Dynamical Mordell-Lang conjecture for totally inseparable liftings of Frobenius

TL;DR

This paper proves that certain inseparable endomorphisms of projective space over a non-archimedean field satisfy the dynamical Mordell-Lang property, extending understanding of dynamical systems in positive characteristic.

Contribution

It establishes the DML property for totally inseparable liftings of Frobenius endomorphisms in a non-archimedean setting, a novel result in arithmetic dynamics.

Findings

Proves DML for totally inseparable Frobenius liftings

Extends dynamical Mordell-Lang results to positive characteristic

Provides corollaries and generalizations of the main theorem

Abstract

We prove that if $K$ is a complete algebraically closed non-archimedian valuation field of positive characteristic and $f$ is an endomorphism of $\mathbb{P}_{K}^{N}$ which is totally inseparable and behaves as the Frobenius on the special fiber, then $f$ satisfies the dynamical Mordell-Lang (DML) property. We also discuss some corollaries and generalizations.