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

Abstract

The typical chaotic maps f(x)=4x(1−x) and g(z)=z2−2 are well known. Veselov generalized these maps. We consider a class of maps PA3d of those generalized maps, view them as holomorphic endomorphisms of P3(C), and make use of methods of complex dynamics in higher dimension developed by Bedford, Fornaess, Jonsson, and Sibony. We determine Julia sets J1,J2,J3,JΠ and the global forms of external rays. Then we have a foliation of the Julia set J2 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 J3 and joins to a Möbius strip, which is the Julia set JΠ in the plane at infinity in P3(C). Rulings of the Möbius strip correspond to rulings of the surface of J3 by external rays.