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

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

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.