Topological sequence entropy and topologically weak mixing

On Local Aspects of Topological Weak Mixing, Sequence Entropy and Chaos

In this paper we show that for every n ≥ 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 nontrivial weakly mixing sets of order 3 always have positive topological sequence entro...

Topological Weak Mixing and Quasi-bohr Systems

A minimal dynamical system (X,T ) is called quasi-Bohr if it is a nontrivial equicontinuous extension of a proximal system. We show that if (X,T ) is a minimal dynamical system which is not weakly mixing then some minimal proximal extension of (X, T ) admits a nontrivial quasi-Bohr factor. (In terms of Ellis groups the corresponding statement is: AG′ = G implies weak mixing.) The converse does ...

Topological Sequence Entropy

Weak hyper semi-quantales and weak hypervalued topological spaces

The purpose of this paper is to construct a weak hyper semi-quantale as a generalization of the concept of semi-quantale and used it as an appropriate hyperlattice-theoretic basis to formulate new lattice-valued topological theories. Based on such weak hyper semi-quantale, we aim to construct the notion of a weak hypervalued-topology as a generalized form of the so-called lattice-valued t...

Brandt extensions and primitive topologically periodic inverse topological semigroups

In this paper we find sufficient conditions on primitive inverse topological semigroup S under which: the inversion inv : (H(S)) (H(S)) is continuous; we show that every topologically periodic countable compact primitive inverse topological semigroups with closed H-classes is topologically isomorphic to an orthogonal sum P i2= Bi (Gi) of topological Brandt extensions Bi (Gi) of countably compac...