Normal numbers, entropy, translations
type: Item
instance of: scholarly article
title: Normal numbers, entropy, translations
zbMATH Open document ID: 0839.11030
DOI: 10.1007/BF02761660
published in: Israel Journal of Mathematics
publication date: 1996-06-19T00:00:00Z
review text: A real number \(x\) is called normal in base \(p\) if the sequence \((p^n x \text{mod} 1)_{n \in \mathbb{N}}\) is uniformly distributed on \([0,1[\), or in an equivalent way, if every block of length \(k\) defined on the alphabet \(\{0,1, \dots, p - 1\}\) appears in the base \(p\) expansion of \(x\) with frequency \(1/p^k\). There exist probability measures \(\mu\), defined on \([0,1]\), such that \(\mu\)-almost every real in \([0,1[\) is normal in base \(p\) and nonnormal in base \(q\), where \(p\) and \(q\) are multiplicatively independent integers. This paper establishes a very interesting criterion for a measure \(\mu\) on the unit circle to satisfy: \(\mu\)-almost every point is normal in a given basis. Moreover this criterion contains all known examples, when \(p\) and \(q\) are coprime and generalizes, in particular, the result of D. Rudolph about measures which are invariant by multiplication by \(p\) and \(q\). The proof of this criterion is based on the study of the relations between the entropy of the multiplication by an integer \(p\) and the conservativity for the translations by the \(p\)-adic rational numbers. Let us note that this paper contains a generalization of these results to the equidistribution of the sequence \((f_k x)_{k \in \mathbb{N}}\), where \((fk)_{k \in \mathbb{N}}\) denotes the Fibonacci sequence (this generalization is due to Kamae) and also to the conservativity for irrational translations.
Mathematics Subject Classification ID: 11K16
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 820112
zbMATH Keywords: normal numbers
zbMATH Keywords: entropy
zbMATH Keywords: measure preserving transformation
zbMATH Keywords: conservativity of a measure
zbMATH Keywords: Fibonacci sequence
author: Bernard Host
reviewed by: Valérie Berthé
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 820112