Holomorphic endomorphisms of P3(C) related to a Lie algebra of type A3 and catastrophe theory

Abstract

The typical chaotic maps $f\left(x\right)=4x(1-x)$ and $g\left(z\right)=z^2-2$ are well known. Veselov generalized these maps. We consider a class of maps $P_{A_3}^d$ of those generalized maps, view them as holomorphic endomorphisms of ${\mathbb{P}}^3\left(\mathbb{C}\right)$, and make use of methods of complex dynamics in higher dimension developed by Bedford, Fornaess, Jonsson, and Sibony. We determine Julia sets $J_1,J_2,J_3,J_{\Pi }$ and the global forms of external rays. Then we have a foliation of the Julia set $J_2$ formed by stable disks that are composed of external rays.

We also show some relations between those maps and catastrophe theory. The set of the critical values of each map restricted to a real three-dimensional subspace decomposes into a tangent developable of an astroid in space and two real curves. They coincide with a cross section of the set obtained by Poston and Stewart where binary quartic forms are degenerate. The tangent developable encloses the Julia set $J_3$ and joins to a Möbius strip, which is the Julia set $J_{\Pi }$ in the plane at infinity in ${\mathbb{P}}^3\left(\mathbb{C}\right)$. Rulings of the Möbius strip correspond to rulings of the surface of $J_3$ by external rays.