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.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2010-12-8848 |
| Date | 2010 December 1900 |
| Creators | Kravchenko, Rostyslav |
| Contributors | Pisier, Gilles |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | application/pdf |
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