Saddle hyperbolicity implies hyperbolicity for polynomial automorphisms of \(\mathbb{C}^2\) (Q1996243)

In higher-dimensional holomorphic dynamics there are several definitions of Julia set. It is a subtle problem to determine whether different Julia sets coincide. In this paper the author studies the dynamics of polynomial automorphisms of \(\mathbb{C}^2\). The Julia set \(J\) is defined as the set of points where iterations of the polynomial automorphism do not form an equicontinuous family. The Julia set \(J^*\) is defined as the closure of saddle periodic points; it is also the support of the measure of maximal entropy. It is obvious that \(J^*\) is contained in \(J\). The author proves that they are equal under the assumption that \(J^*\) is a hyperbolic set for the dynamics. He also points out that this result had appeared in the literature [\textit{J. E. Fornæss}, Math. Ann. 334, No. 2, 457--464 (2006; Zbl 1088.37019)] with a wrong proof.