Random \(\beta\)-expansions with deleted digits
Let \(\beta>1\) be a real number, and let \(A=\{a_1,\dots,a_m\}\) a given set of real numbers with \(a_1<\dots<a_m\) (\(m\geq 2\)). The authors define random \(\beta\)-expansions of the form \(x=\sum_{i=1}^\infty b_i/\beta^i\), with \(b_i\in A\). The author generalize the results obtained in [J. Eur. Math. Soc. 7, No.~1, 51--68 (2005; Zbl 1074.28008)] to a wider class of digit sets. By sketching a dynamical way of generating the greedy expansion, they give simple dynamical proofs of some of the results proved by \textit{M. Pedicini} in [Theor. Comput. Sci. 332, No.~1--3, 313--336 (2005; Zbl 1080.11009)]. They also generalize the greedy and lazy expansions and define a function \(K\) that generates essentially all \(\beta\)-expansions with digits belonging to the set \(A\). The authors also find a \(K\)-invariant measure such that \((K,\nu)\) is isomorphic to the uniform Bernoulli shift with digit set \(A\).
scientific article; zbMATH DE number 5178926