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Teor. Veroyatnost. i Primenen., 2006, Volume 51, Issue 1, Pages 5–21 (Mi tvp143)  

Erdős measures for the goldenshift and Markov chains of the second order

Z. I. Bezhaevaa, V. I. Oseledetsb

a Moscow State Institute of Electronics and Mathematics
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We consider the random variable $\zeta=\omega_1\beta^{-1}+\omega_2\beta^{-2}+\dotsb$, where $\omega_1,\omega_2,…$ is the stationary ergodic 2-step Markov chain with states 0, 1 and $\beta$ is the golden ratio. The paper finds all cases of absolute continuity of the distribution function of the random variable $\zeta$. For other cases the distribution function in continuous and singular. We prove that the respective Erdős measures arise under gluing together the states in a finite Markov chain. Ergodic properties of invariant Erdős measure are studied.

Keywords: 2-step Markov chain, golden ratio, Erdős measure, maximal entropy measure, $K$-automorphism, measure of Hausdorff dimensionality.

DOI: https://doi.org/10.4213/tvp143

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English version:
Theory of Probability and its Applications, 2007, 51:1, 28–41

Bibliographic databases:

Received: 12.10.2005

Citation: Z. I. Bezhaeva, V. I. Oseledets, “Erdős measures for the goldenshift and Markov chains of the second order”, Teor. Veroyatnost. i Primenen., 51:1 (2006), 5–21; Theory Probab. Appl., 51:1 (2007), 28–41

Citation in format AMSBIB
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