RECURRENCE IN LINEAR DYNAMICS
Creators: name: PUIG DE DIOS, YUNIED
Publisher: name: Universit� degli Studi di Milano
Year: 2014

description: A bounded and linear operator is said to be hypercyclic if there
exists a vector such that its orbit under the action of the operator
is dense. The first example of a hypercyclic operator on a
Banach space was given in 1969 by Rolewicz who gave conditions for the unweighted unilateral backward shift on l2 to be hypercyclic. Among its features, we
can mention for example that finite-dimensional spaces cannot
support hypercyclic operators, proved by Kitai. On the
other hand, several people have shown in different contexts, in
the Hilbert space frame, that the set of hypercyclic vectors for
a hypercyclic operator is a G dense set.
This thesis is divided into four chapters. In the first one, we give
some preliminaries by mentioning some definitions and known
results that will be of great help later.
In chapter 2, we introduce a refinement of the notion of hypercyclicity,
relative to the set N(U; V ) when belonging to a certain collection F of subsets of N,
namely a bounded and linear operator T is called F-operator
if N(U; V ) belongs to F, for any pair of non-empty open sets U; V in X. First, we do an analysis of the hierarchy established between
F-operators, whenever F covers those families mostly studied
in Ramsey theory. Second, we investigate which kind of properties
of density can the sets N(x;U) and
N(U; V ) have for a given hypercyclic operator, and classify the
hypercyclic operators accordingly to these properties.
In chapter three, we introduce the following notion: an operator
T on X satisfies property PF if for any U non-empty open set in
X, there exists x in X such that N(x;U) belongs to F. Let BD the collection
of sets in N with positive upper Banach density. We generalize
the main result of a recent paper of Costakis and Parissis using a strong result of Bergelson
and Mccutcheon in the vein of Szemer�di?s theorem, leading
us to a characterization of those operators satisfying property
PBD. It turns out that operators having property PBD satisfy a
kind of recurrence described in terms of essential idempotents of
 N (the Stone-Cech compactification of N). We will discuss the
case of weighted backward shifts satisfying property PBD. On
the other hand, as a consequence we obtain a characterization
of reiteratively hypercyclic operators, i.e. operators for which
there exists x in X such that for any U non-empty open set in
X, the set N(x;U) belongs to BD.
The fourth chapter focuses on a refinement of the notion of disjoint
hypercyclicity. We extend a result of B�s, Martin, Peris
and Shkarin by stating: a weighted backward shift Bw is F-operator if
and only if (Bw; ... ;Bw^r) is d-F, for any r in N, where F runs along some filters strictly containing the family of cofinite sets,
which are frequently used in Ramsey theory. On the other hand,
we point out that this phenomenon does not occur beyond the
weighted shift frame by showing a mixing linear operator T on
a Hilbert space such that the tuple (T; T^2) is not d-syndetic.
We also, investigate the relationship between reiteratively hypercyclic
operators and d-F tuples, for filters F contained in
the family of syndetic sets. Finally, we examine conditions to
impose in order to get reiterative hypercyclicity from syndeticity
in the weighted shift frame.