Graph-directed systems and self-similar measures on limit spaces of self-similar groups
- Bondarenko, Ievgen V.
- Kravchenko, Rostyslav V.
DOIAbstract
AbstractLet 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 (JG,s) and the limit G-space XG together with the covering ⋃g∈GT⋅g by the tile T. We develop the theory of self-similar measures m on these limit spaces. It is shown that (JG,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 o...
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