Introduction to Symbolic dynamics , spring 2025

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: 

1. A brief introduction to topological dynamics.

2. Shift spaces (subshifts).

3. Subshifts of finite type (SFT) and sofic shifts.

4. Cellular automata and sliding block codes, endomorphisms and automorphisms of shift spaces.

5. Topological conjugacy (isomorphism), embedding and factors of subshifts.

6. Topological entropy.

Possible additional topics, possibly as individual projects assigned to enrolled students:

1.  Multidimensional shift spaces

2. Shift equivalence and strong shift equivalence of matrices.

3. Krieger’s embedding theorem.

הודעות:

• שאלות בנוגע לדרישות קדם, תוכן הקורס, חובות והרכב הציון בקורס? מתלבטים?  מוזמנים לפנות אלי במייל או במשרד!  

Resources and links:

• An introduction to symbolic dynamics and coding / Lind and Marcus

• Symbolic Dynamics. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems

• Symbolic Dynamics/ Kitchens 

• Dimension groups and Dynamical systems/ Durand and Perrin (preprint version public on Arxiv here)
  
  

Exercises:

• Exercise 1

• Exercise 2

• Exercise 3

• Exercise 4

• Exercise 5

• Exercise 6

• Exercise 7

• Exercise 8

• Exercise 9

• Exercise 10

Notes:

1. Week 1 notes

2. Week 2 notes

3. Week 3 Notes

4. Slides from week 4

5. Week 5 Notes

6. Week 6 notes

7. Week 7 notes

8. Week 8 notes

9. Week 9 notes

10. Week 10 notes

11. Slides from week 12