We study global questions of iteration for holomorphic self maps of $\mathbb {P}^k$. After discussing some basic properties of holomophic and meromorphic maps in $\mathbb {P}^k$, we describe the maps $f$ in $\mathbb {P}^2$ for which there exists a variety $V$ satisfying $f^{-1} (V) = V$. We show that for a Zariski dense set of holomorphic maps in $\mathbb {P}^2$ the complement of the critical orbit is Kobayashi hyperbolic. We then study expansive properties of the maps in the interior of the complement of the critical orbit, under suitable hyperbolicity assumptions. We finally ify maps in $\mathbb {P}^2$ such that the orbit of the critical set is a variety.