Ergodic Theory and Transfer Operators

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Dates

Lecture Wed, 10:15 - 11:45 Arnimallee 6, SR 009

General information

Description

Ergodic theory is concerned with the behavior of dynamic systems when these are running for a long time. Vaguely speaking, the long-term statistical behavior of an ergodic dynamical system is not going to depend on its initial condition. This course discusses the mathematical characterization of this property. A central role is going to be played by the so-called transfer operator, which describes the action of the dynamics on a distribution of states. We are also going to highlight its importance in applications, when it comes to the numerical approximation of quantities of interest. If time permits, we will introduce entropy, as a notion of complicatedness (or "non-predictability") of a dynamical system.

Target Audience

Students of Bachelor and Master courses in Mathematics and of BMS

Requirements

Contents

I. BASIC DEFINITIONS AND CONSTRUCTIONS (Notes)

   I.1 Motivation
     1.1 Law of the large numbers
     1.2 A simple weather model
     1.3 Long-term behavior
     1.4 Dynamical systems
     1.5 The study of dynamical systems
     1.6 Pioneers
     1.7 Preview

   I.2 The Setup of Ergodic Theory
     1.8 Measure-preserving transformations
     1.9 Poincaré's recurrence theorem (PRT)
     1.10 Remarks to the PRT
     1.11 The probabilistic point of view
     1.12 Invariant sets
     1.13 Ergodicity
     1.14 Ergodicity: invariant of measure-theoretic isomorphism
     1.15 Independence and mixing

   I.3 Examples
     1.16 Circle rotation
     1.17 Angle doubling
     1.18 Bernoulli schemes
     1.19 Subshift of finite type
     1.20 Markov shift

   I.4 Basic Constructions
     1.21 Products
     1.22 Skew-products
     1.23 Factors and extensions
     1.24 Induced transformations

II. ERGODIC THEOREMS (Notes)

   II.1 The Mean Ergodic Theorem
     2.1 Von Neumann’s mean ergodic theorem (MET)
     2.2 Consequences of the MET

   II.2 The Birkhoff Ergodic Theorem
     2.3 The maximal ergodic theorem
     2.4 The Birkhoff ergodic theorem (BET)
     2.5 Consequences of the BET
     2.6 Markov chains
     2.7 Internet search

   II.3 The Ergodic Decomposition
     2.8 Conditional expectation
     2.9 Ergodic limit
     2.10 Conditional probabilities
     2.11 Related topics

   II.4 The Subadditive Ergodic Theorem
     2.12 Random walks on groups
     2.13 Kingman’s subadditive ergodic theorem

III. TRANSFER OPERATORS (Notes)

   III.1 Studying Dynamical Systems with Densities
     3.1 In practice
     3.2 Long trajectories vs densities
     3.3 Frobenius-Perron operator (FPO)
     3.4 Properties of the FPO
     3.5 Ergodicity and mixing

   III.2 Absolutely Continuous Invariant Measures (ACIMs)
     3.6 ACIMs and the FPO
     3.7 Piecewise monotonic transformations
     3.8 Functions of bounded variation
     3.9 A contraction property of the FPO
     3.10 Existence of ACIMs

   III.3 Numerical Approximation
     3.11 Abstract setting
     3.12 Ulam’s method
     3.13 Probabilistic interpretation
     3.14 Computational aspects
     3.15 Convergence: Li’s proof
     3.16 Remarks

IV. ENTROPY (Notes)

   IV.1 Entropy of a Partition
     4.1 Uncertainty
     4.2 Measurable partition
     4.3 Entropy of a partition
     4.4 Conditional entropy

   IV.2 Entropy of Measure Preserving Transformations
     4.5 Metric entropy
     4.6 The Kolmogorov-Sinai generator theorem
     4.7 The Shannon-McMillan-Breiman theorem
     4.8 Examples
     4.9 Historical remarks
     4.10 Topological entropy

Exercise sheets

Working with exercise sheets are meant to be integral part of the lectures. The sheets comprise tasks such as proving results omitted in the lectures, elaborating examples which complement the material, and you gain hands-on experience by solving the programming exercises. For this I recommend to use MATLAB, but feel free to work with the tool of your choice.

Accompanying Material

The handout(s) should serve as a quick reference for definitions and results needed in the lectures and for the exercises, or they comprise additional material not completely covered in the lectures.

Literature

The books listed below are available in the library during the whole term. To get a quick overview, one can go to Primo, and search for „Semesterliste Koltai“.

Acknowledgment

I would like to thank Cecilia González-Tokman and Gary Froyland for making the material from their lecture Math 5175: Ergodic Theory, Dynamical Systems and Applications available to me.

Contact

Péter Koltai Arnimallee 6, Raum 113
email: peter.koltai{at}fu-berlin.de