## Kyoto Journal of Mathematics

### Holomorphic endomorphisms of $\mathbb{P}^{3}(\mathbb{C})$ related to a Lie algebra of type $A_{3}$ and catastrophe theory

Keisuke Uchimura

#### Abstract

The typical chaotic maps $f(x)=4x(1-x)$ and $g(z)=z^{2}-2$ are well known. Veselov generalized these maps. We consider a class of maps $P_{A_{3}}^{d}$ of those generalized maps, view them as holomorphic endomorphisms of ${\mathbb{P}^{3}}({\mathbb{C}})$, and make use of methods of complex dynamics in higher dimension developed by Bedford, Fornaess, Jonsson, and Sibony. We determine Julia sets $J_{1},J_{2},J_{3},J_{\Pi}$ and the global forms of external rays. Then we have a foliation of the Julia set $J_{2}$ 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 $J_{3}$ and joins to a Möbius strip, which is the Julia set $J_{\Pi}$ in the plane at infinity in ${\mathbb{P}}^{3}({\mathbb{C}})$. Rulings of the Möbius strip correspond to rulings of the surface of $J_{3}$ by external rays.

#### Article information

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

Dates
Accepted: 3 March 2016
First available in Project Euclid: 11 March 2017

https://projecteuclid.org/euclid.kjm/1489201237

Digital Object Identifier
doi:10.1215/21562261-3759576

Mathematical Reviews number (MathSciNet)
MR3621786

Zentralblatt MATH identifier
1380.37099

#### 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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