Normal numbers, entropy, translations
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.
scientific article; zbMATH DE number 820112