Geodesic
Ergodic decomposition of group actions on rooted trees
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, geometry, and number theory, Tome 292 (2016), pp. 100-117

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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. 
@article{TM_2016_292_a5,
     author = {Rostislav Grigorchuk and Dmytro Savchuk},
     title = {Ergodic decomposition of group actions on rooted trees},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {100--117},
     publisher = {mathdoc},
     volume = {292},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2016_292_a5/}
}
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Rostislav Grigorchuk; Dmytro Savchuk. Ergodic decomposition of group actions on rooted trees. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, geometry, and number theory, Tome 292 (2016), pp. 100-117. http://geodesic.mathdoc.fr/item/TM_2016_292_a5/