Measures Induced by Automata and Their Actions
Roman Kogan · OakTrust (Texas A&M University Libraries) · 2017
In this thesis we explore the theme of automata, measures on spaces of sequences X^N in a finite alphabet X, and their connections. The notion of a finite-state measure (a measure given by a finite automaton, or equivalently, having a finite number of sections) is introduced, and applied to the problem of studying the images of Markov measures under the action of tree automorphisms given by automata. Another approach, based on prior work by Kravchenko, is also applied to this problem to compute the Radon-Nikodym derivative in the case when the automaton has polynomial growth, and to compute frequencies by using a lift to (S x X)^N. The question of when the image of a finite-state measure under the action of a noninvertible automaton is answered. We also explore when a finite-state measure is Gibbs. For the second part of the thesis, we introduce the notion of the automatic logarithm, and a measure associated with it. We compute this measure for certain interesting examples, in which it turns out to be finite-state.