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

Abstract

Let G be a group and ϕ : H → G be a contracting homomorphism from a subgroup H < G of finite index. V. Nekrashevych (2005) [25] associated with the pair ( G , ϕ ) the limit dynamical system ( J G , s ) and the limit G -space X G together with the covering ⋃ g ∈ G T ⋅ g by the tile T . We develop the theory of self-similar measures m on these limit spaces. It is shown that ( J G , s , m ) is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile T 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 T ∩ ( T ⋅ g ) for g ∈ 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.