Remarks on algebraic dynamics in positive characteristic

TL;DR

This paper explores arithmetic dynamics in positive characteristic, extending fundamental concepts like arithmetic degree and canonical height, and proves several conjectures related to automorphisms and endomorphisms of projective varieties.

Contribution

It generalizes key arithmetic dynamical concepts to positive characteristic and proves conjectures such as the dynamical Mordell-Lang and Zariski dense orbit conjectures for specific classes of automorphisms.

Findings

Proved dynamical Mordell-Lang conjecture for automorphisms of projective surfaces of positive entropy.

Established Zariski dense orbit conjecture for automorphisms of projective surfaces.

Demonstrated equidistribution of backward orbits for finite flat endomorphisms with large topological degree.

Abstract

In this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. We generalise some basic facts on arithmetic degree and canonical height in positive characteristic. As applications, we prove the dynamical Mordell-Lang conjecture for automorphisms of projective surfaces of positive entropy, the Zariski dense orbit conjecture for automorphisms of projective surfaces and for endomorphisms of projective varieties with large first dynamical degree. We also study ergodic theory for constructible topology. For example, we prove the equidistribution of backward orbits for finite flat endomorphisms with large topological degree. As applications, we give a simple proof for weak dynamical Mordell-Lang and prove a counting result for backward orbits without multiplicities. This gives some applications for equidistributions on Berkovich spaces.