Abstracts Category : Other

Hypercyclic Algebras and Affine Dynamics

by Dimitrios Papathanasiou

An operator T on a Frchet space X is said to behypercyclic if it has a dense orbit. In that case, the set HC(T) ofhypercyclic vectors for T is a dense Gdsubset of X. In most cases the set HC(T){0} is not a vector space.However, Herrero and Bourdon showed that if T is hypercyclic thenHC(T) contains a hypercyclic manifold, that is a dense linearsubspace of X except for the origin. In a different direction, agreat amount of research has been carried out in the search ofhypercyclic subspaces, that is infinite dimensional closedsubspaces contained (excluding the origin) in HC(T). It is notalways the case that a hypercyclic operator has a hypercyclicsubspace. For instance, Rolewicz's operator onl2 does not have a hypercyclic subspace, buton the other hand all hypercyclic convolution operators on thespace H(C) of entire functions have hypercyclic subspaces.If thespace X is a Frchet algebra, continuing the search for structurein HC(T) one may ask whether HC(T){0} contains an algebra. In thatdirection, Aron, Conejero, Peris and Seoane-Seplveda showed thatthe translation operators on H(C) do not support a hypercyclicalgebra. On the other hand, Shkarin and independently Bayart andMatheron showed that the complex differentiation operator D on H(C)has a hypercyclic algebra.In the present dissertation we firstcontinue the search for hypercyclic algebras in the setting ofconvolution operators on H(C). Following Bayart and Matheron'stechniques, we extend their above mentioned result with Shkarin, byestablishing that P(D) supports a hypercyclic algebra whenever P isa non-constant polynomial vanishing at 0.With a different approachwe provide a geometric condition on the set {z: |F(z)|=1} whichensures the existence of hypercyclic algebras for F(D) with F H(C) of exponential type. This new approach not only recovers theresult of Shkarin-Bayart and Matheron but also gives hypercyclicalgebras for convolution operators F(D) which do not satisfy theconditions F(0)=0 or that F be a polynomial, such as I+D,DeD, eD-1, orcos(D).Answering a question of Seoanne-Seplveda, we show that theoperator D supports hypercyclic algebras that are not singlygenerated. We next consider hypercyclic algebras beyond the settingof convolution operators. For instance, we provide abstractcriteria for the existence of hypercyclic algebras, which in asense generalize familiar results from Linear Dynamics. We alsoshow that every hypercyclic weighted backward shift operator onl2 supports a hypercyclic algebra.Finally,on a completely different direction we study the dynamic behaviorof affine maps, that is, maps of the form A=T+a where T is a linearmap and a is a vector of the underlying space. We prove that inmany cases the dynamic behavior of A is identical to that of itslinear part T. We also show that if A is hypercyclic then T has tobe hypercyclic as well. The converse is not true however by anexample due to Shkarin, who provided a hypercyclic operator T onAdvisors/Committee Members: Bes, Juan (Advisor).