Totally geodesic homeomorphisms between Teichmüller spaces

First, we show that a projective measured foliation is a Busemann point, in Gardiner–Masur boundary, if and only if it is indecomposable. Let f : Tg,n → Tg,n be a totally geodesic homeomorphism and suppose that f admits a homeomorphic extension to ∂GMTg,n. We show that f induces a simplicial automorphism of curve complex. Moreover, the restriction of f on Tg,n is an isometry. As an application, we obtain an alternative proof of Royden’s Theorem.