Geodesic
Erdős measures, sofic measures, and Markov chains
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIII, Tome 326 (2005), pp. 28-47 
Z. I. Bezhaeva; V. I. Oseledets. Erdős measures, sofic measures, and Markov chains. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIII, Tome 326 (2005), pp. 28-47. http://geodesic.mathdoc.fr/item/ZNSL_2005_326_a3/
@article{ZNSL_2005_326_a3,
     author = {Z. I. Bezhaeva and V. I. Oseledets},
     title = {Erd\H{o}s measures, sofic measures, and {Markov} chains},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {28--47},
     year = {2005},
     volume = {326},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_326_a3/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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. 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.

[1] N. Sidorov, A. Vershik, “Ergodic properties of the Erdős measures, the entropy of the goldenshift, and related problems”, Monatsh. Math., 126:3 (1998), 215–261 | DOI | MR | Zbl

[2] I. Dobeshi, Desyat lektsii po veivletam, Regulyarnaya i khaoticheskaya dinamika, Izhevsk, 2001

[3] B. Markus, K. Petersen, S. Williams, “Transmission rates and factors of Markov chains”, Contemporary Mathematics, 26 (1984), 279–293 | MR