可测动力学中的像集
Image sets in measurable dynamics
AI总结：
针对不可逆可测动力学中像集定义不良的问题，提出一种一致处理零保持映射像集的方法，并应用于基本遍历性质，将模糊陈述转化为定理并消除额外假设。
摘要：
虽然在动力学的其他领域中像集被常规使用，但在一般不可逆的可测动力学中，像集是定义不良的对象。我们提出了一种一致地处理零保持（因此特别是测度保持）映射的像集的方法。这一概念在基本遍历性质（如回复性、遍历性、精确性和生成元的存在性）的背景下得到了说明。它使我们能够将关于集合论像集的各种有说服力但在逻辑上错误的陈述转化为实际定理，并从一些经典结果中消除对像集可测性的额外假设。
英文摘要：
While routinely used in other areas of dynamics, image sets are ill-defined objects in general non-invertible measurable dynamics. We propose a way of consistently working with image sets of null-preserving (and hence, in particular, of measure-preserving) maps. This concept is illustrated in the context of basic ergodic properties like recurrence, ergodicity, exactness and existence of generators. It allows us to turn various suasive but logically false statements about set-theoretic images into actual theorems, and to eliminate extra assumptions on the measurability of images from some classical results.
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