Isometries in the symmetrized bidisc, II

Abstract

We prove that every map $f:U\to\GG$ preserving the Poincar\'e distance and the Kobayashi distance is holomorphic or anti-holomorphic, where $U$ is a connected open subset of the unit disc. We also study the nonroyal automorphism orbits of $\GG$. A $C^1$ map on a connected relatively open part of such an orbit which preserves the restriction of the ambient Kobayashi--Royden metric is the restriction of a global automorphism or anti-automorphism of $\GG$. The same conclusion holds for maps preserving the ambient Kobayashi distance.