Preparations for the resit exam

The notes for how to study for the May exam are equally valid for the resit exam.

Box display issue for problem sheet 4

The Box app (that you are directed to when downloading problem sheets) does not display all of problem sheet 4. Please download the file to your computer and display it with a local pdf viewer. This appears to resolve this issue. 

Past exams and model answers

Someone has pointed out that model answers for the exam in 2007 as well as the questions for 2009 exam are not available on Math Central. They can be obtained from the following links

2007 model answer
2009 exam paper

How to prepare for the exam.

I have now posted on blackboard (in the lecture notes section) a note concerning a brief outline of the structure of the exam and some guidance on how to prepare. Good luck!

PS (18/4): a students has asked me about exercises 6,7,8 on Problem sheet 4. This material (mixing) will not be examined, in line with my comment to follow the guidance for the class tests.

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...

Lecture of Monday 9 December: Mastery question material

For all MSci and MSc students, the fifth "Mastery" question on the exam will concern some additional material on the topological dynamics of circle homeomorphisms. I will discuss the outline of this material in the last lecture of this course on Monday 9 December.