Math 856 - DYNAMICAL SYSTEMS and ERGODIC THEORY; Fall
2024, 3 credits
Instructor: Doğan
Çömez
Lectures: MWF
2:00-2:50 pm, Lebedeff Hall 183
Office: Minard 408 E24
Office hours: 3:00-3:50
pm
Email: Dogan.Comez@ndsu.edu
Prerequisites: Math
750 or the consent of the instructor.
Reference Texts: Introduction to Ergodic Theory,
by P. Walters, Springer-Verlag, 1975
Dynamical Systems and Ergodic
Theory, M. Policott & M. Yuri, London Math. Soc.,
1998
An
Introduction to Symbolic Dynamics and Coding, D. Lind & B. Marcus, Cambridge
University Press, 2003
GENERAL INFORMATION:
Course Description:
This course is a study of measurable dynamical systems in general. Topics
that will be covered include fundamental concepts of measurable dynamical
systems (recurrence, ergodicity, mixing, and spectral analysis of systems),
ergodic theorems and their applications.
During the second part of the course focus will be on symbolic dynamical
systems, which provide rich domain of problems as well as applications to other
mathematical sciences. In both parts of
the course recent developments in the subject as well as some current problems
of research will be discussed.
Course Objectives:
In the broadest sense, the branch of mathematics that attempts to understand
the processes in motion is called dynamical
systems. Depending on the basic structure on
the underlying system, this study is done in various domains (topological,
smooth or measurable). The purpose of
this course is to help students learn fundamental properties of dynamical
systems (in particular, the properties of measurable systems) in a rigorous
manner; develop mathematical skills needed to apply these to the problems
arising in various settings; gain increased understanding of how the concepts
they learned in the course apply within mathematics and to other disciplines.
Grading: There
will be (approximately) six assignments, dates TBD.
Academic Responsibility and
Conduct: All work in this course must be completed in a manner consistent
with NDSU University Senate Policy, Section 335: Code of Academic
Responsibility and Conduct. This policy
applies to cases in which cheating, plagiarism, or other academic misconduct
have occurred in an instructional context. Students found guilty of academic
misconduct are subject to penalties, such as failure of the course, up to and
possibly including suspension and/or expulsion. Student academic misconduct
records are maintained by the Office of Registration and Records.
Special needs:
Any student with disabilities or other special needs, who need special
accommodations in the course, is invited to share these concerns or requests
with Dr. Çömez as soon as possible.
Veterans: Veterans and student soldiers with special
circumstances or who are activated are encouraged to notify Dr. Çömez in advance.
OUTLINE OF THE COURSE:
Chapter I. Fundamentals of measurable dynamical
systems
1.
Measure preserving systems
2.
Ergodicity, recurrence, and mixing,
for measurable systems
3.
Spectral analysis of measurable
systems
4.
Isomorphism and conjugacy of dynamical
systems
5.
Entropy of measurable systems
Chapter
II. Ergodic
theorems
1.
The individual ergodic theorem and the
mean ergodic theorem
2.
Some applications to analytic number
theory
3.
Subsequential and operator theoretical
ergodic theorem
Chapter
III. Symbolic
Dynamical Systems
1.
Symbolic systems and examples
2.
Entropy of shift spaces
3.
Subshifts
of finite type and sofic systems
4.
Other shift spaces of particular
interest
Chapter IV.
Other topics of interest.
1.
Basic constructions in ergodic theory
2.
Non-commutative ergodic theorems
3.
Multiple recurrence and
applications to number theory