On Closed Invariant Sets in Local Dynamics

We investigate the dynamical behaviour of a holomorphic map on a $f-$invariant subset $\mathcal{C}$ of $U,$ where $f:U \to \mathbb{C}^k.$ We study two cases: when $U$ is an open, connected and polynomially convex subset of $\mathbb{C}^k$ and $\mathcal{C} \subset \subset U,$ closed in $U,$ and when $\partial U$ has a p.s.h. barrier at each of its points and $\mathcal{C}$ is not relatively compact in $U.$ In the second part of the paper, we prove a Birkhoff's type Theorem for holomorphic maps in several complex variables, i.e. given an injective holomorphic map $f,$ defined in a neighborhood of $\overline{U},$ 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 $\overline{U}$ which touches $\partial U.$