Normal sequences for Markov shifts and intrinsically ergodic subshifts
Let \(b=2,3,...,t\in (0,1)\) and let \(t=\sum^{\infty}_{j=1}d_ j\cdot b^{-j}\) be the b-ary expansion of t. Then t is called to be normal to the base b if each fixed finite block of an arbitrary length k in the expression \(0.d_ 1d_ 2d_ 3...\) appears with an asymptotic frequency \(b^{-k}\) in this expression. It is known that almost every number in (0,1) is normal to the base 10. The first explicit example of a normal number was given by \textit{D. Champernowne} [J. Lond. Math. Soc. 8, 254-260 (1933; Zbl 0007.33701)], this is the number \(0.123456789 10 11 12 13...\) In the paper under review the authors generalize Champernowne's construction and find explicit normal sequences for finite state Markov processes and ergodic subshifts with unique measure of maximal entropy. If S is the state space of the process, a set \(\Omega_ n\subset S^ n\) is constructed for any \(n\geq 1\). Next, \(w_ n\) is formed by concatenating all elements of \(\Omega_ n\) (in an arbitrary order), and the desired sequence is \(w_ 1w_ 2w_ 3... \).
scientific article; zbMATH DE number 4047831
scientific article