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Zapiski Nauchnykh Seminarov POMI, 2005, Volume 326, Pages 28–47 (Mi znsl336)  

This article is cited in 12 scientific papers (total in 12 papers)

Erdős measures, sofic measures, and Markov chains

Z. I. Bezhaevaa, V. I. Oseledetsb

a Moscow State Institute of Electronics and Mathematics
b M. V. Lomonosov Moscow State University
References:
Abstract: We consider random variable $\zeta=\xi_1\rho+\xi_2\rho^2+\ldots$ where $\xi_1,\xi_2,\ldots$ are independent identically distibuted random variables taking values 0, 1, with $P(\xi_i=0)=p_0$, $P(\xi_i=1)=p_1$, $0<p_0<1$. Let $\beta=1/\rho$ be the golden number.
The Fibonacci expansion for a random point $\rho\zeta$ from $[0,1]$ is of form $\eta_1\rho+\eta_2\rho^2+\ldots$ where random variables $\eta_k=0,1$ and $\eta_k\eta_{k+1}=0$. The infinite random word $\eta=\eta_1\eta_2\ldots\eta_n\ldots$ takes values in the Fibonacci compactum and defines an Erdős measure $\mu(A)=P(\eta\in A)$ on it. The invariant Erdős measure is the shift-invariant measure with respect to which Erdős measure is absolutely continuous.
We show that Erdős measures are sofic. Recall that a sofic system is a symbolic system which 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 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 $\zeta$. Our method is also applicable when $\xi_1,\xi_2,\ldots$ is a stationary Markov chain taking values 0, 1. In particular, we prove that the distribution of $\zeta$ is singular and that Erdős measures appear as result of gluing together states in a regular Markov chain with 7 states.
Received: 08.04.2005
English version:
Journal of Mathematical Sciences (New York), 2007, Volume 140, Issue 3, Pages 357–368
DOI: https://doi.org/10.1007/s10958-007-0445-2
Bibliographic databases:
UDC: 519.217, 517.518.1
Language: Russian
Citation: Z. I. Bezhaeva, V. I. Oseledets, “Erdős measures, sofic measures, and Markov chains”, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIII, Zap. Nauchn. Sem. POMI, 326, POMI, St. Petersburg, 2005, 28–47; J. Math. Sci. (N. Y.), 140:3 (2007), 357–368
Citation in format AMSBIB
\Bibitem{BezOse05}
\by Z.~I.~Bezhaeva, V.~I.~Oseledets
\paper Erd\H os measures, sofic measures, and Markov chains
\inbook Representation theory, dynamical systems, combinatorial and algoritmic methods. Part~XIII
\serial Zap. Nauchn. Sem. POMI
\yr 2005
\vol 326
\pages 28--47
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl336}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2183214}
\zmath{https://zbmath.org/?q=an:1105.60050}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2007
\vol 140
\issue 3
\pages 357--368
\crossref{https://doi.org/10.1007/s10958-007-0445-2}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33845796171}
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  • This publication is cited in the following 12 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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