# American Institute of Mathematical Sciences

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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

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

Received  October 2009 Revised  July 2010 Published  October 2010

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.
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.

show all references

##### 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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