2026/2027
Введение в символическую динамику и кодирование
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Язык:
английский
Кредиты:
6
Контактные часы:
60
Course Syllabus
Abstract
A dynamical system is a law describing position of an object in terms of its previous positions. Dynamical systems play important roles in Physical Sciences, Biology, Engineering, Economics, and Pure Mathematics. Symbolic dynamical systems are the most ``explicit'' ones; their description is based on arrays of symbols - which makes such systems easy to deal with from viewpoints of combinatorics and computer simulation. On the other hand, many interesting, ``real-life'', dynamical systems admit symbolic models. Passing from non-symbolic dynamics to the corresponding symbolic model is known as coding. In this course, we will sketch some basics of symbolic dynamics and demonstrate how these apply to solving interesting problems about billiards, interval (or polygon) exchange maps, hyperbolic torus maps, iterated complex polynomials, etc.
Expected Learning Outcomes
- Students will learn basic structures on spaces of symbolic sequences
- Students will be able to determine whether a given symbolic sequence is recurrent, balanced, uniquely ergodic
- Students will know examples of coding
- Students will be able to build symbolic models for a wide class of dynamical systems
- Students will master concepts and techniques of renormalization as seen from symbolic dynamics perspective
- Students will be able to translate certain questions from geometry and dynamics into a combinatorial language of automata.
- Students will be able to decide whether a given sequence is Sturmian.
Course Contents
- Symbolic dynamical systems: basic definitions. Factor complexity.
- Recurrence, balance, unique ergodicity
- Examples of coding: billiards, interval exchange maps, hyperbolic torus automorphisms.
- Topological conjugacy via coding: expanding maps of the circle, smooth homeomorphisms of the circle, hyperbolic complex polynomials.
- Substution dynamics and self-similarity of piecewise linear maps. Pisot numbers and Pisot substitutions.
- Automata and automatic sequences. Graphs on words.
- Sturmian sequences and circle rotations.
Bibliography
Recommended Core Bibliography
- Harald Uhlig. (1998). A Toolkit for Analysing Nonlinear Dynamic Stochastic Models Easily. QM&RBC Codes. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsrep&AN=edsrep.c.dge.qmrbcd.123
- Katok, A. B., & Hasselblatt, B. (2002). Handbook of Dynamical Systems (Vol. 1st ed). Amsterdam: North Holland. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=207259
- Shilʹnikov, L. P. (2001). Methods Of Qualitative Theory In Nonlinear Dynamics (Part Ii). New Jersey: World Scientific. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=235933
- Strogatz, S. H. (2000). Nonlinear Dynamics and Chaos : With Applications to Physics, Biology, Chemistry, and Engineering (Vol. 1st pbk. print). Cambridge, MA: Westview Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=421098
Recommended Additional Bibliography
- Michael Blank. (2018). Discreteness and Continuity in Problems of Chaotic Dynamics. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1790218
- Scárdua, B., & Rojas, C. A. M. (2017). Geometry, Dynamics And Topology Of Foliations: A First Course. World Scientific Publishing Company.