Holomorphic semigroups of finite shift in the unit disc

TL;DR

This paper characterizes when a parabolic holomorphic semigroup in the unit disc has finite shift by providing three equivalent conditions involving asymptotic convergence speeds, hyperbolic metric properties, and Euclidean domain characteristics.

Contribution

It introduces three new necessary and sufficient conditions for finite shift in parabolic holomorphic semigroups, linking geometric and dynamical properties.

Findings

Three equivalent conditions for finite shift are established.

Conditions relate asymptotic speeds, hyperbolic metrics, and Euclidean properties.

Provides a comprehensive characterization of finite shift semigroups.

Abstract

We give three necessary and sufficient conditions so that a parabolic holomorphic semigroup $(\phi_t)$ in the unit disc is of finite shift. One is in terms of the asymptotic behavior of speeds of convergence, the second one is related to the hyperbolic metric of its Koenigs domain $\Omega$ and the latter one deals with Euclidean properties of $\Omega$.