Completely Invariant Domains of Holomorphic Self-Maps on C

Let/(z) be a holomorph.self-map on C.-G-(0) with essential singularities 0 and It is proved that f(z) has a completdy invariant domain.D.F(f),then D is doubly connected and D contains all the singularities of the inverse of f(z),moreover,if f is of the finite type, then D=F(f). This result implies that f(z) has at most one completely invariant domain in F(f).