Hypercyclic composition operators with automorphisms of the upper half-plane

On the Frechet space H(P) of all holomorphic functions on the open upper half-plane P, we study universal sequences of composition operators C-sigma n : H(P) -> H(P) given by automorphisms sigma(n)(z) =(a(n)z+ b(n))/(c(n)z+ d(n)) of P, where a(n), b(n), c(n), d(n) is an element of R with the normalization a(n)d(n) - b(n)c(n) = 1. We show that a sequence C-sigma n : H(P) -> H(P) is universal if and only if lim sup{|a(n)| +|b(n)| +|c(n)| +|d(n)|} = infinity, which in turn is equivalent to the existence of a point zeta in R boolean OR{infinity} such that some subsequence sigma(nk)(z) -> zeta uniformly on compact subsets of P. Applying these conditions to the case when each sigma(n) is the n-fold composition sigma(n)(z) = sigma omicron center dot center dot center dot omicron sigma(z) of a non-identity automorphism sigma(z) = (az + b)/(cz + d), where a, b, c, d is an element of Rwith the normalization ad - bc = 1, we show that C-sigma 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 zeta for which sigma(n)(z) -> zeta 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. (c) 2022 Elsevier Inc. All rights reserved.