Abstract
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.
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A. Bialynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic variables. Proc. Amer. Math. Soc. 13(1962), 202–203.
R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping. Commun. Math. Phys. 67(1979), 137–146.
B.-S. Du, Bifurcation of periodic points of some diffeomorphisms on ℝ3. Nonlinear Anal. 9(1985), 309–319.
H. R. Dullin and J. D. Meiss, Generalized Hénon maps: the cubic diffeomorphisms of the plane. Phys. D 143(2000), 262–289.
S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorphisms. Ergodic Theory Dynam. Systems 9(1989), 67–99.
J. Milnor, Singular Points of Complex Hypersurfaces. Princeton University Press, Princeton, NJ (1968).
C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. 2nd ed., CRC Press, Boca Raton, FL (1999).
W. Rudin, Injective polynomial maps are automorphisms. Amer. Math. Monthly 102(1995), 540–543.
B. L. van der Waerden, Algebra. Vol. I. Springer-Verlag, New York (1991).
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Li, MC., Malkin, M. Bounded Nonwandering Sets for Polynomial Mappings. Journal of Dynamical and Control Systems 10, 377–389 (2004). https://doi.org/10.1023/B:JODS.0000034436.39278.37
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DOI: https://doi.org/10.1023/B:JODS.0000034436.39278.37