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Let $G$ be a group and $\phi:H\to G$ be a contracting homomorphism from asubgroup $H<G$ 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 limitspaces. It is shown that $(\lims,\si,\mu)$ is conjugated to the one-sidedBernoulli shift. Using sofic subshifts we prove that the tile $\tile$ hasinteger measure and we give an algorithmic way to compute it. In addition wegive 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 measuresfor the rational functions on the Riemann sphere and to the evaluation of theLebesgue measure of integral self-affine tiles.

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