Common Hypercyclic Vectors for Composition Operators

A continuous operator acting on a topological vector space X is called hypercyclic provided there exists a vector x ∈ X such that its orbit {T nx; n ≥ 1} is dense in X. Such a vector is called a hypercyclic vector for T . The set of hypercyclic vectors will be denoted by HC(T ). The first example of hypercyclic operator was given by Birkhoff, 1929 [3], who shows that the operator of translation by a non-zero complex number is hypercyclic on the space of holomorphic functions. For a complete account on hypercyclicity, we refer to [8]. The main focus of our study is the hypercyclic behavior for composition operators. Let us denote by H2(D) the Hardy space on the unit disk D, and by Aut(D) the set of automorphisms of D. For φ in Aut(D), the hypercyclicity of the composition operator Cφ defined on H 2(D) by Cφ(f) = f ◦ φ is well-understood since the work of Bourdon and Shapiro [5] :