Frequently hypercyclic operators

Abstract

We investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators T T on separable complex F \mathcal {F} -spaces: T T is frequently hypercyclic if there exists a vector x x such that for every nonempty open subset U U of X X , the set of integers n n such that T n x T^{n}x belongs to U U has positive lower density. We give several criteria for frequent hypercyclicity, and this leads us in particular to study linear transformations from the point of view of ergodic theory. Several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting.