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A survey on transitivity in discrete time dynamical systems. Application to symbolic systems and related languages. (English) Zbl 1112.37005

Summary: 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 systems: 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 C. Knudsen [Am. Math. Mon. 101, 563–565 (1994; Zbl 0840.54031)] 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.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B10 Symbolic dynamics

Citations:

Zbl 0840.54031
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References:

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