Kyoto Journal of Mathematics

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

Keisuke Uchimura

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Abstract

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.

Article information

Source
Kyoto J. Math., Volume 57, Number 1 (2017), 197-232.

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

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1489201237

Digital Object Identifier
doi:10.1215/21562261-3759576

Mathematical Reviews number (MathSciNet)
MR3621786

Zentralblatt MATH identifier
1380.37099

Subjects
Primary: 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations 58K35: Catastrophe theory
Secondary: 22E10: General properties and structure of complex Lie groups [See also 32M05] 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04] 32H50: Iteration problems

Keywords
dynamical systems catastrophe theory Chebyshev endomorphisms

Citation

Uchimura, Keisuke. Holomorphic endomorphisms of $\mathbb{P}^{3}(\mathbb{C})$ related to a Lie algebra of type $A_{3}$ and catastrophe theory. Kyoto J. Math. 57 (2017), no. 1, 197--232. doi:10.1215/21562261-3759576. https://projecteuclid.org/euclid.kjm/1489201237


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