Graph-directed systems and self-similar measures on limit spaces of self-similar groups

Let G be a group and empty set : H -> G be a contracting homomorphism from a subgroup H < G of finite index. V. Nekrashevych (2005) [25] associated with the pair (G, empty set) the limit dynamical system (partial derivative(G), s) and the limit G-space chi(G) together with the covering boolean OR(g epsilon G) tau center dot g by the tile tau. We develop the theory of self-similar measures m on these limit spaces. It is shown that (partial derivative(G), s, m) is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile tau has integer measure and we give an algorithmic way to compute it. In addition we give an algorithm to find the measure of the intersection of tiles tau boolean AND (tau center dot g) for g epsilon G. We present applications to the invariant measures for the rational functions on the Riemann sphere and to the evaluation of the Lebesgue measure of integral self-affine tiles. (c) 2010 Elsevier Inc. All rights reserved.