Around the dynamical Mordell-Lang conjecture

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 $\bar{\mathbb{Q}}$. 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.