This article is cited in 2 papers

The Erdős–Vershik problem for the golden ratio

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: Properties of the Erdős measure and the invariant Erdős measure for the golden ratio and all values of the Bernoulli parameter are studied. It is proved that a shift on the two-sided Fibonacci compact set with invariant Erdős measure is isomorphic to the integral automorphism for a Bernoulli shift with countable alphabet. An effective algorithm for calculating the entropy of an invariant Erdős measure is proposed. It is shown that, for certain values of the Bernoulli parameter, this algorithm gives the Hausdorff dimension of an Erdős measure to 15 decimal places.

Keywords: hidden Markov chain, Erdős measure, invariant Erdős measure, golden shift, integral automorphism, entropy, Hausdorff dimension of a measure.

UDC: 517.987.5+519.21

Received: 22.08.2008

DOI: 10.4213/faa2990

 English version: Functional Analysis and Its Applications, 2010, 44:2, 83–91

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