Bounded Nonwandering Sets for Polynomial Mappings

In this paper, we consider a class of polynomial mappings on Rm or Cm which is defined by the assumption that the delay equations induced by the mappings have leading monomials in a single variable. We show that for any mapping from this class, the nonwandering set is bounded while for all unbounded orbits, some kind of monotonicity takes place. The class under consideration is proved to contain, in particular, the generalized Hénon mappings and the Arneodo–Coullet–Tresser mappings.