About the absolute continuity of the Erdös measure for the golden ratio, tribonacci number, and second order Markov chains

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 Erdős measure is the distribution law of such a series. The problem of absolute continuity of the Erdős 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 Erdős 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 Erdős measure. For tribonacci numbers and 2-Markov chains, we find only two cases with absolute continuity.