For a sequence {Tn} of continuous linear operators on a separable Fréchet space X, we discuss necessary conditions and sufficient conditions for {Tn} to be topologically mixing, and the relations between topological mixing and the Hypercyclicity Criterion. Among them are: 1) topological mixing is equivalent to being hereditarily densely hypercyclic; 2) the Hypercyclicity Criterion with respect ...

in this paper we give conditions for a tuple of commutative bounded linear operators which holds in the property of the hypercyclicity criterion. we characterize topological transitivity and semi-hereiditarily of a dynamical system given  by an n-tuple of operators acting on a separable infinite dimensional banach space .

We study tensor products of strongly continuous semigroups on Banach spaces that satisfy the hypercyclicity criterion, the recurrent hypercyclicity criterion or are chaotic.

We obtain a Disjoint Frequent Hypercyclicity Criterion and show that it characterizes disjoint frequent hypercyclicity for family of unilateral pseudo-shifts on c0(N) ?p(N), 1?p<?. As an application, we characterize frequently hypercyclic weighted shifts. give analogous results the weaker notions upper reiterative hypercyclicity. Finally, provide counterexamples showing that, although hypercycl...

In this paper we study hereditarily homogeneous generalized topological spaces. Various properties of hereditarily homogeneous generalized topological spaces are discussed. We prove that a generalized topological space is hereditarily homogeneous if and only if every transposition of $X$ is a $mu$-homeomorphism on $X$.

We give necessary and sufficient conditions for an operator on a separable Hilbert space to satisfy the hypercyclicity criterion.

We study hypercyclicity properties of functions of Banach space operators. Generalizations of the results of Herzog-Schmoeger and Bermudez-Miller are obtained. As a corollary we also show that each non-trivial operator commuting with a generalized backward shift is supercyclic. This gives a positive answer to a conjecture of Godefroy and Shapiro. Furthermore, we show that the norm-closures of t...

The backward shift B on the Bergman space of the unit disc is known to be hypercyclic (meaning: it has a dense orbit). Here we ask: “Which operators that commute with B inherit its hypercyclicity?” We show that the problem reduces to the study of operators of the form φ(B) where φ is a holomorphic self-map of the unit disc that multiplies the Dirichlet space into itself, and that the question o...

In this paper, we introduce para-chaotic tuples of operators and we give some relations between para-chaoticity and Hypercyclicity Criterion for a tuple of operators.