 199 (2007), no. 2, 199â262 (French). MR 2358052, DOI 10.1007/s11511-007-0020-1
- Ătienne Ghys and Vlad Sergiescu, Sur un groupe remarquable de diffĂ©omorphismes du cercle, Comment. Math. Helv. 62 (1987), no. 2, 185â239 (French). MR 896095, DOI 10.1007/BF02564445
- Y. Guivarcâh and Y. Le Jan, Asymptotic winding of the geodesic flow on modular surfaces and continued fractions, Ann. Sci. Ăcole Norm. Sup. (4) 26 (1993), no. 1, 23â50. MR 1209912
- Y. GuivarcâČh and C. R. E. Raja, Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces, Preprint arXiv:0908.0637.
- S. Hurder, Exceptional minimal sets and the GodbillonâVey class, Ann. Inst. Fourier (Grenoble), to appear.
- Tomoki Inoue, Ratio ergodic theorems for maps with indifferent fixed points, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 625â642. MR 1452184, DOI 10.1017/S0143385797084952
- Ricardo Mañé, Introdução à teoria ergódica, Projeto Euclides [Euclid Project], vol. 14, Instituto de Matemåtica Pura e Aplicada (IMPA), Rio de Janeiro, 1983 (Portuguese). MR 800092
Bibliographic Information
- D. A. Filimonov
- Affiliation: Moscow State University
- Email: mityafil@gmail.com
- V. A. Kleptsyn
- Affiliation: CNRS, Institut de Recherche Mathématique de Rennes
- MR Author ID: 751650
- Email: victor.kleptsyn@univ-rennes1.fr
- Published electronically: January 24, 2013
- Additional Notes: This work was carried out with the partial support of the Russo-French programme âCooperation network in mathematicsâ, grant RFFI-10-01-00739-a and grant RFFI-CNRS-10-01-93115-NTsNIL-a.
- © Copyright 2013 American Mathematical Society
- 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
- MathSciNet review: 3184966