Measure Theory of Self-Similar Groups and Digit Tiles

This dissertation is devoted to the measure theoretical aspects of the theory of automata and groups generated by them. It consists of two main parts. In the first part we study the action of automata on Bernoulli measures. We describe how a finite-state automorphism of a regular rooted tree changes the Bernoulli measure on the boundary of the tree. It turns out, that a finite-state automorphism of polynomial growth, as defined by Sidki, preserves a measure class of a Bernoulli measure, and we write down the explicit formula for its Radon-Nikodim derivative. On the other hand the image of the Bernoulli measure under the action of a strongly connected finite-state automorphism is singular to the measure itself. The second part is devoted to introduction of measure into the theory of limit spaces of Nekrashevysh. Let G be a group and φ : H → G be a contracting homomorphism from a subgroup H < G of finite index. Nekrashevych 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 selfsimilar measures m on these limit spaces. It is shown that (JG, s,m) is conjugate 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 evaluation of the Lebesgue measure of integral self-affine tiles. Previously the main tools in the theory of self-similar fractals were tools from measure theory and analysis. The methods developed in this disseration provide a new way to investigate self-similar and self-affine fractals, using combinatorics and group theory