Titre de la dissertation: »Frequently Hypercyclic Random Vectors »

Promoteur: Monsieur Karl Grosse-Erdmann

Résumé de la dissertation:Some results concerning the existence of almost surely frequently hypercyclic random vectors have been proved in the literature for certain chaotic weighted shifts. This is of interest for at least two reasons. It is usually difficult to find explicit (frequently) hypercyclic vectors, and random vectors have a probability distribution whose ergodic properties can be studied. The first objective of the thesis is to extend the previously known results. In particular, we prove that every chaotic weighted shift on very general sequence spaces and every operator satisfying the Frequent Hypercyclicity Criterion admits an almost surely frequently hypercyclic random vector.
We also investigate the case of semigroups. The desired random vector is constructed using a stochastic integral. Although our general result requires that this integral is well defined, we can apply it to the translation semigroups on the space of entire functions.
The second part of the thesis deals with the rate of growth of frequently hypercyclic functions. We present two methods. Recently, a probabilistic approach provided a quasi-optimal rate of growth for the dfferentiation operator and the Taylor shift. Based on these results and the first part of the thesis, we obtain a general criterion for chaotic weighted shifts. The rate of growth is expressed as a function depending only on the weights, multiplied by some logarithmic factor. We give several examples of shifts defined on the space of entire functions or the space of holomorphic functions on the unit disk, recovering previous results and finding new ones. We also consider the differentiation operators on the space of harmonic functions on the plane and weighted shifts on Köthe sequence spaces. The possible optimality of the growth is also discussed. On spaces of holomorphic functions, we can also ask whether the growth holds outside some small, but possibly unbounded, set. We give results in this direction, which are stated for general random complex series. This second approach seems to be new in linear dynamics. In particular, we prove that for any chaotic weighted shift, the growth sought by the previous method does hold outside such a set.