Notes on Hypercyclicity of Composition Operator on the Weighted Dirichlet Spaces

In this paper, the hypercyclic behavior of multiples of composition operators induced by linear fractional self-maps of unit disk on the weighted Dirichlet spaces D-alpha(p) will be presented. Using the classification of linear fractional self-maps, we show that composition operators exhibit strikingly diverse types of hypercyclic behavior. Furthermore, Colonna and Martinez-Avebdafio raised the open question: When p - 2 < alpha < p, does it exist hypercyclic composition operators on D-alpha(p)? Guo and Shu answered the question when p > 3. In this paper, we show that under the condition p - 2 < alpha < p, there is hypercyclic composition operators on D-alpha(p) for p > 1.