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

TL;DR

This paper investigates the extension of conjugating homeomorphisms between semigroups of holomorphic self-maps of the unit disc and introduces topological invariants for these semigroups, enhancing understanding of their structure.

Contribution

It proves that conjugating homeomorphisms extend to the boundary outside exceptional arcs and introduces topological invariants for semigroups of holomorphic self-maps.

Findings

Homeomorphisms extend to boundary outside maximal contact arcs.

For elliptic semigroups, extension is to the entire boundary.

Provides new topological invariants for semigroup classification.

Abstract

Let $(\varphi_t)$, $(\phi_t)$ 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 $f \circ \phi_t=\varphi_t \circ f$ for all $t\geq 0$, 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.