Ends of Schreier graphs and cut-points of limit spaces of self-similar groups

Every self-similar group acts on the space X-omega of infinite words over some alphabet X. We study the Schreier graphs Gamma(w) for w is an element of X-omega of the action of self-similar groups generated by bounded automata on the space X-omega. Using sofic subshifts we determine the number of ends for every Schreier graph. w. Almost all Schreier graphs Gamma(w) with respect to the uniform measure on X-omega have one or two ends, and we characterize bounded automata whose Schreier graphs have two ends almost surely. The connection with (local) cut-points of limit spaces of self-similar groups is established.