Automatic walks
type: Item
instance of: scholarly article
title: Automatic walks
zbMATH Open document ID: 0795.11011
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1994-09-13T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93579
review text: Let \(f: A\to G\) be a map from a set \(A\) to a locally compact abelian group \(G\). A sequence \(x= (x_ n)_ n\) determines a walk \((S_ n(x))_ n\) on \(G\) through \(S_ 0(x)=e\) and \(S_ n(x)= f(x_ 1)+ \cdots+ f(x_ n)\). The case \(A=\mathbb{R}\), \(G=\mathbb{C}\), \(f(x)= e^{2\pi ix}\) corresponds to the classical walks associated with exponential sums. The author discusses results for automatic sequences \(x\), that is sequences recognizable by finite automata, concentrating on the classification of the walks as recurrent \((\| S_ n(x)\| <\varepsilon)\), ultimately recurrent, transient (that is, not ultimately recurrent), or bounded. The notion can be extended to a walk on a graph. Some interesting results for the case in which \(A\) has cardinality 2 are discussed.
Mathematics Subject Classification ID: 11B85
zbMATH DE Number: 484163
zbMATH Keywords: recurrent walks
zbMATH Keywords: transient walks
zbMATH Keywords: bounded walks
zbMATH Keywords: locally compact abelian group
zbMATH Keywords: automatic sequences
zbMATH Keywords: ultimately recurrent
author: F. Michel Dekking
reviewed by: John H. Loxton
MaRDI profile type: Publication
OpenAlex ID: W4253599947
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DOI: 10.5802/JTNB.80
description: scientific article; zbMATH DE number 484163