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Conceptions on topological transitivity in products and symmetric products. (English) Zbl 1477.37017

Summary: Having a finite number of topological spaces \(X_i\) and functions \(f_i : X_i \rightarrow X_i\), and considering one of the following classes of functions: exact, transitive, strongly transitive, totally transitive, orbit-transitive, strictly orbittransitive, \( \omega \)-transitive, mixing, weakly mixing, mild mixing, chaotic, exactly Devaney chaotic, minimal, backward minimal, totally minimal, \(TT_{++}\), scattering, Touhey or an \(F\)-system, in this paper, we study dynamical behaviors of the systems \((X_i, f_i)\), \((\prod X_i, \prod f_i)\), \((\mathcal{F}_n (\prod X_i), \mathcal{F}_n (\prod X_i))\), and \((\mathcal{F}_n(X_i), \mathcal{F}_n(f_i))\).

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B02 Dynamics in general topological spaces
37B45 Continua theory in dynamics
54B20 Hyperspaces in general topology
54F15 Continua and generalizations
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