Hypercyclicity is the study of linear and continuous operators that possess a dense orbit. Given a separable, infinite dimensional topological vector space X, we say a continuous linear operator T is hypercyclic if there exists a vector x in X such that its orbit Orb(T,x)={x, Tx, T²x, …} is dense in X.
Many interesting phenomena appear when analyzing the behavior of iterates of linear and continuous operators, in particular we emphasize the existence of several zero-one laws. We first note that, if an operator T has a hypercyclic vector, it has a dense Gδ set of such vectors, and hence the set of hypercyclic vectors for an operator is either empty or very large in a topological sense. Next, by proving that a somewhere dense orbit is everywhere dense, P. S. Bourdon and N. S. Feldman showed a second zero-one law which states that either an orbit Orb(T,x) is nowhere dense or it is dense in the whole space.
In my dissertation we uncovered the existence of another such zero-one law for certain classes of operators. We showed that for a weighted backward shift on ℓp to be hypercyclic it suffices to require the operator to have an orbit Orb(T,x) with a single non-zero limit point, thus relaxing Bourdon and Feldman’s condition of having a dense orbit in some open subset of X. However, our condition does not guarantee that the original orbit Orb(T,x) is dense in X, nonetheless we can demonstrate how to construct a hypercyclic vector for T by using the non-zero limit point of the orbit. Even more interestingly, the condition above can be relaxed to simply requiring that the orbit has infinitely many members in a ball whose closure avoids the zero vector.
To summarize this behavior of weighted backward shifts, we emphasize that a shift T is not hypercyclic if and only if every set of the form Orb(T,x)∪{0} is closed in ℓp . Thus we showed the existence of a zero-one law for the hypercyclicity of these shifts, which states that either no orbit has a non-zero limit point in ℓp; or some orbit has every vector in ℓp as a limit point.
Furthermore we showed that this zero-one law for the hypercyclic behavior of shifts is also shared by other classes of operators, in particular the adjoints of the multiplication operators on the Bergman space A2(Ω) for an arbitrary region Ω. To achieve this we cannot borrow techniques used for the shift operators, but instead we have to take a function theoretical approach.
However, we also showed that this behavior does not generalize to all classes of operators, namely we provided an example of a linear fractional composition operator on the Hardy space H2(𝔻) that is not hypercyclic, and yet it has an orbit with a non-constant limit point.
To summarize the importance of our results, we would like to point out that in our endeavor to study the phenomena of hypercyclicity it is important to understand how an operator fails to be hypercyclic. Having proved that for certain classes of operators, a non-hypercyclic operator can at most have the zero vector as an orbital limit point, we have shown that these operators fail at having a dense orbit in quite a dramatic way. Thus we described the hypercyclic behavior of certain operators as a zero-one law of orbital limit points, and so we have uncovered another facet of hypercyclicity associated with dichotomous behavior.