The action of finite-state tree automorphisms on Bernoulli measures

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July 2010, 4(3): 443-451. doi: 10.3934/jmd.2010.4.443

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, United States

Received October 2009 Revised July 2010 Published October 2010

Abstract
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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: regular rooted tree., Bernoulli measure, Markov chain, finite automata.

Mathematics Subject Classification: Primary: 20E0.

Citation: Rostyslav Kravchenko. The action of finite-state tree automorphisms on Bernoulli measures. Journal of Modern Dynamics, 2010, 4 (3) : 443-451. doi: 10.3934/jmd.2010.4.443

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.

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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.

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