Dynamics of weighted composition operators on Stein manifolds

In the present paper, we investigate the hypercyclicity of weighted composition operators acting on the space of holomorphic functions on a connected finite-dimensional Stein manifold. Let ψ be a holomorphic self-map on a connected Stein n-manifold Ω and ω ∈ H(Ω) a holomorphic function. We study the hypercyclicity of weighted composition operator Πψ,ω : H(Ω) → H(Ω) defined by Πψ,ω (f ) := ω · (f ◦ ψ) for every f ∈ H(Ω). We prove that Πψ,ω is hypercyclic if and only if ω(p)̸ = 0 at each p ∈ Ω, ψ is univalent without fixed points in Ω, ψ(Ω) is a Runge domain and for every compact holomorphically convex set K ⊂ Ω there is an integer n such that K ∩ ψ[n](K) = ∅ and their union is holomorphically convex.