Bull. London Math. Soc. 47 (2015) 136–142 C2014 London Mathematical Society doi:10.1112/blms/bdu104 Mixing Taylor shifts and universal Taylor series

Beise, T. Meyrath and J. Müller It is known that, generically, Taylor series of functions holomorphic in a simply connected complex domain exhibit maximal erratic behaviour outside the domain (so-called universality) and overconvergence in parts of the domain. Our aim is to show how the theory of universal Taylor series can be put into the framework of linear dynamics. This leads to a unified approach to universality and overconvergence and yields new insight into the boundary behaviour of Taylor series. 1. Mixing Taylor shifts By C∞, we denote the extended complex plane endowed with the spherical metric. For Ω ⊂ C∞ open, H(Ω) is the space of holomorphic functions on Ω that vanish at infinity in case that ∞ ∈ Ω. This space is endowed with the usual topology of uniform convergence on compact sets. In case 0 ∈ Ω, we define T = TΩ: H(Ω) → H(Ω) by Tf(z):= f(z) − f(0