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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
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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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http://mi.mathnet.ru/eng/tvp143https://doi.org/10.4213/tvp143 http://mi.mathnet.ru/eng/tvp/v51/i1/p5
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