Semiflows with arbitrary acting topological semigroups

该项目属于拓扑动力系统领域。动力系统是物理系统的数学模型。现实生活中有各种各样的情况，都可以以这样或那样的方式被视为动态系统。动力系统有三个组成部分：(A)相空间，它的点代表物理系统的所有可能的物理状态；(B)时间，粗略地说，它可以是离散的；连续的；或组合的；一维的或具有任何有限(甚至无限)维度的；等等；(C)一组物理规则，根据这组规则，当前处于某种状态的系统在任何给定时刻假定另一种状态。在经典的拓扑动力系统理论中，时间由实数组(连续时间)或整数组(离散时间)表示。然后，所讨论的群被称为作用群，动力系统被称为流。PI正在发展涉及非负整数半群的半流的相关理论。获得各种作用半流的统一表述将推进拓扑动力学领域，并可应用于其他分析和拓扑学领域。它对数学生物学和数学物理的某些领域也很重要。这个项目的目标是促进一般半流理论的发展。作用半群T的泛包络半群(拓扑学上是拓扑空间T的Stone-Cech紧化Beta T)是用来刻画半流轨迹的渐近性和常返性的一个重要对象。Beta T的代数结构起着至关重要的作用。主要研究人员计划利用T上的极大完全正则超滤子(假设T是完全正则空间)给出Beta T的一个详细构造，包括对Beta T上的一个运算的定义，然后利用这种显式描述详细地分析它的代数结构。主要研究人员预计，对于一般的半流条件，如“Beta T减去T是半群”、“Beta T减去T包含一个不变集”、“Beta T减去T包含一个幂等元”，以及最小和最大幂等元的概念以及其他代数概念，将出现在关于半流动力学的陈述中。首席研究员将研究一般半流中的各种常返性概念。它们将根据它们到给定邻域(无限、同步、完全等)的返回时间集合的广度来定义。然后根据固定它们的Beta T元素的性质来表征它们。主要的研究人员还计划研究以下概念：近端的点对、远端的点、积递归点及其与远端的点的关系、Ellis半群(作用半群的另一个重要的包络半群)的结构、IP集、Ramsey型定理等。首席调查员还将致力于将敏感性和混乱推广到一般半流的情况。

The project is in the area of Topological Dynamical Systems. A dynamical system is a mathematical model of a physical system. There is a huge variety of real life situations that can be considered as dynamical systems in one way or another. Dynamical systems have three components: (a) the phase space, whose points represent all possible physical states of the physical system; (b) the time, which, roughly speaking, can be discrete; continuous; or combined; one-dimensional or having any finite (or even infinite) number of dimensions; etc; (c) the set of physical rules according to which the system that is currently at some state assumes at any given moment another state. In the classical theory of Topological Dynamical Systems the time is represented by the group of real numbers (continuous time) or by the group of whole numbers (discrete time). The group in question is then said to be the acting group and the dynamical system is called a flow. The PI is developing the related theory of semiflows involving semigroups of nonnegative whole numbers. Obtaining unifying statements for various kinds of acting semiflows will advance the area of topological dynamics and have applications to other areas of analysis and topology. It is also important for some areas of mathematical biology and mathematical physics.The goal of this project is to advance the development of a theory of general semiflows. The universal enveloping semigroup of the acting semigroup T (topologically that is the Stone-Cech compactification Beta T of the topological space T) is an important object that is used to capture asymptotics and recurrence of the trajectories of the semiflow. The algebraic structure of Beta T plays a crucial role. The principal investigator plans on giving a detailed construction of Beta T in terms of maximal completely regular ultrafilters on T (assuming that T is a completely regular space), including the definition of an operation on Beta T, and then analyze in detail its algebraic structure using this explicit description. The principal investigator expects that for general semiflows conditions like "Beta T minus T is a semigroup", "Beta T minus T contains an invariant set", "Beta T minus T contains an idempotent", together with the notions of minimal and maximal idempotents and other algebraic notions, will be appearing in the statements about the dynamics of the semiflow. The principal investigator will study various notions of recurrence in general semiflows. They will be defined in terms of the largeness of the set of their return times to a given neighborhood (infinite, syndetic, replete, etc.) and then characterized in terms of properties of elements of Beta T that fix them. The principal investigator plans also to study the following notions: the proximal pairs of points, distal points, product-recurrent points and their relation with distal points, the structure of the Ellis semigroup (another important enveloping semigroup of the acting semigroup), IP sets, Ramsey type theorems and others. The Principal investigator will also work on generalizing sensitivity and chaos to the case of general semiflows.