  We first introduce the class of quasi-algebraically stable meromorphic maps
of $\P^k.$ This class is strictly larger than that of algebraically stable
meromorphic self-maps of $\P^k.$
  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.
