Mastery question material

The mastery material this year (for question 5 on the exam for M4 and MSc students) consists of the (topological) Poincare classification of circle homeomorphisms. The relevant self-study text is [BS], section 7.1. This can be downloaded from Black Board.  Section (7.2) on circle diffeomorphisms will not be examined, but I included it out of general interest. I also include [HK] section 4.3, which deals with the same material as [BS] section 7.1, but the treatment is less condensed and it may be useful as an alternative source of basically the same material.

This material will be discussed in the last lecture on Monday 9 December.

PS: In class, there was a question regarding useful exercises. I would think that the exercises in the handout, [BS, p159], [HK, p134-135, 4.3.2-4.3.7]. I am sure you can find more exercises in other textbooks and online as the material is rather commonly taught.

How to prepare for the third class test on 6 December

The test will concern the following material:

Class notes of 11 until 29 November concerning ergodic theory:

• crash course in measure theory
• invariant measures
• Poincaré-recurrence
• Birkhoff's Ergodic Theorem; almost sure existence of the Birkhoff sum of an observable and it being equal to a conditional expectation of the observable wrt the sigma-subalgebra of measurable invariant sets. 
• ergodic invariant measures and Birkhoff's Ergodic Theorem in the context
• Problem sheet #4: questions 1,2,3,4,5

Please note that you while not being the focus of the test, you will still be expected to be familiar with the elementary notions of topological dynamics introduced earlier in the course.

As mentioned already various times, while we need to know some important and useful facts from measure theory, this course is not a course on measure theory, so in the exam and in this test I will not test your understanding of measure theory. I may, of course, test your understanding of the application of measure theory in the context of dynamical systems problems in this course, where relevant.

I discussed the proof of Birkhoff's Ergodic Theorem in some detail in the lectures since I consider it important to de-mystify this central result. However, as already announced in class, I will not test your understanding of this proof in the exam or in this test. However, you should understand the statement of the Theorem and how it applies to suitable examples, as discussed in the course.

Relevant lecture notes (see Blackboard)

• [MR] Dynamical systems lecture notes 2017: chapter 5 (ergodic theory), sections 2,3,4,5,6,7,8,9 (but not prop 5.38), 10,11,12, 13 (but not the proofs)

The aim of the test is to get an impression of understanding, not whether you know the notes by heart. Please keep this in mind...