Ergodic Theory and Transfer Operators



Lecture Wed, 10:15 - 11:45 Arnimallee 6, SR 009
Lecture (between May 24th - June 28th) Wed, 14:15 - 15:45 ZIB, Room 4023
Exercise Wed, 8:30 - 10:00 Arnimallee 6, SR 009

General information


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.

Target Audience

Students of Bachelor and Master courses in Mathematics and of BMS




   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 Preview

   I.2 The Setup of Ergodic Theory

     1.7 Measure-preserving transformations
     1.8 Poincaré's recurrence theorem (PRT)
     1.9 Remarks to the PRT
     1.10 Invariant sets
     1.11 Ergodicity
     1.12 Ergodicity: invariant of measure-theoretic isomorphism
     1.13 Independence and mixing

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

   I.4 Basic Constructions

     1.19 Products
     1.20 Skew-products
     1.21 Factors and extensions


   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


   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
     3.6 Absolutely continuous invariant measures and the FPO

   III.2 Numerical Approximation
     3.7 Abstract setting
     3.8 Ulam’s method
     3.9 Probabilistic interpretation
     3.10 Computational aspects
     3.11 Remarks
     3.12 Generalised Galerkin methods
     3.13 Extended Dynamic Mode Decomposition (EDMD)
     3.14 Convergence to a Galerkin method
     3.15 EDMD for the FPO


   Transfer operators

   Chapter 1 from Adam Nielsen’s thesis + "Make it reversible" (Sections 3.1, 3.2 therein)

   Spectral analysis of transfer operators

   Section 4 from this work of Wilhelm Huisinga and Marcus Weber’s ZIB report.


   Section 3 until Section 3.4.2 in Marcus Weber’s thesis.

   Projection on sets vs invariant subspaces

   Section 3.2 from this work of Marco Sarich.

   Cyclic behavior and Schur decomposition

   Marcus Weber’s talk.

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 we 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.


The books listed below are available in the library.


Péter Koltai thanks Cecilia González-Tokman and Gary Froyland for making the material from their lecture Math 5175: Ergodic Theory, Dynamical Systems and Applications available to me.


Péter Koltai Arnimallee 6, Room 113
email: peter.koltai{at}
Marcus Weber ZIB, Room 4023
email: weber{at}
Martin Plonka Arnimallee 6, Room 114
email: plonka{at}