Holomorphic endomorphisms of \(\mathbb{P}^{3}(\mathbb{C})\) related to a Lie algebra of type \(A_{3}\) and catastrophe theory. (English) Zbl 1380.37099

Uchimura, Keisuke

Kyoto J. Math. 57, No. 1, 197-232 (2017).

The paper under review deals with dynamics of a class of generalized Chebyshev maps in three complex variables, denoted by \(P^d_{A_3}\). The considered maps are seen as holomorphic endomorphisms on the complex projective space \(\mathbb{P}^{3}(\mathbb{C})\), and their dynamical properties can be deduced thanks to the fact that they are related to the map \((t_1,t_2,t_3)\mapsto(t_1^d,t_2^d,t_3^d)\) via a commutative diagram.
Given a holomorphic endomorphism of \(\mathbb{P}^{k}(\mathbb{C})\), denote by \(T\) its Green current and set \(T^l\) the product \(T\wedge\cdots \wedge T\) with \(l\) terms. The \(l\)-th Julia set \(J_l\) is defined as the support of \(T^l\). The author is able to explicitly determine for the considered maps four types of Julia sets: \(J_1, J_2, J_3\), and the second Julia set \(J_\Pi\) of the restriction to the hyperplane \(\Pi\) at infinity.
The descriptions of \(J_1\) and \(J_2\) are achieved by studying the external rays and applying the results of E. Bedford and M. Jonsson [Am. J. Math. 122, No. 1, 153–212 (2000; Zbl 0941.37027)] on regular polynomial endomorphisms. The key result to obtain a description of \(J_3\) is given by J.-Y. Briend and J. Duval’s equidistribution result [Acta Math. 182, No. 2, 143–157 (1999; Zbl 1144.37436)]. To describe \(J_\Pi\) the author uses M. Jonsson’s results on polynomial skew products [Math. Ann. 314, No. 3, 403–447 (1999; Zbl 0940.37018)].
The author also investigates the relation between the set of critical values of the considered maps and catastrophe theory.

Reviewer: Jasmin Raissy (Toulouse)

MSC:

37F45	Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
32H50	Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
58K35	Catastrophe theory

Keywords:

dynamical system; catastrophe theory; Chebyshev endomorphism

Citations:

Zbl 0941.37027; Zbl 1144.37436; Zbl 0940.37018