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

Let $G$ be a group and $\phi:H\to G$ be a contracting homomorphism from a subgroup $H<G$ of finite index. V.Nekrashevych [25] associated with the pair $(G,\phi)$ the limit dynamical system $(\lims,\si)$ and the limit $G$-space $\limGs$ together with the covering $\cup_{g\in G}\tile\cdot g$ by the tile $\tile$. We develop the theory of self-similar measures $\mu$ on these limit spaces. It is shown that $(\lims,\si,\mu)$ is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile $\tile$ 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 $\tile\cap (\tile\cdot g)$ for $g\in G$. 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.