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

Abstract

Let (φt), (ϕt) be two one-parameter semigroups of holomorphic self-maps of the unit disk D⊂C. Let f:D→D be a homeomorphism. We prove that, if f∘ϕt=φt∘f for all t≥0, then f extends to a homeomorphism of D‾ outside exceptional maximal contact arcs (in particular, for elliptic semigroups, f extends to a homeomorphism of D‾). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disk.