Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems

AbstractIn this paper we show that for every$n\geq 2$there are minimal systems with perfect weakly mixing sets of order$n$and all weakly mixing sets of order$n+ 1$trivial. We present some relations between weakly mixing sets and topological sequence entropy; in particular, we prove that invertible minimal systems with non-trivial weakly mixing sets of order three always have positive topological sequence entropy. We also study relations between weak mixing of sets and other well-established notions from qualitative theory of dynamical systems like (regional) proximality, chaos and equicontinuity in a broad sense.

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Weak forms of topological and measure-theoretical equicontinuity: relationships with discrete spectrum and sequence entropy

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.83 ◽

2016 ◽

Vol 37 (4) ◽

pp. 1211-1237 ◽

Cited By ~ 14

Author(s):

FELIPE GARCÍA-RAMOS

Keyword(s):

Dynamical Systems ◽

Discrete Spectrum ◽

Haar Measure ◽

Topological Category ◽

Sequence Entropy ◽

Ergodic Measures ◽

Factor Map ◽

Topological Dynamical Systems ◽

Theoretical Sensitivity ◽

Topological Sequence Entropy

We define weaker forms of topological and measure-theoretical equicontinuity for topological dynamical systems, and we study their relationships with sequence entropy and systems with discrete spectrum. We show that for topological systems equipped with ergodic measures having discrete spectrum is equivalent to$\unicode[STIX]{x1D707}$-mean equicontinuity. In the purely topological category we show that minimal subshifts with zero topological sequence entropy are strictly contained in diam-mean equicontinuous systems; and that transitive almost automorphic subshifts are diam-mean equicontinuous if and only if they are regular (i.e. the maximal equicontinuous factor map is one–one on a set of full Haar measure). For both categories we find characterizations using stronger versions of the classical notion of sensitivity. As a consequence, we obtain a dichotomy between discrete spectrum and a strong form of measure-theoretical sensitivity.

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On some kinds of dense orbits in general nonautonomous dynamical systems

Nonautonomous Dynamical Systems ◽

10.1515/msds-2020-0116 ◽

2020 ◽

Vol 7 (1) ◽

pp. 163-175

Author(s):

Mehdi Pourbarat

Keyword(s):

Dynamical Systems ◽

Initial Conditions ◽

Point Of View ◽

Topological Conjugacy ◽

Topological Transitivity ◽

Nonautonomous Systems ◽

Nonautonomous Dynamical Systems ◽

Sensitive Dependence ◽

Dense Orbits

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

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