The g-periodic subvarieties for an automorphism g of positive entropy on a compact Kahler manifold

TL;DR

This paper investigates the structure of g-periodic subvarieties on compact Kähler manifolds with automorphisms of positive entropy, establishing bounds on prime divisors and characterizing those containing infinitely many g-periodic curves.

Contribution

It proves that the number of g-periodic prime divisors is bounded by the rank of the Néron-Severi group and characterizes prime divisors with infinitely many g-periodic curves on threefolds.

Findings

Bound on the number of g-periodic prime divisors by NS(X) rank

Prime divisors with infinitely many g-periodic curves are themselves g-periodic on threefolds

Extension of ideas related to the Dynamic Manin-Mumford conjecture

Abstract

For a compact Kahler manifold X and a strongly primitive automorphism g of positive entropy, it is shown that X has at most rank NS(X) of g-periodic prime divisors B_i (i.e., g^s(B_i) = B_i for some s > 0). When X is a projective threefold, every prime divisor containing infinitely many g-periodic curves, is shown to be g-periodic itself (a result in the spirit of the Dynamic Manin-Mumford conjecture).