"Introduction to Symbolic Dynamics"

Spring 2016

Lecturer: Tom Meyerovitch
Announcements

Note change of schedule: Thursday class moved to Tuesday.

Lecture Time and place:
- Mondays, 14-16, Building 28 room 107
- Tuesday 16-18, Building 58 room 201

Course description

Symbolic dynamics is a branch of mathematics that deals with sequences of characters letters or "symbols" form the point of view of dynamical systems. The basic guiding philosophy is that sometimes it is possible to code and understand complicated systems by a sequence of discrete samples. The decimal expansion of real numbers is a simple example of this kind of procedure. Techniques and ideas from symbolic dynamics have found significant applications in data storage and transmission as well as other parts of mathematics. In this course we will introduce basic notions and results in symbolic dynamics, via interesting examples. We will illustrate relations to other fields and relate to the more general frameworks of topological dynamics and ergodic theory. Basic topics to be covered:

A brief introduction to topological dynamics
Shift spaces and Languages
Shifts of finite type and sofic shifts
Cellular automata and sliding block codes, endomorphisms and automorphisms of shift spaces.
Topological entropy

Possible additional topics (subject to time, participants background and participants preferences):

Krieger's topological embedding theorem
On the isomorphism problem for shifts of finite type (strong shift equivalence and shift equivalence)
Multidimensional shift spaces and shift spaces over countable groups

We will introduce and study the notions of Measure-preserving transformations, Ergodicity, Recurrence The formal prerequisite is basic knowledge of measure theory. The background in other fields (such as functional and harmonic analysis and probability) is not a prerequisite, but will be introduced as needed.

Textbooks and other resources

An introduction to Symbolic Dynamics and Coding - Marcus and Lind
Symbolic dynamics and its applications. Lectures of the American Mathematical Society Short Course held in San Diego, CA, January 4–5, 2002. Edited by Susan G. Williams. Proceedings of Symposia in Applied Mathematics, 60. AMS Short Course Lecture Notes. American Mathematical Society, Providence, RI, 2004. viii+156 pp. ISBN: 0-8218-3157-7 37-06 (37B10)
Symbolic dynamics. One-sided, two-sided and countable state Markov shifts- B. Kitchens
Introduction to the Modern Theory of Dynamical systems - A. Katok and B. Hasselbtatt
Algebraic Combinatorics on Words M. Lothaire
Ergodic Theory via Joinings - E. Glasner
I will post some notes in This blog

Grading scheme:

The grade will based on a take-home exam (assignment) to be submitted in the end of the course. The assignment will consist of questions that will be given throughout the course.