This dissertation is devoted to groups generated by bounded automata and
geometric objects related to these groups (limit spaces, Schreier graphs, etc.).
It is shown that groups generated by bounded automata are contracting. We
introduce the notion of a post-critical set of a finite automaton and prove that the
limit space of a contracting self-similar group generated by a finite automaton is
post-critically finite (finitely-ramified) if and only if the automaton is bounded.
We show that the Schreier graphs on levels of automaton groups can be
constructed by an iterative procedure of inflation of graphs. This was used to associate
a piecewise linear map of the form fK(v) = minA[set]KAv, where K is a finite set of
nonnegative matrices, with every bounded automaton. We give an effective criterium
for the existence of a strictly positive eigenvector of fK. The existence of nonnegative
generalized eigenvectors of fK is proved and used to give an algorithmic way for finding
the exponents λmax and λmin of the maximal and minimal growth of the components
of fK(n)(v). We prove that the growth exponent of diameters of the Schreier graphs is
equal to λmax and the orbital contracting coefficient of the group is equal to 1/λmin
. We
prove that the simple random walks on orbital Schreier graphs are recurrent.
A number of examples are presented to illustrate the developed methods with
special attention to iterated monodromy groups of quadratic polynomials. We present
the first example of a group whose coefficients λmin and λmax have different values.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/85845 |
| Date | 10 October 2008 |
| Creators | Bondarenko, Ievgen |
| Contributors | Grigorchuk, Rostislav |
| Publisher | Texas A&M University |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | electronic, born digital |
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