Growth of the number of periodic points for meromorphic maps

TL;DR

This paper proves that dominant meromorphic maps on compact Kähler manifolds have at most exponential growth in their isolated periodic points, with growth rate bounded by the algebraic entropy, and introduces new techniques involving positive closed currents.

Contribution

It establishes that such meromorphic maps are Artin-Mazur maps and provides growth estimates for periodic points, including new methods involving h-dimension of currents.

Findings

Periodic points grow at most exponentially with period n.

Growth rate is bounded by the algebraic entropy of the map.

New techniques involving the h-dimension of positive closed currents are introduced.

Abstract

We show that any dominant meromorphic self-map f of a compact Kaehler manifold X is an Artin-Mazur map. More precisely, if P_n(f) is the number of its isolated periodic points of period n (counted with multiplicity), then P_n(f) grows at most exponentially fast with respect to n and the exponential rate is at most equal to the algebraic entropy of f. Further estimates are given when X is a surface. Among the techniques introduced in this paper, the h-dimension of the density between two arbitrary positive closed currents on a compact Kaehler surface is obtained.