Professor Qiudong (Don) Wang, Department of Mathematics, University of Arizona
In these lectures I will start with the studies of two simple dynamical systems: the iteration of a given piecewise linear mapping and that of a rigid rotation. We will then observe how, by asking rather natural and innocent questions, these relatively “simple” discussions motivates, and lead to the development of major mathematical theories in the subject of dynamical systems. If time allows, I will also give an introductory lecture on Celestial Mechanics, especially on the study of integral manifolds of a given gravitational problem.
I will use the study of a piecewise linear map to introduce the fundamental concepts of coding and symbolic dynamics. We will then discuss the possibility of generalizing our method of study, which will naturally lead to the study of quadratic family, unimodal maps, and the kneading theory, etc.
I will use the study of maps of rigid rotations to demonstrate the connection between the study of certain dynamical systems and that of the property of real numbers. We will then move to the study of two dimensional maps to discuss the contents of KAM theory and the theory on Aubry-Mather set.
I will then move to the subject of Celestial Mechanics. Here I intend to drive the equation of motion for the restricted three-body problem, find the first integral (Jacob's integral), and study the geometric structure of the integral manifold. We will then discuss the study of the integral manifold for the three, and more body problems.
I will provide a list of introductory books on dynamical systems and celestial mechanics to the interested students. I will ask them to read selected materials from these book, and report to me and to each other on weekly basis. The books I have in my mind are:
Depending on the level of the individual students, I will also, if possible, include certain research articles as part of their reading assignment.