Compactness of the chain recurrent sets of polynomial maps

In this paper, a class of polynomial maps on $\Bbb R^m$ or $\Bbb C^m$ is considered; this class is defined by the assumption that the difference equation induced by the map has leading monomial of a single variable. It is shown that for any map from this class, the nonwandering set and also the chain recurrent set are bounded while for all unbounded orbits, some kind of monotonicity takes place. Results of this paper generalize those of on boundedness of the nonwandering set for real polynomial maps for which the difference equation has leading monomial of a single variable.