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Ergodic decomposition of group actions on rooted trees

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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.

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Authors and Affiliations

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

    Rostislav Grigorchuk

  2. Department of Mathematics and Statistics, University of South Florida, 4202 East Fowler Ave., Tampa, FL, 33620-5700, USA

    Dmytro Savchuk

Authors
  1. Rostislav Grigorchuk
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  2. Dmytro Savchuk
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Correspondence to Rostislav Grigorchuk.

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Dedicated to Vladimir Petrovich Platonov on the occasion of his 75th birthday

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Grigorchuk, R., Savchuk, D. Ergodic decomposition of group actions on rooted trees. Proc. Steklov Inst. Math. 292, 94–111 (2016). https://doi.org/10.1134/S0081543816010065

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