  For convex domains with $C^{1,\epsilon}$ boundary we give a precise
description of the automorphism group: if an orbit of the automorphism group
accumulates on at least two different closed complex faces of the boundary,
then the automorphism group has finitely many components and the connected
component of the identity is the almost direct product of a compact group and a
non-compact connected simple Lie group with real rank one and finite center. In
this case, we also show the limit set is homeomorphic to a sphere and prove a
gap theorem: either the domain is biholomorphic to the unit ball (and the limit
set is the entire boundary) or the limit set has co-dimension at least two in
the boundary.

Complex Variables;Differential Geometry