Erdős measures, sofic measures, and Markov chains

Bezhaeva, Z. I.; Oseledets, V. I.

doi:10.1007/s10958-007-0445-2

Cited by 8 publications

(11 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Among the extensive literature that we do not cite elsewhere, we can mention in addition [47,70,35,10,88] Recall that a monoid is a set S with a binary operation S × S → S which is associative and has a neutral element (identity). This means we can think of A * as the multiplicative free monoid generated by A, where the operation is concatenation and the neutral element is ǫ.…”

Section: Identification Of Hidden Markov Measuresmentioning

confidence: 99%

Hidden Markov processes in the context of symbolic dynamics

Boyle

Petersen²

2011

Entropy of Hidden Markov Processes and Connections to Dynamical Systems

View full text Add to dashboard Cite

Abstract. In an effort to aid communication among different fields and perhaps facilitate progress on problems common to all of them, this article discusses hidden Markov processes from several viewpoints, especially that of symbolic dynamics, where they are known as sofic measures, or continuous shift-commuting images of Markov measures. It provides background, describes known tools and methods, surveys some of the literature, and proposes several open problems.

show abstract

Section: Identification Of Hidden Markov Measuresmentioning

confidence: 99%

Hidden Markov processes in the context of symbolic dynamics

Boyle

Petersen²

2011

Entropy of Hidden Markov Processes and Connections to Dynamical Systems

View full text Add to dashboard Cite

show abstract

“…Below we give the definition of an invariant Erdős measure on the Fibonacci compactum borrowed from [1]. In [1], the problem about the ergodic properties of an invariant Erdős measure was reduced to the study of the hidden Markov chain {η i = f (ξ i )} generated by a Markov chain {ξ i } with 5 states 1, 2, 3, 4, and 5 and transition matrix P of the form…”

Section: An Invariant Erdős Measure On the Fibonacci Compactummentioning

confidence: 99%

“…This set is compact with respect to the 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. The measure µ is an invariant Erdős measure on the Fibonacci compactum [1].…”

Section: An Invariant Erdős Measure On the Fibonacci Compactummentioning

confidence: 99%

See 1 more Smart Citation

The Erdős-Vershik problem for the golden ratio

Bezhaeva

Oseledets

2010

Funct Anal Its Appl

Self Cite

View full text Add to dashboard Cite

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...

show abstract

“…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), .…”

mentioning

confidence: 99%

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

Olivier,

Thomas

2009

Preprint

View full text Add to dashboard Cite

Let Pn be the n-step right product A1 • • • An, where A1, A2, . . . is a given infinite sequence of d×d matrices with nonnegative entries. In a wide range of situations, the normalized matrix product Pn/ Pn does not converge and we shall be rather interested in the asymptotic behavior of the normalized columns PnUi/ PnUi , where U1, . . . , U d are the canonical d × 1 vectors. Our main result in Theorem A gives a sufficient condition (C) over the sequence A1, A2, . . . ensuring the existence of dominant columns of Pn, having the same projective limit V : more precisely, for any rank n, there exists a partition of {1, . . . , d} made of two subsets Jn = ∅ and J c n such that each one of the sequences of normalized columns, say PnUj n / PnUj n with jn ∈ Jn tends to V as n tends to +∞ and are dominant in the sense that the ratio PnU j n / PnUj n tends to 0, as soon as j n ∈ J c n . The existence of sequences of such dominant columns implies that for any probability vector X with positive entries, the probability vector PnX/ PnX , converges as n tends to +∞. Our main application of Theorem A (and our initial motivation) is related to an Erdős problem concerned with a family of probability measures µ β (for 1 < β < 2 a real parameter) fully supported by a subinterval of the real line, known as Bernoulli convolutions. For some parameters β (actually the so-called PV-numbers) such measures are known to be 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 PnX, where An takes only finitely many values, say A(0), . . . , A(a), and X is a probability vector with positive entries. Because, An = A(ξn), where ξ = ξ1ξ2 • • • is a sequence (one-sided infinite word) with ξn ∈ {0, . . . , a}, we shall write Pn = Pn(ξ) the dependence of the n-step product with ξ: when the convergence of Pn(ξ)X/ Pn(ξ)X is uniform w.r.t. ξ, a sharp analysis of the measure µ β (Gibbs structures and multifractal decomposition) becomes possible. However, most of the matrices involved in the decomposition of these Bernoulli convolutions are large, sparse and it is usually not easy to prove the condition (C) of Theorem A. To illustrate the technics, we consider one parameter β for which the matrices are neither too small nor too large and we verify condition (C): this leads to the Gibbs properties of µ β .

show abstract

Erdős measures, sofic measures, and Markov chains

Cited by 8 publications

References 3 publications

Hidden Markov processes in the context of symbolic dynamics

Hidden Markov processes in the context of symbolic dynamics

The Erdős-Vershik problem for the golden ratio

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

Contact Info

Product

Resources

About