Erdős measures, sofic measures, and Markov chains

We consider random variable $\zeta=\xi_1\rho+\xi_2\rho^2+\ldots$ where $\xi_1,\xi_2,\ldots$ are independent identically distibuted random variables taking values 0, 1, with $P(\xi_i=0)=p_0$, $P(\xi_i=1)=p_1$, $0<p_0<1$. Let $\beta=1/\rho$ be the golden number.
The Fibonacci expansion for a random point $\rho\zeta$ from $[0,1]$ is of form $\eta_1\rho+\eta_2\rho^2+\ldots$ where random variables $\eta_k=0,1$ and $\eta_k\eta_{k+1}=0$. The infinite random word $\eta=\eta_1\eta_2\ldots\eta_n\ldots$ takes values in the Fibonacci compactum and defines an Erdős measure $\mu(A)=P(\eta\in A)$ on it. The invariant Erdős measure is the shift-invariant measure with respect to which Erdős measure is absolutely continuous.
We show that Erdős measures are sofic. Recall that a sofic system is a symbolic system which is a continuous factor of a topological Markov chain. A sofic measure is a one-block (or symbol to symbol) factor of the measure corresponding to a homogeneous Markov chain. For Erdős measures the corresponding regular Markov chain has 5 states. This gives ergodic properties of the invariant Erdős measure.
We give a new ergodic theory proof of the singularity of the distribution of the random variable $\zeta$. Our method is also applicable when $\xi_1,\xi_2,\ldots$ is a stationary Markov chain taking values 0, 1. In particular, we prove that the distribution of $\zeta$ is singular and that Erdős measures appear as result of gluing together states in a regular Markov chain with 7 states.