A survey on transitivity in discrete time dynamical systems. application to symbolic systems and related languages

Issue		RAIRO-Theor. Inf. Appl. Volume 40, Number 2, April-June 2006 Alberto Bertoni: Climbing summits


Page(s)		333 - 352
DOI		10.1051/ita:2006016
Published online		20 July 2006

Theoret. Informatics Appl. 40, 333-352 (2006)
DOI: 10.1051/ita:2006016

A survey on transitivity in discrete time dynamical systems. application to symbolic systems and related languages

Gianpiero Cattaneo, Alberto Dennunzio and Fabio Farina

Dipartimento di Informatica, Sistemistica e Comunicazione, Università degli Studi di Milano-Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano, Italy; cattang@disco.unimib.it; dennunzio@disco.unimib.it; farina@disco.unimib.it

(Published online 20 July 2006)

Abstract
The main goal of this paper is the investigation of a relevant property which appears in the various definition of deterministic topological chaos for discrete time dynamical system: transitivity. Starting from the standard Devaney's notion of topological chaos based on regularity, transitivity, and sensitivity to the initial conditions, the critique formulated by Knudsen is taken into account in order to exclude periodic chaos from this definition. Transitivity (or some stronger versions of it) turns out to be the relevant condition of chaos and its role is discussed by a survey of some important results about it with the presentation of some new results. In particular, we study topological mixing, strong transitivity, and full transitivity. Their applications to symbolic dynamics are investigated with respect to the relationships with the associated languages.

Mathematics Subject Classification. 37B05, 37B10

Key words: Transitivity -- chaos -- symbolic dynamics -- formal languages.

© EDP Sciences 2006

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