Strong submeasures and applications to non-compact dynamical systems

TL;DR

This paper introduces strong submeasures, a new mathematical tool, to extend ergodic theory concepts to non-compact dynamical systems where traditional measures are insufficient.

Contribution

It develops the theory of strong submeasures and demonstrates their application in establishing ergodic properties for non-extendable maps.

Findings

Strong submeasures can replace measures in ergodic theory for non-compact systems.

They enable the proof of existence of invariant measures and entropy definitions.

Application to intersection theory on Kähler manifolds is also discussed.

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. There are many natural examples of continuous maps of the forms $f:U\rightarrow X$, where $X$ is a compact metric space and $U\subset X$ is an open dense subsets, where $f$ cannot extend to a reasonable function on $X$. We can mention cases such as: transcendental maps of $\mathbb{C}$, meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where $X$ is a compact metric space. To the maps mentioned in the previous paragraph, the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure theoretic entropy and topological entropy. In this paper, we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. The same idea can also be applied to improve on \'etale topological entropy, previously defined by the author. In another paper we apply strong submeasures to intersection of positive closed $(1,1)$ currents on compact K\"ahler manifolds.