Henon mappings in the complex domain II: projective and inductive limits of polynomials

TL;DR

This paper explores the topological structure of key invariant sets in Henon mappings in the complex domain, especially under small perturbations, using projective and inductive limits related to the polynomial p.

Contribution

It characterizes the topological structure of invariant sets for hyperbolic polynomials under small Henon perturbations using advanced limit constructions.

Findings

Identifies the structure of invariant sets when p is hyperbolic and |a| is small

Uses projective and inductive limits to describe these sets

Provides a topological description of the sets K, J, K_+/-, J_+/-

Abstract

Let H: C^2 -> C^2 be the Henon mapping given by (x,y) --> (p(x) - ay,x). The key invariant subsets are K_+/-, the sets of points with bounded forward images, J_+/- = the boundary of K_+/-, J = the union of J_+ and J_-, and K = the union of K_+ and K_-. In this paper we identify the topological structure of these sets when p is hyperbolic and |a| is sufficiently small, ie, when H is a small perturbation of the polynomial p. The description involves projective and inductive limits of objects defined in terms of p alone.