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

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 $\overline{\mathbb D}$ outside exceptional maximal contact arcs (in particular, for elliptic semigroups, $f$ extends to a homeomorphism of $\overline{\mathbb D}$). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disc.