Strong submeasures and applications to non-compact dynamical systems

Tuyen Trung Truong

doi:10.1017/etds.2020.132

Abstract

AbstractA 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 the 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 form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, 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. For the aforementioned mentioned 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. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

Highlights

In dynamical systems and ergodic theory, measures play a crucial role
A crucial property is that we can push forward a probability measure by a continuous map and obtain another probability measure, and that this pushforward is linear on the space of measures
When we work with compact complex varieties, usually it is very difficult to construct dynamically interesting holomorphic maps f : X → X, and one must be willing to deal with dominant meromorphic maps f : X X in order to go forward

Summary

Introduction

In dynamical systems and ergodic theory, measures play a crucial role. At least since Henri Poincaré, the first fundamental step for studying the dynamics of a continuous map f : X → X of a compact metric space X is to construct invariant probability measures, that is, those measures μ for which f∗(μ) = μ, and in particular those with measure entropy equal to the topological entropy. If f : U → Y is a continuous function (where Y is an another compact metric space) and μ is a positive finite Borel measure on X, we can define a pushforward f∗(μ) as a positive strong submeasure on X in the following manner.

Results

Conclusion