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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.
@article{ITA_2006__40_2_333_0,
author = {Cattaneo, Gianpiero and Dennunzio, Alberto and Farina, Fabio},
title = {A survey on transitivity in discrete time dynamical systems. {Application} to symbolic systems and related languages},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {333--352},
publisher = {EDP-Sciences},
volume = {40},
number = {2},
year = {2006},
doi = {10.1051/ita:2006016},
mrnumber = {2252643},
zbl = {1112.37005},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ita:2006016/}
}
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 - Informatique Théorique et Applications PY - 2006 SP - 333 EP - 352 VL - 40 IS - 2 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ita:2006016/ DO - 10.1051/ita:2006016 LA - en ID - ITA_2006__40_2_333_0 ER -
%0 Journal Article %A Cattaneo, Gianpiero %A Dennunzio, Alberto %A Farina, Fabio %T A survey on transitivity in discrete time dynamical systems. Application to symbolic systems and related languages %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2006 %P 333-352 %V 40 %N 2 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ita:2006016/ %R 10.1051/ita:2006016 %G en %F ITA_2006__40_2_333_0
Cattaneo, Gianpiero; Dennunzio, Alberto; Farina, Fabio. A survey on transitivity in discrete time dynamical systems. Application to symbolic systems and related languages. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 2, pp. 333-352. doi: 10.1051/ita:2006016
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