Normal sequences for Markov shifts and intrinsically ergodic subshifts
type: Item
instance of: scholarly article
title: Normal sequences for Markov shifts and intrinsically ergodic subshifts
zbMATH Open document ID: 0643.10041
DOI: 10.1007/BF02787263
author: Benjamin Weiss
author: Meir Smorodinsky
published in: Israel Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: 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 \(w_ 1w_ 2w_ 3... \).
Mathematics Subject Classification ID: 11K16
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 4047831
zbMATH Keywords: normal number
zbMATH Keywords: Champernowne's construction
zbMATH Keywords: normal sequences
zbMATH Keywords: finite state Markov processes
zbMATH Keywords: ergodic subshifts
zbMATH Keywords: measure of maximal entropy
reviewed by: Luchezar Stoyanov
MaRDI profile type: Publication
cites work: A construction of a normal number for the continued fraction transformation
cites work: Q4068740
cites work: The Construction of Decimals Normal in the Scale of Ten
cites work: \(\beta\)-shifts have unique maximal measure
cites work: Ergodic theory, entropy
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description: scientific article; zbMATH DE number 4047831