The action of finite-state tree automorphisms on Bernoulli measures

Rostyslav Kravchenko

doi:10.3934/jmd.2010.4.443

Article Contents

2010, Volume 4, Issue 3: 443-451. Doi: 10.3934/jmd.2010.4.443

This issue Previous Article Linear cocycles over hyperbolic systems and criteria of conformality Next Article The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis

The action of finite-state tree automorphisms on Bernoulli measures

Rostyslav Kravchenko^1,

1.
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368

Received: September 30, 2009

Revised: June 30, 2010

Published: September 2010

Abstract Related Papers Cited by

Abstract

We describe how a finite-state automorphism of a regular rooted tree changes the Bernoulli measure on the boundary of the tree. It turns out that a finite-state automorphism of polynomial growth, as defined by S. Sidki, preserves a measure class of a Bernoulli measure, and we write down the explicit formula for its Radon-Nikodym derivative. On the other hand, the image of the Bernoulli measure under the action of a strongly connected finite-state automorphism is singular to the measure itself.

Keywords:

Mathematics Subject Classification: Primary: 20E08.

Citation:

Related Papers

Cited by

References

[1]	L. Bartholdi and R. I. Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups, Tr. Mat. Inst. Steklova, 231 (2000), 5-45.
[2]	Laurent Bartholdi and Volodymyr Nekrashevych, Thurston equivalence of topological polynomials, Aca Math., 197 (2006), 1-51.doi: doi:10.1007/s11511-006-0007-3.
[3]	Patrick Billingsley, "Ergodic Theory and Information," Robert E. Krieger Publishing Co., Huntington, N.Y., 1978.
[4]	R. I. Grigorchuk, On the Milnor problem of group growth, Dokl. Akad. Nauk SSSR, 271 (1983), 30-33.
[5]	R. I. Grigorchuk, V. V. Nekrashevich and V. I. Sushchanskiĭ, Automata, dynamical systems, and groups, Tr. Mat. Inst. Steklova, 231 (2000), 134-214.
[6]	V. B. Kudryavtsev, S. V. Aleshin and A. S. Podkolzin, Vvedenie v teoriyu avtomatov, (Russian) [Introduction to automata theory], "Nauka," Moscow, 1985.
[7]	J. Milnor, Problem 5603, Amer. Math. Monthly, 75 (1968), 685-686,doi: doi:10.2307/2313822.
[8]	Volodymyr Nekrashevych, "Self-similar Groups," American Mathematical Society, 2005.
[9]	A. V. Ryabinin, Stochastic functions of finite automata, in "Algebra, Logic and Number Theory" (Russian), 77-80, Moskov. Gos. Univ., Moscow, 1986.
[10]	Said Sidki, Automorphisms of one-rooted trees: Growth, circuit structure, and acyclicity, J. Math. Sci. (New York), 100 (2000), 1925-1943.doi: doi:10.1007/BF02677504.
[11]	V. A. Ufnarovskii, A growth criterion for graphs and algebras defined by words, Math. Notes, 31 (1982), 238-241.doi: doi:10.1007/BF01145476.
[12]	Mariya Vorobets and Yaroslav Vorobets, On a free group of transformations defined by an automaton, Geom. Dedicata, 124 (2007), 237-249.doi: doi:10.1007/s10711-006-9060-5.