Holomorphic endomorphisms of \(\mathbb{P}^{3}(\mathbb{C})\) related to a Lie algebra of type \(A_{3}\) and catastrophe theory (Q522120)

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 \textit{E. Bedford} and \textit{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 \textit{J.-Y. Briend} and \textit{J. Duval}'s equidistribution result [Acta Math. 182, No. 2, 143--157 (1999; Zbl 1144.37436)]. To describe \(J_\Pi\) the author uses \textit{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.