Frédéric Bayart

Frequently hypercyclic operators

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.

Difference sets and frequently hypercyclic weighted shifts

We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on

HYPONORMAL OPERATORS, WEIGHTED SHIFTS AND WEAK FORMS OF SUPERCYCLICITY

\ell^p(\mathbb Z)

The linear fractional model on the ball

,

Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

p\geq 1

On the Hausdorff Dimension of Graphs of Prevalent Continuous Functions on Compact Sets

. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is

Multifractal spectra of typical and prevalent measures

\mathcal U

Porosity and hypercyclic operators

-frequently hypercyclic, yet not frequently hypercyclic and that there exists an operator which is frequently hypercyclic, yet not distributionally chaotic. These (surprizing) counterexamples are given by weighted shifts on

Similarity to an isometry of a composition operator

c_0

Topologically transitive skew-products of operators

. The construction of these shifts lies on the construction of sets of positive integers whose difference sets have very specific properties.