Hypercyclicity of Composition Operators on Banach Spaces of Analytic Functions

It is well known that there exist hypercyclic composition operators on the Hardy spaces for and a host of other Banach spaces of analytic functions. In this work, we give a classification of a large class of separable Banach spaces X of analytic functions on the open unit disk according to whether or not hypercyclic composition operators on X exist. We highlight how the use of the Hypercyclicity Comparison Principle and information on the relationship among such spaces, paired with known results in the literature, allow us to extend them to many other spaces. Examples of spaces for which no hypercyclic composition operators exist are the little -Bloch spaces for , the analytic Besov spaces, the space of analytic functions of vanishing mean oscillation, the spaces of analytic functions whose derivative belongs to the Hardy space for , and some weighted Dirichlet spaces. By contrast, the little -Bloch spaces for , all weighted Bergman spaces with weight (, ), and a class of weighted Dirichlet spaces, mimic the behavior of the Hardy spaces.