Hypercyclic linear fractional composition operators on weighted Dirichlet spaces Dαp

Abstract

Recently, F. Colonna and R. A. Martínez-Avendaño raised the open question that whether or not the weighted Dirichlet spaces D α p can support hypercyclic composition operators when p − 2 < α < p . In this paper, we investigate the hypercyclicity of composition operators on D α p and partially answer the question. Specifically, under the assumption p − 2 < α < p , we show that the composition operator induced by a parabolic automorphism or a hyperbolic automorphism of the unit disk is hypercyclic on D α p if p > 3 . Furthermore, the composition operator induced by a hyperbolic non automorphism is hypercyclic on D α p for all p > 1 and p − 1 < α < p .