Topological invariants for semigroups of holomorphic self-maps of the unit disc

Abstract

Let $({\phi }_t)$, $({\varphi }_t)$ be two one-parameter semigroups of holomorphic self-maps of the unit disc $\mathbb{D}\subset \mathbb{C}$. Let $f:\mathbb{D}\to \mathbb{D}$ be a homeomorphism. We prove that, if $f\circ {\varphi }_t={\phi }_t\circ f$ for all $t\ge 0$, then $f$ extends to a homeomorphism of $\bar{\mathbb{D}}$ outside exceptional maximal contact arcs (in particular, for elliptic semigroups, $f$ extends to a homeomorphism of $\bar{\mathbb{D}}$). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disc.

arXiv comment28 pages, final version, to appear in J. Math. Pures Appl