On Lattès mappings of \({\mathbb P}^k\) (Q1606244)

The author proves the following result: Let \(f\) be a polynomial endomorphism of degree \(d\geq 2\) of \(\mathbb{C}^k\) \((k\geq 2)\) that extends to a holomorphic endomorphism of \(\mathbb{P}^k\). Suppose that the maximal-order Julia set of \(f\) is laminated by real hypersurfaces in some open set. Then \(f\) is homogeneous and it is a polynomial lift of a Lattès map of \(\mathbb{P}^k\).