math.DS — Dynamical Systems
Stable algebraic families of polarized dynamical systems and exceptional extensions
The notion of exceptional maps in arbitrary dimension was recently introduced by Ji, Song, and Xie. In this paper, we introduce exceptional extensions of polarized dynamical systems: a family of systems admitting a positive-dimensional exceptional factor whose generic fiber dynamics have simultaneous local product coordinates with an isotrivial system. We then prove that an algebraic family of polarized endomorphisms of normal projective varieties over a quasi-projective curve is -stable on a nonempty Zariski open subset if and only if it is isotrivial or it is an exceptional extension after a finite base change. This extends McMullen's theorem for stable algebraic families of rational maps to higher dimension. One ingredient is inspired by Ji and Xie's proof of McMullen's theorem: the use of homoclinic orbits and the multipliers of adjoint periodic points to control local return maps. The new ingredients are a higher-dimensional algebraization method and an inductive rigidity argument for the fiber dynamics, which together yield the exceptional extension structure.