Compact Groups of Differentiable Transformation

H. Cartan' has investigated automorphisms and general transformations into itself of a domain in several complex variables by means of complex analytic functions. His main results are as follows: (i) If a compact group of automorphisms has a fix point then in suitably chosen local coordinates around the fix point all transformations are linear. (ii) In a compact group of automorphisms all sufficiently small subgroups are the identity.2 (iii) If a transformation of a bounded domain into (part of) itself has an invariant point 0, and is of the special form