Common boundary regular fixed points for holomorphic semigroups in strongly convex domains

TL;DR

This paper investigates boundary regular fixed points for holomorphic semigroups in strongly convex domains, establishing that isolated fixed points for one semigroup element are fixed points for all times, and also explores backward iteration sequences.

Contribution

It proves that isolated boundary regular fixed points for a semigroup element are fixed points for all times, extending understanding of boundary behavior in complex dynamics.

Findings

Isolated boundary fixed points are fixed for all semigroup times.

Backward iteration sequences are studied for elliptic self-maps.

Boundary regularity is preserved across the semigroup.

Abstract

Let $D$ be a bounded strongly convex domain with smooth boundary in $\mathbb C^N$. Let $(\phi_t)$ be a continuous semigroup of holomorphic self-maps of $D$. We prove that if $p\in \partial D$ is an isolated boundary regular fixed point for $\phi_{t_0}$ for some $t_0>0$, then $p$ is a boundary regular fixed point for $\phi_t$ for all $t\geq 0$. Along the way we also study backward iteration sequences for elliptic holomorphic self-maps of $D$.