Symbolic Dynamics-1
TIFR-CAM, Even semester, 2021-2022.

Classes will be held from 9:00 AM to 10:20 AM on Tuesdays and Thursdays starting from 1st of February. The classes will be held over zoom.
Recitations, Discussions and Office Hours will be held from 9:30 AM to 10:30 AM on Wednesdays (Room number 112). If the students have questions, they are encouraged to ask their queries by Tuesday so that they can be taken up on Wednesday. If there is enough interest outside TIFR, we will try to make this available online as well.
Grading Scheme: 40 (Final term)+30 (midterm)+ 30 (assignments and recitation); solutions from bonus problems can be used to improve the net midterm+assignment score.
Here is the conversion chart from marks to letter grades. More about the grading scheme can be found by following this link.
Letter Grade Marks range
O 95-100
A+ 90-94
A 85-89
B+ 75-84
B 60-74
C 50-59
P 40-49
F Below 40
Finally, this is the first part of a (tentatively) two semester course. The syllabus for this course and for the tentative second course can be found via the following link. We will follow the book by Douglas Lind and Brian Marcus An Introduction to Symbolic Dynamics and Coding.
The midterm will be held on 8th April from 2:00 PM-4:00 PM, student lectures will be from 16th to 20th May and the final term will be on 25th May from 2:00 PM onwards.
Lecture notes (videos will be available only for 2-3 days. Please download them before I delete them.)

Lecture 1, 1st February: Notes A brief introduction and motivation.

Lecture 2, 3rd February: Notes Full Shifts

Lecture 3, 8th February: Notes Forbidden lists

Lecture 4, 10th February: Notes More Examples and Morse-Hedlund Theorem

Lecture 5, 15th February: Notes Sturmian shifts

Lecture 6, 17th February: Notes Minimal Shifts

Lecture 7, 22nd February: Notes Irreducible Shifts

Lecture 8, 24nd February: Notes Maps between shifts

Lecture 9, 2nd March: Notes Maps between shifts

Lecture 10, 3rd March: Notes Maps between shifts

Lecture 11, 8th March: Notes SFT

Lecture 12, 10th March: Notes Conjugacy of SFT to edge shifts

Lecture 13, 15th March: Notes Topological entropy

Lecture 14, 17th March: Notes Topological entropy

Lecture 15, 22nd March: Notes Topological entropy for SFTs

Lecture 16, 24th March: Notes Perron Frobenius Theorem and Cyclic Structures

Lecture 17, 5th April: Notes Eigenvalues of Primitive Matrices

Lecture 18, 12th April: Notes Entropy via periodic points

Lecture 19, 19th April: Notes Right-resolving labelled graphs

Lecture 20, 21st April: Notes Sofic shifts; a very brief introduction

Lecture 21, 26th April: Notes Invariant probability measures on a shift space

Lecture 22, 28th April: Notes Ergodic measures and extreme points of probability measures

Lecture 23, 3rd May: Notes Ergodicity of iid processes and the statement of the ergodic theorem

Lecture 24, 5th May: Notes An introduction to Shannon Entropy

Lecture 25, 10th May: Notes Video Entropy of a process

Lecture 26, 12th May: Notes Video Entropy of a process

Assignments

Please feel free to discuss with your fellow students but the solution must be understood and written on your own. At any point the student can be asked to explain a given solution during the office hours. Please do not search for the solutions on the internet. It helps no one (including you). The assignment needs to submitted by email to my Tifr address before the class (scan/tex either will do).

Assignment 1, due on 1st March: Link

Assignment 2, due on 15th March: Link

Assignment 3, due on 5th April: Link

Assignment 4, due on 29th April: Link

Assignment 5, due on 20th May: Link

Some suggestions for student presentations : Feel free to choose something not on this list after discussions with the instructor.

Realisation theorem for the entropy of SFTs, In the book by Lind and Marcus, Theorems 11.1.4 and 11.1.6

About the symbolic coding of interval exchange transformations: Link

About the symbolic coding of billiards on polygons: Link Ideally I would like it if the speaker can cover Section 4.1 and parts of Section 5. Also look at the following youtube video Link

About symbolic dynamics and storage on the disk: Link

Existence of characteristic measures for zero-entropy symbolic systems: Link

Minimal systems with multiple invariant measures: Link

Shift spaces with linear growth rate: Link

Automorphism group of shift spaces with quadratic growth rate: Link

Automorphism group of shifts of finite type: Link I would like it if the speaker can cover results from Section 2.

Some Resources

Open Problems in Symbolic Dynamics: Link