Hypercyclic Algebras and Affine Dynamics

An operator T on a Fréchet space X is said to be hypercyclic if it has a dense orbit. In that case, the set HC(T) of hypercyclic vectors for T is a dense Gδ subset 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 then HC(T) contains a hypercyclic manifold, that is a dense linear subspace of X except for the origin. In a different direction, a great amount of research has been carried out in the search of hypercyclic subspaces, that is infinite dimensional closed subspaces contained (excluding the origin) in HC(T). It is not always the case that a hypercyclic operator has a hypercyclic subspace. For instance, Rolewicz\u27s operator on ℓ2 does not have a hypercyclic subspace, but on the other hand all hypercyclic convolution operators on the space H(ℂ) of entire functions have hypercyclic subspaces. If the space X is a Fréchet algebra, continuing the search for structure in HC(T) one may ask whether HC(T)∪{0} contains an algebra. In that direction, Aron, Conejero, Peris and Seoane-Sepúlveda showed that the translation operators on H(ℂ) do not support a hypercyclic algebra. On the other hand, Shkarin and independently Bayart and Matheron showed that the complex differentiation operator D on H(ℂ) has a hypercyclic algebra. In the present dissertation we first continue the search for hypercyclic algebras in the setting of convolution operators on H(ℂ). Following Bayart and Matheron\u27s techniques, we extend their above mentioned result with Shkarin, by establishing that P(D) supports a hypercyclic algebra whenever P is a non-constant polynomial vanishing at 0. With a different approach we provide a geometric condition on the set {z: |Φ(z)|≤1} which ensures the existence of hypercyclic algebras for Φ(D) with Φ ∈ H(ℂ) of exponential type. This new approach not only recovers the result of Shkarin-Bayart and Matheron but also gives hypercyclic algebras for convolution operators Φ(D) which do not satisfy the conditions Φ(0)=0 or that Φ be a polynomial, such as I+D, DeD, eD-1, or cos(D). Answering a question of Seoanne-Sepúlveda, we show that the operator D supports hypercyclic algebras that are not singly generated. We next consider hypercyclic algebras beyond the setting of convolution operators. For instance, we provide abstract criteria for the existence of hypercyclic algebras, which in a sense generalize familiar results from Linear Dynamics. We also show that every hypercyclic weighted backward shift operator on ℓ2 supports a hypercyclic algebra. Finally, on a completely different direction we study the dynamic behavior of affine maps, that is, maps of the form A=T+a where T is a linear map and a is a vector of the underlying space. We prove that in many cases the dynamic behavior of A is identical to that of its linear part T. We also show that if A is hypercyclic then T has to be hypercyclic as well. The converse is not true however by an example due to Shkarin, who provided a hypercyclic operator T on ℓ2 and a specific a ∈ ℓ2 such that A=T+a is not hypercyclic. Furthermore, we generalize several results from linear dynamics to the affine setting, as well as discuss some open questions and provide partial answers to those