Symmetric random walks on \(\mathrm{Homeo}^+ (\mathbb{R})\). (English) Zbl 1300.60104

Deroin, B.; Kleptsyn, V.; Navas, A.; Parwani, K.

Ann. Probab. 41, No. 3B, 2066-2089 (2013).

Summary: We study symmetric random walks on finitely generated groups of orientation-preserving homeomorphisms of the real line. We establish an oscillation property for the induced Markov chain on the line that implies a weak form of recurrence. Except for a few special cases, which can be treated separately, we prove a property of “global stability at a finite distance”: roughly speaking, there exists a compact interval such that any two trajectories get closer and closer whenever one of them returns to the compact interval. The probabilistic techniques employed here lead to interesting results for the study of group actions on the line. For instance, we show that under a suitable change of the coordinates, the drift of every point becomes zero provided that the action is minimal. As a byproduct, we recover the fact that every finitely generated group of homeomorphisms of the real line is topologically conjugate to a group of (globally) Lipschitz homeomorphisms. Moreover, we show that such a conjugacy may be chosen in such a way that the displacement of each element is uniformly bounded.

MSC:

60J99	Markov processes
37H99	Random dynamical systems
06F15	Ordered groups

Keywords:

random walks; groups; homeomorphisms; real line; stationary measure

References:

[1]	Babillot, M., Bougerol, P. and Elie, L. (1997). The random difference equation \(X_{n}=A_{n}X_{n-1}+B_{n}\) in the critical case. Ann. Probab. 25 478-493. · Zbl 0873.60045 · doi:10.1214/aop/1024404297
[2]	Brofferio, S. (2003). How a centred random walk on the affine group goes to infinity. Ann. Inst. Henri Poincaré Probab. Stat. 39 371-384. · Zbl 1016.60006 · doi:10.1016/S0246-0203(02)00015-8
[3]	Chacon, R. V. and Ornstein, D. S. (1960). A general ergodic theorem. Illinois J. Math. 4 153-160. · Zbl 0134.12102
[4]	Deroin, B. (2005). Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive. Ann. Sci. École Norm. Sup. (4) 38 57-75. · Zbl 1070.37031 · doi:10.1016/j.ansens.2004.10.004
[5]	Deroin, B., Kleptsyn, V. and Navas, A. (2007). Sur la dynamique unidimensionnelle en régularité intermédiaire. Acta Math. 199 199-262. · Zbl 1139.37025 · doi:10.1007/s11511-007-0020-1
[6]	Deroin, B. and Navas, A. Groups, orders, and dynamics. Unpublished manuscript.
[7]	Derriennic, Y. (1984). Une condition suffisante de récurrence pour des chaînes de Markov sur la droite. In Probability Measures on Groups , VII ( Oberwolfach , 1983). Lecture Notes in Math. 1064 49-55. Springer, Berlin. · Zbl 0536.60075 · doi:10.1007/BFb0073633
[8]	Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41 45-76. · Zbl 0926.60056 · doi:10.1137/S0036144598338446
[9]	Fenley, S., Feres, R. and Parwani, K. (2009). Harmonic functions on \(\mathbb{R}\)-covered foliations. Ergodic Theory Dynam. Systems 29 1141-1161. · Zbl 1178.53027 · doi:10.1017/S0143385708000643
[10]	Fisher, D. (2011). Groups acting on manifolds: Around the Zimmer program. In Geometry , Rigidity , and Group Actions 72-157. Univ. Chicago Press, Chicago, IL. · Zbl 1264.22012 · doi:10.7208/chicago/9780226237909.001.0001
[11]	Garnett, L. (1983). Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. 51 285-311. · Zbl 0524.58026 · doi:10.1016/0022-1236(83)90015-0
[12]	Ghys, É. (2001). Groups acting on the circle. Enseign. Math. (2) 47 329-407. · Zbl 1044.37033
[13]	Glass, A. M. W. (1999). Partially Ordered Groups. Series in Algebra 7 . World Scientific, River Edge, NJ. · Zbl 0933.06010
[14]	Kellerer, H. G. (2006). Random dynamical systems on ordered topological spaces. Stoch. Dyn. 6 255-300. · Zbl 1108.60063 · doi:10.1142/S0219493706001797
[15]	Malyutin, A. V. (2008). Classification of the group actions on the real line and circle. St. Petersburg Math. J. 19 279-296. · Zbl 1209.37009 · doi:10.1090/S1061-0022-08-00999-0
[16]	Navas, A. (2010). A finitely generated, locally indicable group with no faithful action by \(C^{1}\) diffeomorphisms of the interval. Geom. Topol. 14 573-584. · Zbl 1197.37022 · doi:10.2140/gt.2010.14.573
[17]	Navas, A. (2011). Groups of Circle Diffeomorphisms . Univ. Chicago Press, Chicago, IL. · Zbl 1236.37002
[18]	Pólya, G. (1921). Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz. Math. Ann. 84 149-160. · JFM 48.0603.01 · doi:10.1007/BF01458701