Graph-directed systems and self-similar measures on limit spaces of self-similar groups |
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Authors: | Ievgen V. Bondarenko Rostyslav V. Kravchenko |
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Affiliation: | aNational Taras Shevchenko University of Kyiv, Mechanics and Mathematics Faculty, vul. Volodymyrska 64, 01033 Kyiv, Ukraine;bTexas A&M University, Department of Mathematics, College Station, TX 77843-3368, USA |
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Abstract: | Let G be a group and ?:H→G be a contracting homomorphism from a subgroup H<G of finite index. V. Nekrashevych (2005) [25] associated with the pair (G,?) 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 of the Lebesgue measure of integral self-affine tiles. |
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Keywords: | MSC: primary, 28A80, 20F65 secondary, 37B10, 37D40 |
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