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Markovian random iterations of homeomorphisms of the circle

Part of: Ergodic theory Measure-theoretic ergodic theory Low-dimensional dynamical systems

Published online by Cambridge University Press:  21 July 2021

EDGAR MATIAS Open the ORCID record for EDGAR MATIAS [Opens in a new window]
EDGAR MATIAS*
Affiliation:
Departamento de Matemática, Universidade Federal da Bahia, Av. Adhemar de Barros s/n, Salvador40170-110, Brazil
*
e-mail: edgar.matias@ufba.br
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Abstract

In this paper we prove a local exponential synchronization for Markovian random iterations of homeomorphisms of the circle $S^{1}$ , providing a new result on stochastic circle dynamics even for $C^1$ -diffeomorphisms. This result is obtained by combining an invariance principle for stationary random iterations of homeomorphisms of the circle with a Krylov–Bogolyubov-type result for homogeneous Markov chains.

Keywords

random iteration of mapssynchronizationstationary measureskew productinvariant measure

MSC classification

Primary: 37E10: Maps of the circle
Secondary: 37A50: Relations with probability theory and stochastic processes 28D05: Measure-preserving transformations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 9 , September 2022 , pp. 2935 - 2956
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Alves, J. F. and Araújo, V.. Random perturbations of nonuniformly expanding maps. Geometric methods in dynamics. I. Astérisque 286 (2003), xvii, 25–62.Google Scholar
Antonov, V. A.. Modelling of processes of cyclic evolution type. Synchronization by a random signal. Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 2 (1984), 67–76.Google Scholar
Araújo, V.. Attractors and time averages for random maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 17(3) (2000), 307–369.CrossRefGoogle Scholar
Araújo, V. and Aytaç, H.. Decay of correlations and laws of rare events for transitive random maps. Nonlinearity 30(5) (2017), 1834–1852.CrossRefGoogle Scholar
Arnold, L.. Random Dynamical Systems (Springer Monographs in Mathematics ). Springer-Verlag, Berlin, 1998.Google Scholar
Ávila, A. and Viana, M.. Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math. 181(1) (2010), 115–178.CrossRefGoogle Scholar
Barnsley, M. F., Elton, J. H. and Hardin, D. P.. Recurrent iterated function systems. Constr. Approx. 5(1) (1989), 3–31.CrossRefGoogle Scholar
Bocker-Neto, C. and Viana, M.. Continuity of Lyapunov exponents for random two-dimensional matrices. Ergod. Th. & Dynam. Sys. 37(5) (2017), 1413–1442.CrossRefGoogle Scholar
Breiman, L.. The strong law of large numbers for a class of Markov chains. Ann. Math. Statist. 31 (1960), 801–803.CrossRefGoogle Scholar
Crauel, H.. Extremal exponents of random dynamical systems do not vanish. J. Dynam. Differ. Equat. 2(3) (1990), 245–291.CrossRefGoogle Scholar
Crauel, H.. Markov measures for random dynamical systems. Stoch. Stoch. Rep. 37(3) (1991), 153–173.CrossRefGoogle Scholar
Crauel, H.. Random Probability Measures on Polish Spaces (Stochastics Monographs, 11). Taylor & Francis, London, 2002.CrossRefGoogle Scholar
Díaz, L. J. and Matias, E.. Stability of the Markov operator and synchronization of Markovian random products. Nonlinearity 31(5) (2018), 1782–1806.CrossRefGoogle Scholar
Edalat, A.. Power domains and iterated function systems. Inform. Comput. 124(2) (1996), 182–197.CrossRefGoogle Scholar
Furstenberg, H.. Noncommuting random products. Trans. Amer. Math. Soc. 108 (1963), 377–428.CrossRefGoogle Scholar
Hairer, M.. Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Probab. 33(2) (2005), 703–758.CrossRefGoogle Scholar
Hairer, M. and Ohashi, A.. Ergodic theory for SDEs with extrinsic memory. Ann. Probab. 35(5) (2007), 1950–1977.CrossRefGoogle Scholar
Kakutani, S.. Random ergodic theorems and Markov processes with a stable distribution. Proc. 2nd Berkeley Symp on Mathematical Statistics and Probability. Ed. Neyman, J.. University of California Press, Berkeley, CA. 1951.Google Scholar
Kifer, Y.. Ergodic Theory of Random Transformations (Progress in Probability and Statistics, 10). Birkhäuser, Boston, 1986.CrossRefGoogle Scholar
Kleptsyn, V. and Volk, D.. Physical measures for nonlinear random walks on interval. Mosc. Math. J. 14(2) (2014), 339–365, 428.CrossRefGoogle Scholar
Kleptsyn, V. A. and Nal’skii, M. B.. Convergence of orbits in random dynamical systems on a circle. Funktsional. Anal. i Prilozhen. 38(4) (2004), 36–54, 95–96.CrossRefGoogle Scholar
Ledrappier, F.. Positivity of the exponent for stationary sequences of matrices. Lyapunov Exponents (Bremen, 1984) (Lecture Notes on Mathematics, 1186). Springer, Berlin, 1986, pp. 565–73.Google Scholar
Malheiro, E. C. and Viana, M.. Lyapunov exponents of linear cocycles over Markov shifts. Stoch. Dyn. 15(3) (2015), 1550020.CrossRefGoogle Scholar
Malicet, D.. Random walks on $\mathrm{Homeo} \left({S}^1\right)$ . Comm. Math. Phys. 356(3) (2017), 1083–1116.CrossRefGoogle Scholar
Marie, P. and Rousseau, J. Recurrence for random dynamical systems. Discrete Contin. Dyn. Syst. 30(1) (2011), 1–16.CrossRefGoogle Scholar
Ohno, T.. Asymptotic behaviours of dynamical systems with random parameters. Publ. R.I.M.S. Kyoto Univ. 19 (1983), 83–98.CrossRefGoogle Scholar
Revuz, D.. Markov Chains (North-Holland Mathematical Library, 11), 2nd edn. North-Holland Publishing Co., Amsterdam, 1984.Google Scholar
Rokhlin, V. A. On the fundamental ideas of measure theory. Amer. Math. Soc. Transl. 1952(71) (1952), 55pp.Google Scholar
Rudin, W.. Real and Complex Analysis. McGraw-Hill, New York, 1986.Google Scholar
Szarek, T. and Zdunik, A.. The central limit theorem for function systems on the circle. Mosc. Math. J. 1(21) (2021), 175–190.CrossRefGoogle Scholar
Viana, M.. Lectures on Lyapunov exponents (Cambridge Studies in Advanced Mathematics, 145). Cambridge University Press, Cambridge, 2014.CrossRefGoogle Scholar