Characterization of Dynamics of a Class of Polynomial Automorphisms in ℂn

A kind of higher-dimensional complex polynomial mappings [Formula: see text] is considered: [Formula: see text] where [Formula: see text], [Formula: see text] are polynomials with degrees higher than one, and [Formula: see text] are nonzero complex numbers, [Formula: see text]. Assume that each [Formula: see text] is hyperbolic on its Julia set and [Formula: see text] is sufficiently small, [Formula: see text], then there exists a bounded set on which the dynamics on the forward and backwards Julia sets are described by using the inductive and the projective limits, respectively. These results are a natural higher-dimensional generalization of the work of Hubbard and Oberste-Vorth on two-dimensional complex Hénon mappings. The combination of the symbolic dynamics and the crossed mapping is also applied to study the complicated dynamics of a class of polynomial mappings in [Formula: see text].

Attractors of Compactly Generated Semigroups of Regular Polynomial Mappings

We investigate the metric space of pluriregular sets as well as the contractions on that space induced by infinite compact families of proper polynomial mappings of several complex variables. The topological semigroups generated by such families, with composition as the semigroup operation, lead to the constructions of a variety of Julia-type pluriregular sets. The generating families can also be viewed as infinite iterated function systems with compact attractors. We show that such attractors can be approximated both deterministically and probabilistically in a manner of the classic chaos game.

Contribution to the Jacobian Conjecture: Polynomial Mapping Having Two Zeros at Infinity

This article contains the theorems concerning the algebraic dependence of polynomial mappings with the constant Jacobian having two zeros at infinity. The work is related to the issues of the classical Jacobian Conjecture. This hypothesis affirm that the polynomial mapping of two complex variables with constant non-zero Jacobian is invertible. The Jacobian Conjecture is equivalent to the fact that polynomial mappings with constant non-zero Jacobian do not have two zeros at infinity, therefore it  is equivalent to the two theorems given in the work. The proofs of these theorems proceeds by induction.

ON EQUATIONS GENERATED BY NONLINEAR NILPOTENT MAPPINGS

A generalization of a nilpotent linear operator concept is proposed for nonlinear mapping acting from R^2 to R^2. The properties of nonlinear nilpotent mappings are investigated. Criterions of nilpotence for differentiable and polynomial mappings are obtained.