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The Poisson boundary of a locally discrete group of diffeomorphisms of the circle

Published online by Cambridge University Press:  12 March 2012

BERTRAND DEROIN
BERTRAND DEROIN*
Affiliation:
Université Paris-Sud & CNRS, Laboratoire de Mathématiques, Bât 425, 91405 Orsay Cedex, France (email: [email protected])
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Abstract

We compute the Poisson boundary of locally discrete groups of diffeomorphisms of the circle.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013 , pp. 400 - 415
Copyright
©2012 Cambridge University Press

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References

[1]Antonov, V.. Model of processes of cyclic evolution type. Synchronisation by a random signal. Vestn. Leningr. Univ. Ser. Mat. Mekh. Astron. 2(7) (1984), 67–76.Google Scholar
[2]Avez, A.. Entropie des groupes de type fini. C. R. Acad. Sci. Paris Sér. A–B 275 (1972), 1363–1366.Google Scholar
[3]Baxendale, P.. Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms. Probab. Theory Related Fields 81 (1989), 521–554.CrossRefGoogle Scholar
[4]Chaperon, M.. Invariant manifolds revisited. Proc. Steklov Inst. Math. 236(1) (2002), 415–433.Google Scholar
[5]Deroin, B. and Kleptsyn, V.. Random conformal dynamical systems. Geom. Funct. Anal. 17(4) (2007), 1043–1105.Google Scholar
[6]Deroin, B., Kleptsyn, V. and Navas, A.. Sur la dynamique unidimensionnelle en régularité intermédiaire. Acta Math. 199 (2007), 199–262.Google Scholar
[7]Derriennic, Y.. Entropie, Théorèmes Limite, et Marches Aléatoire (Lecture Notes in Mathematics, 1210). Springer, Berlin, 1986.Google Scholar
[8]Ghys, É.. Sur les groupes engendrés par des difféomorphismes proches de l’identité. Bull. Braz. Math. Soc. (N.S.) 24(2) (1993), 137–178.Google Scholar
[9]Ghys, É.. Groups acting on the circle. Enseign. Mat. 47 (2001), 329–407.Google Scholar
[10]Ghys, É. and Sergiescu, V.. Sur un groupe remarquable de difféomorphismes du cercle. Comment. Math. Helv. 62(2) (1987), 185–239.Google Scholar
[11]Furstenberg, H.. Boundary Theory and Stochastic Processes on Homogeneous Spaces (Proceedings of Symposia in Pure Mathematics, 26). American Mathematical Society, Providence, RI, 1973, pp. 193–229.Google Scholar
[12]Kaimanovich, V.. The Poisson formula for groups with hyperbolic properties. Ann. of Math. (2) 152 (2000), 659–692.Google Scholar
[13]Kaimanovich, V. and Vershik, A.. Random walks on discrete groups: boundary and entropy. Ann. Probab. 11(3) (1983), 457–490.CrossRefGoogle Scholar
[14]Kleptsyn, V. and Nal’ski, M.. Convergence of orbits in random dynamical systems on the circle. Funct. Anal. Appl. 38(4) (2004), 267–282.Google Scholar
[15]Nakai, I.. Separatrices for non solvable dynamics on (C,0). Ann. Inst. Fourier 4(2) (1994), 569–599.Google Scholar
[16]Ledrappier, F.. Quelques propriétés des exposants caractéristiques. École d’été de St Flour XII – 1982 (Lecture Notes in Mathematics, 1097). Springer, Berlin, 1984, pp. 305–396.Google Scholar
[17]Ledrappier, F.. Une relation entre entropie, dimension et exposant pour certaines marches aléatoires. C. R. Acad. Sci. Sér. I Math. 296(8) (1983), 369–372.Google Scholar
[18]Loray, F. and Rebelo, J.. Minimal rigid foliations by curves of CP n. J. Eur. Math. Soc. 5 (2003), 147–201.CrossRefGoogle Scholar
[19]Malliavin, P.. The canonic diffusion above the diffeomorphism group of the circle. C. R. Acad. Sci. Sér. I Math. 329 (1999), 325–329.Google Scholar
[20]Navas, A.. Groups of Circle Diffeomorphisms (Chicago Lectures in Mathematics). Chicago University Press, Chicago, 2010.Google Scholar
[21]Rebelo, J.. A theorem of measurable rigidity in Diffω(S 1). Ergod. Th. & Dynam. Sys. 21(5) (2001), 1525–1561.Google Scholar
[22]Sternberg, S.. Local contractions and a theorem of Poincaré. Amer. J. Math. 79 (1957), 809–824.Google Scholar
[23]Yoccoz, J. C.. Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Petits diviseurs en dimension 1. Astérisque 231 (1995), 89–242.Google Scholar