On Absolute Continuity of the Erdös Measure for the Golden Ratio, Tribonacci Numbers, and Second-Order Markov Chains

Abstract

Theory of Probability &Its Applications, Volume 69, Issue 2, Page 265-280, August 2024.
We consider a power series at a fixed point $\rho \in (0.5,1)$, where random coefficients assume a value $0$ or $1$ and form a stationary ergodic aperiodic process. The Erdös measure is the distribution law of such a series. The problem of absolute continuity of the Erdös measure is reduced to the problem of determining when the corresponding hidden Markov chain is a Parry--Markov chain. For the golden ratio and a 1-Markov chains, we give necessary and sufficient conditions for absolute continuity of the Erdös measure and, using Blackwell--Markov chains, provide a new proof that the necessary conditions obtained earlier by Bezhaeva and Oseledets [Theory Probab. Appl., 51 (2007), pp. 28--41] are also sufficient. For tribonacci numbers and 1-Markov chains, we give a new proof of the theorem on singularity of the Erdös measure. For tribonacci numbers and 2-Markov chains, we find only two cases with absolute continuity.