On closed invariant sets in local dynamics

We investigate the dynamical behaviour of a holomorphic map on an f-invariant subset C of U, where f : U → C k . We study two cases: when U is an open, connected and polynomially convex subset of C k and C ⋐ U , closed in U, and when ∂U has a p.s.h. barrier at each of its points and C is not relatively compact in U. In the second part of the paper, we prove a Birkhoffʹs type theorem for holomorphic maps in several complex variables, i.e. given an injective holomorphic map f, defined in a neighborhood of U ¯ , with U star-shaped and f ( U ) a Runge domain, we prove the existence of a unique, forward invariant, maximal, compact and connected subset of U ¯ which touches ∂U.