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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References
N. Sidorov and A. Vershik, “Ergodic properties of the Erdős measures, the entropy of the goldenshift, and related problems,” Monatsh. Math., 126, 215–261 (1998).
I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia (1992).
B. Marcus, K. Petersen, and S. Williams, “Transmission rates and factors of Markov chains,” Contemp. Math., 26, 279–293 (1984).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 28–47.
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Bezhaeva, Z.I., Oseledets, V.I. Erdős measures, sofic measures, and Markov chains. J Math Sci 140, 357–368 (2007). https://doi.org/10.1007/s10958-007-0445-2
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DOI: https://doi.org/10.1007/s10958-007-0445-2