On the critical dimension and AC entropy for Markov odometers

Reference:

Dooley, A. H. and Mortiss, G., 2006. On the critical dimension and AC entropy for Markov odometers. Monatshefte fur Mathematik, 149 (3), pp. 193-213.

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Official URL:

http://dx.doi.org/10.1007/s00605-005-0372-6

Abstract

Markov odometers are natural models for non-homogeneous Markov chains, and are natural generalisations of infinite product measures. We show how to calculate the critical dimension of these measures: this is an invariant which describes the asymptotic growth rate of sums of Radon-Nikodym derivatives. This interesting invariant appears to give a kind of entropy for non-singular odometer actions. The techniques require a law of large numbers for inhomogeneous Markov chains.

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