Holomorphic endomorphisms of P^3(C) related to a Lie algebra of type A_3 and catastrophe theory

TL;DR

This paper studies a class of holomorphic endomorphisms on complex projective 3-space, analyzing their Julia sets and external rays, revealing geometric structures like Moebius strips and connections to binary quartic forms.

Contribution

It extends complex dynamics to higher dimensions by explicitly determining Julia sets and exploring geometric and algebraic relations of these maps.

Findings

Julia sets include Moebius strips and ruled surfaces

Exact determination of external rays and Julia sets

Connections established between maps and binary quartic forms

Abstract

Chebyshev maps in the complex plane are typical chaotic maps. Veselov generalized these map. We consider a class of those maps and view them as holomorphic endomorphisms on the 3-dimensional complex projective space and make use of the theory of complex dynamics in higher dimension. We determine Julia sets and external rays of those maps, exactly. In the Julia sets, Moebius strip and a special ruled surface appear. We also show some relation between those maps and binary quartic forms.