About a Markov chain connected with the number system

Abstract

This work addresses the problem of finding a stationary Markov chain with maximum
entropy on the set of infinite binary sequences that forbid three consecutive ones. A connection is established between this problem and the non-integer base numeration system studied by A.
O. Gelfond. A probabilistic interpretation of Gelfond’s distribution of remainders is provided
in terms of a three-state ergodic Markov chain. As the main result, the parameters of the
extremal chain are explicitly determined. It is shown that the transition probability matrix and the stationary distribution are expressed via the root 𝜃 > 1 of the equation 1 = 𝜃−1+𝜃−2+𝜃−3.
The entropy of the resulting chain equals ln 𝜃, which defines the maximum achievable entropy in the class of sequences with this forbidden pattern.