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

Gianpiero Cattaneo; Alberto Dennunzio; Fabio Farina

RAIRO - Theoretical Informatics and Applications (2006)

  • Volume: 40, Issue: 2, page 333-352
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Cattaneo, Gianpiero, Dennunzio, Alberto, and Farina, Fabio. "A survey on transitivity in discrete time dynamical systems. application to symbolic systems and related languages." RAIRO - Theoretical Informatics and Applications 40.2 (2006): 333-352. <http://eudml.org/doc/249686>.

@article{Cattaneo2006,
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. },
author = {Cattaneo, Gianpiero, Dennunzio, Alberto, Farina, Fabio},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Transitivity; chaos; symbolic dynamics; formal languages.; formal languages; topological mixing},
language = {eng},
month = {7},
number = {2},
pages = {333-352},
publisher = {EDP Sciences},
title = {A survey on transitivity in discrete time dynamical systems. application to symbolic systems and related languages},
url = {http://eudml.org/doc/249686},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Cattaneo, Gianpiero
AU - Dennunzio, Alberto
AU - Farina, Fabio
TI - A survey on transitivity in discrete time dynamical systems. application to symbolic systems and related languages
JO - RAIRO - Theoretical Informatics and Applications
DA - 2006/7//
PB - EDP Sciences
VL - 40
IS - 2
SP - 333
EP - 352
AB - 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.
LA - eng
KW - Transitivity; chaos; symbolic dynamics; formal languages.; formal languages; topological mixing
UR - http://eudml.org/doc/249686
ER -

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