This article is cited in 1 paper

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

Z. I. Bezhaeva^a, V. I. Oseledets<sup>b

a</sup> 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 $\omega_1,\omega_2,\dots$ 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.

Received: 12.10.2005

DOI: 10.4213/tvp143

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

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