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

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.