The action of finite-state tree automorphisms on Bernoulli measures
Rostyslav Kravchenko - Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States (email)
Abstract: 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 S. Sidki, preserves a measure class of a Bernoulli measure, and we write down the explicit formula for its Radon-Nikodym 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.
Keywords: Bernoulli measure, Markov chain, finite automata, regular rooted
Received: October 2009; Revised: July 2010; Published: October 2010.