Ideas for projects

Here follows a first list of possible directions for projects:

• Approximation of invariant measures with Ulam's method (suggested starting point: Ding & Zhou, chapter 6/7)
• Ergodic decomposition (suggested starting point: Viana & Oliveira, chapter 5, and M. Klünger's lecture notes)
• Decay of correlations (suggested starting point: Viana & Oliveira, chapter 7)
• Entropy (suggested starting point: Viana & Oliveira, chapter 9)
• Ergodic theory of chaotic billiards (suggested starting point: Chernov & Markarian)
• Lyapunov exponents and Oseledets multiplicative ergodic theorem (suggested starting point: Viana)
• Conditionally invariant measures
  (suggested starting point: M.F. Demers and L.-S. Young, Escape rates and conditionally invariant measures, Nonlinearity 19 (2006) 377–397 doi:10.1088/0951-7715/19/2/008

Or other topics from the books

• A. Boyarsky and P. Gora. Laws of chaos (1997)
• N. Chernov and R. Markarian. Chaotic billiards (2006)
• J. Ding and A. Zhou, Statistical properties of deterministic systems (2009)
• A. Lasota and  M. Mackey, Chaos, Fractals and Noise (1994) 
• M. Viana and K. Oliveira. Foundations of ergodic theory (2016)
• M. Viana. Lectures on Lyapunov exponents (2014)

Or anything else you may find of interest...

Material covered 30/1

We completed the lecture notes material of Chapter 3, and reached Chapter 4 until proposition 4.3 on page 24.

Material covered 23/1

We completed the lecture notes material of Chapter 2, and reached Chapter 3 until subsection 3 on page 18.

Material covered 16/1

We discussed lecture notes material Chapter 1, until subsection 6 on page 9.

Lectures tuesdays 11:00-13:00

The lectures for Ergodic Theory will be on tuesdays, 11:00-13:00 in my office (Huxley 638), starting tuesday 16 January. The lectures will initially follow the 2017 lecture notes of Martin Rasmussen (that can be downloaded from right-hand side margin). My rough aim is to cover 10-20 pages per lecture. It is advisable to read ahead in advance of the lecture. I will note the material we covered after each lecture.

Time table

I have just become aware of the fact that there are 3 contact hours scheduled in the official time-table for this course. This has happened without my knowledge. In previous years no rooms were booked for this course. We discuss the schedule of lectures and meetings on Fri 12 January, 12pm in room 638.

First meeting, 12pm on Friday 12 January in Huxley 638

Students interested in taking this course are advised to contact me (jeroen.lamb@imperial.ac.uk). There will be a first meeting on Friday 12 January 2017 at 12pm in my office (Huxley 638) where the course material and the scheduling of the weekly meetings will be discussed.

Welcome

Ergodic theory has strong links to analysis, probability theory, (random and deterministic) dynamical systems, number theory, differential and difference equations and can be motivated from many different angles and applications. In contrast to topological dynamics, ergodic theory focusses on a probabilistic description of dynamical systems. Importantly, the "ergodic hypothesis" lies at the basis of statistical mechanics. A background in probability and measure  theory is required to understand even the basic material in ergodic theory. For this reason, the first part of the course will concentrate on a self-contained  review of the required background. The second part of the course will focus on selected topics in ergodic theory. 

The course is organised as a reading course leading to an individual project. There will be weekly meetings, where selected material will be presented and discussed within the group; this will guide the independent study. 

Recommended literature:

Martin Rasmussen, Ergodic Theory lecture notes (2017) as attached in the right-hand side margin.

Marcelo Viana and Krerley Oliveira. Foundations of Ergodic Theory (2016); full access online from Imperial at https://doi.org/10.1017/CBO9781316422601

Assessment:
M4 and MSc students can take this course for credit. Students taking the course for credit are to prepare an essay based on an individual project (counting for 60%); there will be a 30-minute oral exam on the project and the content of the course (counting for 40%). PhD students can also take this module for credit. Any interested PhD student is advised to discuss the method of assessment directly with me.