Abstract
In this paper, we study random walks \({g_n=f_{n-1}\ldots f_0}\) on the group Homeo (S 1) of the homeomorphisms of the circle, where the homeomorphisms f k are chosen randomly, independently, with respect to a same probability measure \({\nu}\). We prove that under the only condition that there is no probability measure invariant by \({\nu}\)-almost every homeomorphism, the random walk almost surely contracts small intervals. It generalizes what has been known on this subject until now, since various conditions on \({\nu}\) were imposed in order to get the phenomenon of contractions. Moreover, we obtain the surprising fact that the rate of contraction is exponential, even in the lack of assumptions of smoothness on the f k ’s. We deduce various dynamical consequences on the random walk (g n ): finiteness of ergodic stationary measures, distribution of the trajectories, asymptotic law of the evaluations, etc. The proof of the main result is based on a modification of the Ávila-Viana’s invariance principle, working for continuous cocycles on a space fibred in circles.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Antonov, V.A.: Modeling Cyclic Evolution Processes: Synchronization by Means of a Random Signal. Leningradskii Universitet Vestnik Matematika Mekhanika Astronomiia, pp. 67–76 (1984)
Ávila A., Viana M.: Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math. 181(1), 115–178 (2010)
Baxendale P.H.: Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms. Probab. Theory Relat. Fields 81(4), 521–554 (1989)
Crauel H.: Extremal exponents of random dynamical systems do not vanish. J. Dyn. Differ. Equ. 2(3), 245–291 (1990)
Deroin, B.: Propriétés ergodiques des groupes de difféomorphismes du cercle par rapporta la mesure de lebesgue. Course, disponible at http://www.math.ens.fr/~deroin/Publications/coursneuchatel.pdf
Deroin B., Kleptsyn V., Navas A.: Sur la dynamique unidimensionnelle en régularité intermédiaire. Acta Math. 199(2), 199–262 (2007)
Deroin B., Kleptsyn V., Navas A., Parwani K.: Symmetric random walks on \({{\rm Homeo}(\mathbb{R})}\). Ann. Probab. 41(3B), 2066–2089 (2013)
Furman A.: Random walks on groups and random transformations. Handb. Dyn. Syst. 1, 931–1014 (2002)
Furstenberg, H.: Noncommuting random products. Trans. Am. Math. Soc. 108, 377–428 (1963)
Gorodetski A., Kleptsyn V.: Synchronization properties of random piecewise isometries. Commun. Math. Phys. 345(3), 781–796 (2016)
Homburg, A.J.: Synchronization in iterated function systems. arXiv preprint arXiv:1303.6054 (2013)
Hu H.: Dimensions of invariant sets of expanding maps. Commun. Math. Phys. 176(2), 307–320 (1996)
Kaijser T.: On stochastic perturbations of iterations of circle maps. Phys. D Nonlinear Phenomena 68(2), 201–231 (1993)
Kifer Y.: Ergodic Theory of Random Transformations. Springer, Berlin (1986)
Kleptsyn V.A., Nalskii M.B.: Contraction of orbits in random dynamical systems on the circle. Funct. Anal. Appl. 38(4), 267–282 (2004)
Kudryashov Y.G.: Bony attractors. Funct. Anal. Appl. 44(3), 219–222 (2010)
Le Jan, Y.: Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants. In: Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques, vol. 23, pp. 111–120. Gauthier-Villars (1987)
Ledrappier, F.: Positivity of the exponent for stationary sequences of matrices. In: Lyapunov Exponents, pp. 56–73. Springer (1986)
Ledrappier F.: Some relations between dimension and lyapounov exponents. Commun. Math. Phys. 81(2), 229–238 (1981)
Navas A.: Groups of Circle Diffeomorphisms. University of Chicago Press, Chicago (2011)
Young L.-S.: Dimension, entropy and lyapunov exponents. Ergodic Theory Dyn. Syst. 2(01), 109–124 (1982)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Liverani
Rights and permissions
About this article
Cite this article
Malicet, D. Random Walks on Homeo(S 1). Commun. Math. Phys. 356, 1083–1116 (2017). https://doi.org/10.1007/s00220-017-2996-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-017-2996-5