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.
