On the dynamics of \(n\)-dimensional quadratic endomorphisms (Q1290542)

Let \(F= (f_1,\dots, f_n)\) be a \(C^2\)-convex endomorphism; for \(\nu\in \mathbb{R}^n\) fixed, consider the one parameter family \(F_\mu= F- \mu\nu\). The authors find sufficient conditions on the geometry of intersections of the level sets of the functions \(f_i\) such that for large values of \(\mu\), the map \(F_\mu\) belongs to some class \({\mathcal H}_0\), for which the dynamical behaviour is completely described, namely: either the nonwandering set \(\Omega(F_\mu)\) is empty or \(F_\mu\) restricted to \(\Omega(F_\mu)\) is an expanding map. The authors show that these conditions (sufficient) are generic in the space of quadratic endomorphisms.