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Journal of Modern Dynamics (JMD)
 

The action of finite-state tree automorphisms on Bernoulli measures

Pages: 443 - 451, Issue 3, July 2010      doi:10.3934/jmd.2010.4.443

 
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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 tree.
Mathematics Subject Classification:  Primary: 20E08.

Received: October 2009;      Revised: July 2010;      Available Online: October 2010.

 References