Some applications of the minimal model program in arithmetic dynamics

TL;DR

This paper explores the application of the minimal model program to arithmetic dynamics, establishing results on pre-periodic points, reducing conjectures to special varieties, and linking automorphisms with positive entropy to the structure of the automorphism group.

Contribution

It introduces a new approach connecting minimal model techniques with arithmetic dynamics, addressing the Medvedev-Scanlon conjecture and automorphism entropy.

Findings

Density results on pre-periodic points for varieties with int-amplified endomorphisms

Reduction of certain Medvedev-Scanlon conjecture cases to Q-abelian varieties

Characterization of automorphisms with positive entropy via the automorphism group structure

Abstract

We describe a general program for studying the dynamics of surjective endomorphisms of algebraic varieties that are amenable to techniques from the minimal model program. We obtain density results on the pre-periodic points of surjective endomorphisms of varieties admitting an int-amplified endomorphism, and reduce certain cases of the Medvedev-Scanlon conjecture to so called Q-abelian varieties using our approach. We also provide a connection between the existence of an automorphism with positive entropy and group of connected components of a variety. In particular, we show that if $X$ is normal and projective with finitely generated nef cone then $X$ has an automorphism of positive entropy if and only if the group of connected components $\pi_0\Aut(X)$ has an element of infinite order.