Hypercyclic composition operators with automorphisms of the upper half-plane

Abstract

On the Fréchet space H(P) of all holomorphic functions on the open upper half-plane P, we study universal sequences of composition operators Cσn:H(P)→H(P) given by automorphisms σn(z)=(anz+bn)/(cnz+dn) of P, where an,bn,cn,dn∈R with the normalization andn−bncn=1. We show that a sequence Cσn:H(P)→H(P) is universal if and only if limsup{|an|+|bn|+|cn|+|dn|}=∞, which in turn is equivalent to the existence of a point ζ in R∪{∞} such that some subsequence σnk(z)→ζ uniformly on compact subsets of P. Applying these conditions to the case when each σn is the n-fold composition σn(z)=σ∘⋯∘σ(z) of a non-identity automorphism σ(z)=(az+b)/(cz+d), where a,b,c,d∈R with the normalization ad−bc=1, we show that Cσ is hypercyclic if and only if |a+d|≥2. Furthermore we obtain an explicit formula, in terms of a,b,c,d, of the point ζ for which σn(z)→ζ uniformly on compact subsets of P. Motivated by our results for the region P, we obtain analogous results when the region is the open unit disk. Finally we generalize the aforementioned limit-point result to hypercyclic composition operators with automorphisms on a simply connected region whose complement has a nonempty interior.