Symbolic Dynamics

Brian Marcus (University of British Columbia)

Jan 5, 2027 — Apr 27, 2027

About the course

Symbolic Dynamics is a part of the theory of dynamical systems, with strong connections to ergodic theory and information theory. This course will serve as an introduction to symbolic dynamics, including shifts of finite type, sofic shifts, entropy, Gibbs meauures, equilibrium meaasures, and problems of encoding and decoding from one symbolic system to another. Emphasis on Z-symbolic dynamics (i.e. spaces of one-dimensional sequences), with some material on G-symbolic dynamics (where G is a countable group). There is a very rich history of major results in this area, and many open problems.

Registration

This course is available for registration under the Western Dean's Agreement. To register, you must obtain the approval of the course instructor and you must complete the Western Dean's agreement form , using the details below. The completed form should be signed by your home institution department and school of graduate studies, then returned to the host institution of the course.

Enrollment Details

Course Name
Symbolic Dynamics
Date
Jan 5, 2027 — Apr 27, 2027
Course Number
MATH601D
Section Number
201
Section Code
MATH_V 601D:201

Instructor(s)

For help with completing the Western Dean’s agreement form, please contact the graduate student program coordinator at your institution. For more information about the agreement, please see the Western Dean's Agreement website

Other Course Details

Remote Access

The instructor will write on overhead projectors which will be shared via zoom. Lecture notes will be posted on the course website.

Availability

This course is available to students within the PIMS network, universities beyond the PIMS network and from industry/government.

Lecture Schedule

  • MWF 1-2pm CHEM C124
  • There will be three homework sets and each student is expected to give one course lecture

Course Topics

  1. Shift spaces [LM, Ch 1 and Sec. 6.2, 6.3]
  2. Shifts of Finite Type (SFTs) [LM, Ch 2]
  3. Sofic shifts [LM, Ch 3]
  4. Minimal shifts: Morse shift and Sturmian shifts [LM, Sec 13.7]
  5. Classical Perron Frobenius Theory [LM, Ch 4]
  6. Topological entropy [LM, Ch 4]
  7. A quick introduction to ergodic theory [W, Ch. 1,2,4]
  8. Measures of maximal entropy for SFTs [LM, Sec. 13.3]
  9. Ruelle Perron Frobenius Theory [B, Sec. 1.A,B]
  10. Gibbs measures and equilibrium states [B, Sec 1.C,D]
  11. Classification of SFTs up to topological conjugacy [LM, Ch. 7]
  12. Other coding problems for SFTs and sofic shifts in symbolic dynamics (e.g., almost topological conjugacy, embeddings, factor maps) [LM, parts of Ch. 5, 8, 9, 10, 12]
  13. Markov partitions [LM. Sec. 6.5]
  14. Symbolic dynamics for G-shifts, where G is a countable group [LM, sec. A.8]

References:

  • [B] Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, R. Bowen (with revisions by Jean-Rene Chazottes), 2nd edition, 2017, Springer Lecture Notes in Mathematics, 470, Springer
  • [LM] An Introduction to Symbolic Dynamics and Coding, D. Lind and B. Marcus, 2nd edition, 2021, Cambridge Mathematical Library, Cambridge University Press
  • [W] An Introduction to Ergodic Theory, P. Walters, 1982, Springer Graduate Texts in Mathematics, 79, Springer
2026-2027