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Regular version of the site
2020/2021

Research Seminar "Introduction to Ergodic Theory"

Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type: Optional course (faculty)
When: 1, 2 module
Instructors: Mikhail Blank
Language: English
ECTS credits: 3

Course Syllabus

Abstract

Is it possible to distinguish deterministic chaotic dynamics from a purely random and whether this question makes sense? Does irreversibility influence qualitative characteristics of the process? Ergodic theory studies these and other statistical properties of dynamical systems. Interest in this subject stems from the fact that «typical» deterministic dynamical systems (eg, differential equations) exhibit chaotic behavior: their trajectories look similar to the implementation of random processes. We begin with the classical results by Poincare, Birkhoff, Khinchin, Kolmogorov, and get to modern productions (including yet unresolved) problems. This is an introductory course designed for 2–4 bachelors and graduate students. Prior knowledge except for the course in mathematical analysis is not required (although it is desirable).
Learning Objectives

Learning Objectives

  • This course is aimed at providing students with a solid working knowledge in the basic concepts, important techniques and examples in Ergodic Theory of dynamical systems.
Expected Learning Outcomes

Expected Learning Outcomes

  • At the end of the course the student is expected be able to analyze statistical properties of dynamical systems, in particular to be familiar with the ergodic theorem and its numerous applications, e.g. in number theory
Course Contents

Course Contents

  • Dynamical systems: trajectories, invariant sets, simple and strange attractors and their classification, randomness
  • The action in the space of measures, transfer operator, invariant measures. Comparison with Markov chains
  • Ergodicity, Birkhoff ergodic theorem, mixing, CLT. Sinai–Bowen–Ruelle measures and natural/observable measure
  • Basic ergodic structures: direct and skew products, Poincare and integral maps, a natural extension and the problem of irreversibility
  • Ergodic approach to number theoretical problems
  • Entropy: metric and topological approaches
  • Operator formalism. Spectral theory of dynamical systems. Banach space of measures, random perturbations
  • Multicomponent systems: synchronization and phase transitions
  • Mathematical foundations of numerical simulations
Assessment Elements

Assessment Elements

  • non-blocking cumulative assessment
  • non-blocking exam
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    0.4 * cumulative assessment + 0.6 * exam
Bibliography

Bibliography

Recommended Core Bibliography

  • Hasselblatt, B., Takens, F., & Broer, H. W. (2010). Handbook of Dynamical Systems. Amsterdam: North Holland. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=344991
  • 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

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