The purpose of the present work is to treat a new notion related to linear dynamics, which can be viewed as a “localization” of the notion of hypercyclicity. In particular, let T be a bounded linear operator acting on a Banach space X and let x be a non-zero vector in X such that for every open neighborhood U ⊂ X of x and every non-empty open set V ⊂ X there exists a positive integer n such tha...

We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space `(N), any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for c0(N) or `(N), 1 < p <∞. Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.

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 give a sufficient condition for two operators to be disjointly frequently hypercyclic. apply this criterion composition acting on H(D) or the Hardy space H2(D). simplify result disjoint frequent hypercyclicity of pseudo shifts recent paper Martin et al. and we exhibit hypercyclic weighted shifts.