# Ergodic Theory and Transfer Operators

## News

- There is
**no lecture on May 20, 2015**. It will be held on**Thursday, June 11, 2015**, at**4:00PM - 5:30PM**in the usual lecture room. - There is
**no lecture on July 1, 2015**. It will be held on**Thursday, July 9, 2015**, at**4:00PM - 5:30PM**in the usual lecture room.

## 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

- Basics of calculus (Analysis I,II)
- Linear algebra (Lineare Algebra I,II)
- Measure theory

## 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.

- Exercise sheet 01 from 4/15/2015
- Exercise sheet 02 from 4/22/2015
- Exercise sheet 03 from 4/29/2015 (updated)
- Exercise sheet 04 from 5/06/2015
- Exercise sheet 05 from 5/13/2015
- Exercise sheet 06 from 5/27/2015
- Exercise sheet 07 from 6/03/2015 (updated)
- Exercise sheet 08 from 6/11/2015
- Exercise sheet 09 from 6/17/2015
- Exercise sheet 10 from 6/24/2015
- Exercise sheet 11 from 7/09/2015
- Exercise sheet 12 from 7/15/2015

## 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“.

- [Sa] Omri Sarig;
*Lecture Notes on Ergodic Theory* - [BG] Abraham Boyarsky, Pawel Góra;
*Laws of Chaos*. Springer Science+Business Media New York, 1997 - [BS] Michael Brin and Garrett Stuck;
*Introduction to Dynamical Systems*. Cambridge University Press, 2003 - [LM] Andrzej Lasota and Michael C. Mackey;
*Chaos, Fractals, and Noise*. Springer, 1994 - [Wa] Peter Walters;
*An Introduction to Ergodic Theory*. Springer, 1982 - [Ma] Ricardo Mañé;
*Ergodic Theory and Differentiable Dynamics*. Springer, 1983

## 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 |