Hypercyclic composition operators on the $$S^p $$ S p space with automorphism symbols

Abstract

Let \(S^p\) be the space of holomorphic functions whose derivative lies in the classical Hardy space \(H^p\) over the unit disk. We prove in this paper that the composition operator \(C_\varphi \) with \(\varphi \) an automorphism is hypercyclic on \(S^p\), \(0<p<1\), if and only if \(\varphi \) has no interior fixed point. This answers affirmatively a problem posed by Colonna and Martinez-Avendano in the paper “Hypercyclicity of composition operators on Banach spaces of analytic functions” (Complex Anal Oper Theory 12(1): 305–323, 2018).