Polynomial endomorphisms of $\A^2$ with many periodic curves
Xiao Zhong
TL;DR
The paper addresses the behavior of constant-parameter families of regular polynomial endomorphisms on ${\mathbb P}^2$ that admit a Zariski-dense collection of periodic curves. It proves that such a family must be invariant under some iterate of the fixed endomorphism, and that the degrees of generic members stabilize under the action, situating the result as a specialized, degree-stabilized form of a Dynamical Manin–Mumford-type statement on the moduli of divisors. The work develops a blend of blow-up analysis at the line at infinity, local transversality arguments, and moduli-space techniques to derive invariance and degree-stabilization, and it connects these results to broader conjectures in the dynamical Manin–Mumford program. As an application, it provides a complete classification of regular polynomial endomorphisms with infinitely many periodic curves of bounded degree, showing that such maps preserve either a k-web or a pencil of curves, with explicit normal forms arising from established classification theorems.
Abstract
In this paper, we prove that for a regular polynomial endomorphism of positive degree on $\mathbb{P}^2$, a family of curves containing a Zariski dense set of periodic curves is invariant under some iterate of the endomorphism. The setting is closely related to the Relative Dynamical Manin-Mumford Conjecture, recently proposed by DeMarco and Mavraki, which concerns a parametrized family of endomorphisms and varieties. Our result proves a weaker version of the conjecture where the endomorphism is a regular polynomial endomorphism on $\mathbb{P}^2$ that remains fixed in the family, and the family of curves contains a dense set of periodic curves. This result can also be viewed as a Dynamical Manin-Mumford type statement on the moduli space of divisors, and it proves a special case of the Dynamical Manin-Mumford Conjecture with a stronger assumption. Moreover, our result specifically implies a uniform degree stabilization statement for a generic set of curves in a family under the transformation of a regular polynomial endomorphism. We demonstrate that a more general degree stabilization statement for a family of positive dimension subvarieties in $\mathbb{P}^K$ under the transformation of a family of endomorphisms is predicted by the Relative Dynamical Manin-Mumford Conjecture. We then prove that it is true when $K=2$ for families of regular polynomial endomorphisms under certain restrictions on the ramifications at the line at infinity. Finally, we demonstrate an application of our result to classify all regular polynomial endomorphisms that admit infinitely many periodic curves of bounded degree.