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Strong submeasures and applications to non-compact dynamical systems
Ergodic Theory and Dynamical Systems ( IF 0.9 ) Pub Date : 2020-12-14 , DOI: 10.1017/etds.2020.132
TUYEN TRUNG TRUONG

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 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.

中文翻译:

非紧致动力系统的强子测度和应用

紧凑度量空间上的强子度量X是连续函数空间上的次线性和有界算子X. 如果一个强子度量不递减,则它是正的。根据 Hahn-Banach 定理,正强子测度是其质量从上方均匀界定的非空测度集合的上确界。形式的连续映射有很多自然的例子$f:U\rightarrow X$, 在哪里X是紧度量空间,并且$U\子集 X$是一个开密集子集,其中F不能扩展到合理的功能X. 我们可以举出先验地图等案例$\mathbb {C}$,紧复变体上的亚纯映射,或连续自映射$f:U\rightarrow U$密集开子集的$U\子集 X$在哪里X是紧度量空间。对于前面提到的测度的使用并不足以建立遍历理论的基本性质,例如不变测度的存在或测度论熵和拓扑熵的合理定义。在本文中,我们展示了强子度量可用于完全解决问题并建立这些基本属性。在另一篇论文中,我们将强子度量应用于正闭的交集$(1,1)$紧凑型 Kähler 歧管上的电流。
更新日期:2020-12-14
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