Geodesic
Erd\H os measures for the goldenshift and Markov chains of the second order
Teoriâ veroâtnostej i ee primeneniâ, Tome 51 (2006) no. 1, pp. 5-21

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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, measure of Hausdorff dimensionality.
Mots-clés : $K$-automorphism 
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     title = {Erd\H os measures for the goldenshift and {Markov} chains of the second order},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     publisher = {mathdoc},
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     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2006_51_1_a1/}
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Z. I. Bezhaeva; V. I. Oseledets. Erd\H os measures for the goldenshift and Markov chains of the second order. Teoriâ veroâtnostej i ee primeneniâ, Tome 51 (2006) no. 1, pp. 5-21. http://geodesic.mathdoc.fr/item/TVP_2006_51_1_a1/