Orbit Equivalences in Borel Dynamics

遍历理论是研究动力系统平均行为的数学分支，例如天体或气体粒子的运动。对所有这样的系统进行详尽的分类是不可能的，当动力系统共享轨道但以不同的顺序遍历它们或共享路径但以不同的速度跟随它们时，研究等效的概念是富有成效的。这样的关系叫做轨道等价关系。由于它们的统计性质，遍历的理论结论总是在一些外围轨道上留下异常行为的可能性。Borel动力学是一门数学学科，它努力适应和增强遍历理论方法来研究动力系统的所有轨道，排除任何例外。PI将在Borel动力学范围内研究轨道等效的变体。本项目为研究生提供研究训练机会。本课题主要研究波雷尔流的轨道等价关系。已知保测度变换的完全分类是一项不可行的任务，遍历理论随着等效概念的引入而得到了发展，等效概念弱于同构。值得注意的例子包括轨道等价、（甚至）Kakutani等价和α等价，所有这些都可以在受限轨道等价的统一框架下进行观察。项目计划在Borel动力学范围内研究轨道等效的相应概念。在这个项目中调查的问题可能与Borel组合学、遍历理论、符号动力学和波兰群体行为的描述集理论有关。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Ergodic theory is the branch of mathematics that studies average behavior of dynamical systems, examples of which are movements of celestial bodies or gas particles. Exhaustive classification of all such systems is impossible, and it is fruitful to study notions of equivalence when dynamical systems share orbits but traverse them in a different order or share paths but follow them with different speeds. Such relations are called orbit equivalence relations. Due to their statistical nature, ergodic theoretical conclusions always leave the possibility of an exceptional behavior on some outlying orbits. Borel dynamics is a mathematical discipline that strives to adapt and enhance ergodic theoretical methods to study all orbits of a dynamical system, ruling out any exceptions. The PI will investigate variants of orbit equivalence within the scope of Borel dynamics. The project provides research training opportunities for graduate students.This project concentrates on the study of orbit equivalence relations of Borel flows. Complete classification of measure-preserving transformations is known to be an infeasible task, and ergodic theory has been advanced with the introduction of notions of equivalence that are weaker than the isomorphism. Notable examples include orbit equivalence, (even) Kakutani equivalence, and alpha equivalence, all of which can be viewed under the unified umbrella of restricted orbit equivalence formalism. The PI intends to investigate corresponding notions of orbit equivalence within the scope of Borel dynamics. Problems investigated in this project are likely to have connections with Borel combinatorics, ergodic theory, symbolic dynamics, and descriptive set theory of Polish group actions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.