Algebraic degrees for iterates of meromorphic self-maps of $\mathbb{P}^k$

We first introduce the class of quasi-algebraically stable meromorphic maps of
$\mathbb{P}^k$. This class is strictly larger than that of algebraically stable meromorphic self-maps of $\mathbb{P}^k$. Then we prove that all maps in the new class enjoy a recurrent property. In particular, the algebraic degrees for iterates of these maps can be computed and their first dynamical degrees are always algebraic integers.