Recently, I am studying polynomial-like map in higher dimension, according to the book "Holomorphic Dynamical Systems" (Lecture Notes in Mathematics), page 235.
Suppose $U, V$ are open subsets of $\mathbb{C}^{n}$, where $n\geq 1$. We assume $U\subset \subset V$ and $V$ is a convex open subset. A holomorphic map $f: U\to V$ is called polynomial-like if it is proper, i.e. the preimage of a compact set is a compact set. This property gives me a feeling that points closed to the boundary of $U$ are mapped to points closed to the boundary of $V$.
Is $f$ between such $U$ and $V$ is an open map? If it is an open map, we know $f$ is surjective (properness implies closed map). Then by openness of $f$, we know critical points of $f$ are at most hypersurfaces and then $f$ is a ramified covering map.
But I cannot figure out whether $f$ is open or not, and I also don't know how to use convexity of $V$.