Measures of maximal entropy for random \(\beta\)-expansions
Summary: Let \(\beta>1\) be a non-integer. We consider \(\beta\)-expansions of the form \(\sum^\infty_{i=1}d_i/\beta^i\), where the digits \((d_i)_{i\geq 1}\) are generated by means of a Borel map \(K_\beta\) defined on \(\{0,1\}^\mathbb{N}\times [0,\lfloor\beta\rfloor/(\beta-1)]\). We show that \(K_\beta\) has a unique mixing measure \(\nu_\beta\) of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure \(\nu_\beta\) the digits \((d_i)_{i\geq 1}\) form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of \(\beta\)-expansions.
scientific article; zbMATH DE number 2148276