Abstract. We investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators T on separable complex F -spaces: T is frequently hypercyclic if there exists a vector x such that for every nonempty open subset U of X, the set of integers n such that T n x belongs to 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.

We give conditions for an operator T on a complex separable Banach space X with sufficiently many eigenvectors associated to eigenvalues of modulus 1 to admit a non-degenerate invariant Gaussian measure with respect to which it is weak-mixing. The existence of such a measure depends on the geometry of the Banach space and on the possibility of parametrizing the T-eigenvector fields of T in a regular way. We also investigate the connection with frequent hypercyclicity.

Given a Polish topology τ on B1pXq, the set of all contraction operators on X " ℓp, 1 ď p ă 8 or X " c0, we prove several results related to the following question: does a typical T P B1pXq in the Baire Category sense has a non-trivial invariant subspace? In other words, is there a dense G δ set G Ď pB1pXq, τ q such that every T P G has a non-trivial invariant subspace? We mostly focus on the Strong Operator Topology and the StrongO perator Topology.

We show that an operator on a separable complex Banach space with sufficiently many eigenvectors associated to eigenvalues of modulus 1 is hypercyclic. We apply this result to construct hypercyclic operators with prescribed K unimodular point spectrum. We show how eigenvectors associated to unimodular eigenvalues can be used to exhibit common hypercyclic vectors for uncountable families of operators, and prove that the family of composition operators C on H 2 (D), where is a disk automorphism having +1 as attractive fixed point, has a residual set of common hypercyclic vectors.