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.

Third and last test: friday 6 December 13:15-13:45 (end of the friday lecture) in room H342.

Just to confirm the time of the final test, which will be concerning some of the material on Ergodic Theory. I will post details of the latter asap.

The lecture on Monday 2 December will be mostly devoted to (selected problems on the) fourth problem sheet.

Class test weightings

I have been informed that the department's admin system is unable to cope with equal weightings on the three class tests (as it apparently can only deal with limited digital numerals in the weighting formula; I am not kidding :-)). So as a result, the weighting for the tests will be only so slightly skewed:
Test 1 : 3.3% of total exam mark
Test 2: 3.3% of total exam mark
Test 3: 3.4% of total exam mark
(still adding up to the announced overal 10% weight on the exam)

Fourth and final problem sheet

Please find a link to the fourth and final problem sheet in the right-hand-side margin. This problem sheet concerns the material on ergodic theory.

Birkhoff Ergodic Theorem notes

Please note that you can now find on Black Board a chapter on the Birkhoff Ergodic Theorem (by Martin Rasmussen, 2017) that broadly covers the material in our lectures.

Test 2 model answers

These can now be found in the right hand side margin.

Extra office hour wed 20/11 12:00

Someone requested this opportunity in view of the upcoming class test.

Problem sheet 2, question 2

Please note that the model answer shows that $NW(f)$ and $\overline{R(f)}$ are forward invariant, i.e. $f(NW(f))\subset NW(f)$ and $f(\overline{R(f)})\subset \overline{R(f)}$. I adjusted the questions accordingly. In fact, proving the opposite including $\supset$ is a little more involved and in the case of $NW(f)$ requires some additional condition (that $f$ is open). 

How to prepare for the second class test on 22 November

The test will concern the following material:

Class notes of 18 October and 1 November concerning topological dynamics (starting at page 21 of the 18/11 class notes).

• attractors
• topological conjugacy
• symbolic dynamics 
• topological markov chains 
• topological entropy
• Problem sheet #2: questions 1(a,b), 2(a,b), 3, 4, 5
• Problem sheet #3: questions 1,2,3,5,6

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

Relevant lecture notes (see Blackboard)

• [HK] chap 7; section 7.3 (not section 7.3.3), section 7.4 1 (not sections 7.4.4 and 7.4.5)
• [MR] Dynamical systems lecture notes 2017: chapter 3 section 4 (topological entropy)
• [VG] Dynamical systems lecture notes 2013: chapter 4 (topological entropy) and chapter 5 (topological Markov chains)

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...

Topological entropy notes

You can now find relevant notes by Martin Rasmussen (2017) and Vasili Gelfreich (2013) on topological entropy on blackboard. I also extended the [BS] topological dynamics notes there to include their section on topological entropy.

For your agenda: Friday 22 November - different room and time [11am-1pm in room HXLY311] and 2nd class test

Please note that I have just moved the second class test from Friday 15/11 to Friday 22/11, so as to give us a little more time to absorb the topological dynamics material that is to feature in this test [detailed instructions on how to prepare will be posted soon at the course webpage]. On Friday 22/11 we have a different time and place of the lecture, 11am-1pm in room HXLY 311. In order to avoid any clashes people may have 11-12, the class test will be held at towards the end of the lecture, from 12:15-12:45. Please note this in your agendas.

Problem sheet #3

A new problem sheet has now been posted in the right-hand-side margin of this blog. With the exemption of question 4, the questions concern topological entropy, which will be the topic of friday's lecture (by Martin Rasmussen).

Class notes 1 Nov 2019; problem sheet #2 model answers

I posted class notes for the lecture of 1 Nov. Please note that I changed the first example slightly so as to let the map $f$ be continuous. The remainder of the discussion is unaltered.

I also have posted model answers to problem sheet #2.  I apologise for the fact that I apparently at some point had posted only an unfinished draft of this problem sheet, without questions 4 and 5 (which are relatively important).

Symbolic dynamics: topological Markov chains

In addition to the class notes, see also [HK] sections 7.3.4-7.3.7 for a treatment of topological Markov chains. The notes in [HK] consider predominantly invertible (two-sided) shifts on bi-infinite sequences of finite numbers of symbols instead of the non-invertible (one-sided) shift on semi-infinite sequences that we have been considering in the lectures. But both treatments are very similar and all the statements in [HK] section 7.3.7 equally hold for the one-sided shift.

Test 1 results

The test 1 results are now on Blackboard. I handed most of the papers out in class, please collect from the central office in case you did not pick it up in class today.

The questions and model answers can now also be consulted through the link in the right-hand-side margin of this blog.

Please note that all marks on blackboard are 4/3 times the mark you find on your script (with rounding off) as question 4 was done poorly by everyone. I thought questions 1-3 should have been within easy reach (many of you got most of this right). Question 4 was a little more challenging and required you to take some more risk/initiative. Please have a good look at the model answers.

The test serves to all of us as a reality check on how things are moving along. Please review your study practices, if the test results have given you reason ..