On absolute continuity of the Erdős measure for the golden ratio, Tribonacci numbers, and second-order Markov chains
type: Item
instance of: scholarly article
title: On absolute continuity of the Erdős measure for the golden ratio, Tribonacci numbers, and second-order Markov chains
author: Q6096071
author: Q6096072
author: Valery Oseledets
published in: Theory of Probability and its Applications
publication date: 2024-08-19T00:00:00Z
review text: For a stationary \(\{0,1\}\)-valued process \((\epsilon_k)\) the associated Erdős measure \(\mu\) is the distribution law of the random variable \(\zeta=\sum_{k=1}^{\infty}\epsilon_k\beta^{-k}\) for \(\beta\in(1,2)\). If the process is a \(0\)-Markov chain (that is, independent and identically distributed) then \(\mu\) is an infinite Bernoulli convolution. The `Erdős problem' in this context is to understand when \(\mu\) is absolutely continuous. The main results here give explicit conditions in terms of transition probabilities for \(1\)-Markov and \(2\)-Markov processes to have absolutely continuous Erdős measures when \(\beta\) is the golden mean and for \(2\)-Markov processes and the Tribonacci number. This completes earlier partial results finding some of the conditions that guarantee absolute continuity.
reviewed by: Thomas B. Ward
Mathematics Subject Classification ID: 37A44
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 37A50
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 7898580
zbMATH Keywords: Erdős measure
zbMATH Keywords: invariant Erdős measure
zbMATH Keywords: hidden Markov chain
zbMATH Keywords: sofic measure
zbMATH Keywords: Blackwell-Markov chains
zbMATH Keywords: golden ratio
zbMATH Keywords: Tribonacci number
zbMATH Keywords: Fibonacci compact set
zbMATH Keywords: Tribonacci compact set
zbMATH Keywords: Markov partition
MaRDI profile type: Publication
cites work: Erdös Measures for the Golden Ratio and 2‐step Markov Chains
cites work: Remarks on the entropy of sofic dynamical system of Blackwell's type
cites work: Hidden Markov processes in the context of symbolic dynamics
cites work: On a family of symmetric Bernoulli convolutions.
DOI: 10.1137/S0040585X97T991908
description: scientific article; zbMATH DE number 7898580