Strong submeasures and several applications

TL;DR

This paper introduces strong submeasures, a classical but overlooked concept, and demonstrates their diverse applications in complex dynamics, intersection theory, and entropy, providing new results in these areas.

Contribution

The paper develops the theory of strong submeasures and applies them to complex dynamics, intersection theory, and entropy, yielding several novel results.

Findings

Positive strong submeasures can be represented as supremums of bounded measures.

Existence of invariant strong submeasures for meromorphic maps on complex varieties.

Explicit calculations of self-intersection of certain currents in Kähler surfaces.

Abstract

A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By Hahn-Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. We give several applications of strong submeasures in various diverse topics, thus illustrate the usefulness of this classical but largely overlooked notion. The applications include: - Pullback and pushforward of all measures by meromorphic selfmaps of compact complex varieties. - The existence of invariant positive strong submeasures for meromorphic maps between compact complex varieties, a notion of entropy for such submeasures (which coincide with the classical ones in good cases) and a version of the Variation Principle. - Intersection of every positive closed (1,1) currents on compact K\"ahler manifolds. Explicit calculations are given for self-intersection of the current of integration of some curves $C$ in a compact K\"ahler surface where the self-intersection in cohomology is negative. All of these points are new and have not been previously given in work by other authors. In addition, we will apply the same ideas to entropy of transcendental maps of $\mathbb{C}$ and $\mathbb{C}^2$.