Ergodic decomposition of group actions on rooted trees
- 28 Downloads
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.
KeywordsSTEKLOV Institute Rooted Tree Ergodic Component Ergodic Decomposition Uniform Probability Measure
Unable to display preview. Download preview PDF.
- 10.S. V. Fomin, “On measures invariant under a group of transformations,” Izv. Akad. Nauk SSSR, Ser. Mat. 14 (3), 261–274 (1950) [Am. Math. Soc. Transl., Ser. 2, 51, 317–332 (1966)].Google Scholar
- 14.R. I. Grigorchuk, “Just infinite branch groups,” in New Horizons in Pro-p Groups (Birkhäuser, Boston, MA, 2000), Prog. Math. 184, pp. 121–179.Google Scholar
- 17.R. Grigorchuk and P. de la Harpe, “Amenability and ergodic properties of topological groups: From Bogolyubov onwards,” arxiv: 1404.7030 [math.GR].Google Scholar
- 24.I. Klimann, “The finiteness of a group generated by a 2-letter invertible–reversible Mealy automaton is decidable,” in Proc. 30th Int. Symp. on Theoretical Aspects of Computer Science (STACS 2013), Ed. by N. Portier and T. Wilke (Schloss Dagstuhl–Leibniz-Zent. Inform., Wadern, 2013), Leibniz Int. Proc. Inform. 20, pp. 502–513.Google Scholar
- 25.I. Klimann, M. Picantin, and D. Savchuk, “A connected 3-state reversible Mealy automaton cannot generate an infinite Burnside group,” in Developments in Language Theory: Proc. 19th Int. Conf., DLT 2015, Liverpool, 2015 (Springer, Cham, 2015), Lect. Notes Comput. Sci. 9168, pp. 313–325; arxiv: 1409.6142 [cs.FL].CrossRefGoogle Scholar
- 32.D. M. Savchuk and S. N. Sidki, “Affine automorphisms of rooted trees,” Geom. Dedicata (in press); arXiv: 1510.08434 [math.GR].Google Scholar