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 Tr. Mat. Inst. Steklova, 2016, Volume 292, Pages 100–117 (Mi tm3685)

Ergodic decomposition of group actions on rooted trees

Rostislav Grigorchuka, Dmytro Savchukb

a Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
b Department of Mathematics and Statistics, University of South Florida, 4202 East Fowler Ave., Tampa, FL 33620-5700, USA

Abstract: We prove a general result about the decomposition into ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree associated with the action, and show that the canonical system of ergodic invariant probability measures coincides with the system of uniform measures on the boundaries of minimal invariant subtrees of the tree. Special attention is paid to the case of groups generated by finite automata. Few examples, including the lamplighter group, Sushchansky group, and so-called universal group, are considered in order to demonstrate applications of the theorem.

 Funding Agency Grant Number National Science Foundation DMS-1207699 University of South Florida Simons Foundation #317198 The first author was partially supported by the NSF grant DMS-1207699. The second author was partially supported by the New Researcher Grant and Proposal Enhancement Grant from the USF Internal Awards Program, and also by Simons Collaboration Grant #317198 from the Simons Foundation.

DOI: https://doi.org/10.1134/S0371968516010064

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 292, 94–111

Bibliographic databases:

UDC: 512+517.98+519.1
Received: December 30, 2014
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Citation: Rostislav Grigorchuk, Dmytro Savchuk, “Ergodic decomposition of group actions on rooted trees”, Algebra, geometry, and number theory, Collected papers. Dedicated to Academician Vladimir Petrovich Platonov on the occasion of his 75th birthday, Tr. Mat. Inst. Steklova, 292, MAIK Nauka/Interperiodica, Moscow, 2016, 100–117; Proc. Steklov Inst. Math., 292 (2016), 94–111

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3685
• https://doi.org/10.1134/S0371968516010064
• http://mi.mathnet.ru/eng/tm/v292/p100

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This publication is cited in the following articles:
1. I. Klimann, M. Picantin, D. Savchuk, “A connected 3-state reversible Mealy automaton cannot generate an infinite Burnside group”, Developments in language theory, Lecture Notes in Comput. Sci., 9168, Springer, Cham, 2015, 313–325
2. I. Klimann, M. Picantin, D. Savchuk, “A connected 3-state reversible Mealy automaton cannot generate an infinite Burnside group”, Internat. J. Found. Comput. Sci., 29:2 (2018), 297–314
3. T. Godin, I. Klimann, “On bireversible Mealy automata and the Burnside problem”, Theoret. Comput. Sci., 707 (2018), 24–35
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