Dynamical families of regular polynomial endomorphisms on $\mathbb{P}^2$ with many periodic curves

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.