Automorphisms of Blowups and the Dynamical Mordell-Lang Conjecture

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We use $p$-adic analytic methods to analyze automorphisms of smooth projective varieties. We prove a version of the dynamical Mordell-Lang conjecture for arbitrary subschemes of a variety. We apply this result (1) to classify automorphisms $\phi: X\to X$ such that there is a divisor $D$ in $X$ whose intersections with its iterates are not dense in $D$, and (2) to show that various properties of $\mathrm{Aut}(X)$ (for example, finiteness of its component group) are not altered by blowups in high codimension.
This is joint work with John Lesieutre.