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April 2017 Holomorphic endomorphisms of P3(C) related to a Lie algebra of type A3 and catastrophe theory
Keisuke Uchimura
Kyoto J. Math. 57(1): 197-232 (April 2017). DOI: 10.1215/21562261-3759576


The typical chaotic maps f(x)=4x(1x) and g(z)=z22 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.


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Keisuke Uchimura. "Holomorphic endomorphisms of P3(C) related to a Lie algebra of type A3 and catastrophe theory." Kyoto J. Math. 57 (1) 197 - 232, April 2017.


Received: 25 December 2014; Accepted: 3 March 2016; Published: April 2017
First available in Project Euclid: 11 March 2017

zbMATH: 1380.37099
MathSciNet: MR3621786
Digital Object Identifier: 10.1215/21562261-3759576

Primary: 37F45, 58K35
Secondary: 22E10, 32H50, 37F10

Rights: Copyright © 2017 Kyoto University


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Vol.57 • No. 1 • April 2017
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