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

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