We study transitive non-minimal ℕ-actions and ℤ-actions. We show that there are such actions whose non-transitive points are periodic and whose topological entropy is positive. It turns out that such actions can be obtained by perturbing minimal systems under some reasonable assumptions.
In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family ℱ (of subsets of ℤ₊) and a MDS (X,,μ,T), several notions of ergodicity related to ℱ are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is ℱ-convergence ergodic of order k if for any of positive...
In this paper we explore topological factors in between the Kronecker factor and the maximal equicontinuous factor of a system. For this purpose we introduce the concept of sequence entropy -tuple for a measure and we show that the set of sequence entropy tuples for a measure is contained in the set of topological sequence entropy tuples [H- Y]. The reciprocal is not true. In addition, following topological ideas in [BHM], we introduce a weak notion and a strong notion of complexity pair for a...
The properties of topological dynamical systems which are disjoint from all minimal systems of zero entropy, , are investigated. Unlike the measurable case, it is known that topological -systems make up a proper subset of the systems which are disjoint from . We show that has an invariant measure with full support, and if in addition is transitive, then is weakly mixing. A transitive diagonal system with only one minimal point is constructed. As a consequence, there exists a thickly syndetic...
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