Around the dynamical Mordell-Lang conjecture

TL;DR

This paper advances the understanding of the dynamical Mordell-Lang conjecture by generalizing known results, providing new proofs, and reformulating methods using Berkovich spaces, applicable over various fields and for partial orbits.

Contribution

It generalizes the DML conjecture results to all endomorphisms of ^2 over , extends the weak DML theorem to a uniform and partial orbit version, and reformulates the p-adic method in Berkovich space.

Findings

Generalized DML for all endomorphisms of ^2 over .

Proved a uniform upper bound for arithmetic degree applicable in multiple settings.

Reformulated the p-adic method using Berkovich space for broader applicability.

Abstract

There are three aims of this note. The first one is to report some advances around the dynamical Mordell-Lang (=DML) conjecture. Second, we generalize some known results. For example, the Dynamical Mordell-lang conjecture was known for endomorphisms of $\mathbb{A}^2$ over $\overline{\mathbb{Q}}$. We generalize this result to all endomorphisms of $\mathbb{A}^2$ over $\mathbb{C}$. We generalize the weak DML theorem to a uniform version and to a version for partial orbit. Using this, we give a new proof of the Kawaguchi-Silverman-Matsuzawa' upper bound for arithmetic degree. We indeed prove a uniform version which works in both number field and function field case in any characteristic and it works for partial orbits. We also reformulate the ``$p$-adic method", in particular the $p$-adic interpolation lemma in language of Berkovich space and get more general statements. The third aim is to propose some further questions.