We study multiply recurrent and hypercyclic operators as a special case of $\mathcal F$-hypercyclicity, where F$ is the family subsets natural numbers containing arbitrarily long arithmetic progressions. prove several properties operators, we characterize those which are weakly mixing recurrent, show that there but not mixing.

We characterize disjoint hypercyclic and supercyclic tuples of unilateral Rolewicz-type operators on $c_0(\N)$ $\ell^p(\N)$, $p \in [1, \infty)$, which are a generalization the backward shift operator. show that hypercyclicity supercyclicity equivalent among subfamily these always satisfy Disjoint Hypercyclicity Criterion. also simultaneous \infty)$.

We prove the spaceability of set hypercyclic vectors for shift-like operators. Shift-like operators appear naturally as composition on Lp(X), when underlying space X is dissipative. In process proving main theorem, we provide, among other results independent interest, a characterization weakly mixing dissipative bounded distortion.

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...