Erdős measures,sofic measures,and Markov chains |
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Authors: | Z I Bezhaeva V I Oseledets |
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Institution: | (1) Moscow Institute of Electronics and Mathematics, Russia;(2) Moscow State University, Russia |
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Abstract: | We consider the random variable ζ = ξ1ρ+ξ2ρ2+…, where ξ1, ξ2, … are independent identically distibuted random variables taking the values 0 and 1 with probabilities P(ξi = 0) = p0, P(ξi = 1) = p1, 0 < p0 < 1. Let β = 1/ρ be the golden number.
The Fibonacci expansion for a random point ρζ from 0, 1] is of the form η1ρ + η2ρ2 + … where the random variables ηk are {0, 1}-valued and ηkηk+1 = 0. The infinite random word η = η1η2 … ηn … takes values in the Fibonacci compactum and determines the so-called Erdős measure μ(A) = P(η ∈ A) on it. The invariant
Erdős measure is the shift-invariant measure with respect to which the Erdős measure is absolutely continuous.
We show that the Erdős measures are sofic. Recall that a sofic system is a symbolic system that is a continuous factor of
a topological Markov chain. A sofic measure is a one-block (or symbol-to-symbol) factor of the measure corresponding to a
homogeneous Markov chain. For the Erdős measures, the corresponding regular Markov chain has 5 states. This gives ergodic
properties of the invariant Erdős measure.
We give a new ergodic theory proof of the singularity of the distribution of the random variable ζ. Our method is also applicable
when ξ1, ξ2, … is a stationary Markov chain with values 0, 1. In particular, we prove that the distribution of ζ is singular and that
the Erdős measures appear as the result of gluing together states in a regular Markov chain with 7 states. Bibliography: 3
titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 28–47. |
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