The Erdős-Vershik problem for the golden ratio

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.Key words: hidden Markov chain, Erdős measure, invariant Erdős measure, golden shift, integral automorphism, entropy, Hausdorff dimension of a measure.Almost seventy years ago Erdős posed the following problem: What distribution function the random variable ζ = ζ 1 ρ + ζ 2 ρ 2 + . . . , where ζ 1 , ζ 2 , . . . are independent identically distributed random variables taking the values 0 and 1 with P (ζ i = 0) = 1/2 and 0 < ρ < 1, can have?We refer to the distribution of such a random variable ζ as an Erdős measure on the real line. The problem of Erdős has been the subject of many papers. The authors of [3] defined an Erdős measure on the unit interval [0, 1] and on the Fibonacci compactum, as well as an invariant Erdős measure on the Fibonacci compactum for the case ρ = ( √ 5 − 1)/2 (the reciprocal of the golden ratio β = ( √ 5 + 1)/2). It was also proved in [3] that an Erdős measure is equivalent to an invariant Erdős measure on the Fibonacci compactum.Vershik posed the problem about the ergodic properties of an invariant Erdős measure on the Fibonacci compactum. This problem was solved in [3].In the previous paper [1], we discovered a connection between the Erdős-Vershik problem and the class of hidden Markov chains for the more general case of 0 < P (ζ i = 0) = q < 1, P (ζ i = 1) = p, and ρ = ( √ 5 − 1)/2. In what follows we consider this case. With the help of this connection, we shall prove an analog of one of the main results of [3]. Namely, we shall prove that a shift on the two-sided Fibonacci compactum with invariant Erdős measure is isomorphic to the integral automorphism for a Bernoulli shift with countable alphabet. We also obtain a formula for the entropy of an invariant Erdős measure.The ratio of the entropy of an invariant Erdős measure to ln β is the Hausdorff dimension of this invariant Erdős measure on the Fibonacci compactum with metric d(x, y) = ρ n(x,y) , where n(x, y) is the length of the longest common prefix of the words x and y. This dimension is equal to the Hausdorff dimension of the corresponding Erdős measure on the real line. Recall that the Hausdorff dimension of a probability measure is the infimum of the Hausdorff dimensions of all sets of measure 1. The above statement follows from the equivalence of an Erdős measure and the corresponding ergodic invariant Erdős measure. (See [2], where a method for calculating the Hausdorff dimension of ergodic measures for symbolic dynamical systems and maps of the unit interval is desc...

Projective convergence of columns for inhomogeneous products of matrices with nonnegative entries

“…Lemma 9.7 (Doubling property). Let W = ω 1 • • • ω n ∈ {0, 1, 2} n be factorized by a concatenation of 3 13 words in W but not factorized by 001001; then, for any nonnegative vector V ,…”

“…However the technics developed in these papers use specificities of the 2 × 2 matrices and seems to us difficult to generalize. Our motivation comes from several works concerned with the geometry of fractal/multifractal sets and measures (see [34][1] [28,29][53] [38,39][18]), with a special importance for the family of the Bernoulli convolutions µ β (1 < β < 2): the mass distributions µ β , whose support is a subinterval of the real line, arise in an Erdős problem [13] and are related to measure theoretic aspects of Gibbs structures for numeration with redundant digits (see [52] [11] [22] [16][41] [2,3][40] [21]). For some parameters β (actually the so-called Pisot-Vijayaraghavan (PV) numbers) such a measure is linearly representable: the µ β -measure of a suitable family of nested generating intervals may be computed by means of matrix products of the form P n X, where A n takes only finitely many values, say A(0), .…”