Period three implies all periods

Notes covering the material presented by Martin Rasmussen on Monday 28/10 can be found on Blackboard.

ps: There are no notes concerning the lecture by Dmitry Turaev on Friday 25/10 and the material that he covered will not be examinable (although I hope it has provided some insights of matters in dynamical systems that go beyond the material in this course). See also [HK] section 7.4.4 (blackboard) for a demonstration of symbolic dynamics of the "linear horseshoe".

Office hour this week

With apologies for the late notice, due to my absence, this week the office hour will be Fri 1 Nov, 2-3pm.

New problem sheet

Just to point out that the second problem sheet is now available to download in the right-hand side margin. Selected problems from this sheet will be discussed in a problem class on Monday 4 November. I will post model answers in the weekend of 2-3 November.

problem sheet 1 question 7(ii) model answer

The model answer starts with an assertion that uses the fact that $f(\overline{A})=\overline{f(A)}$ where $f:X\to X$ is continuous,  $A\subset X$ and $X$ is compact. I present here the proof for completeness.

The inclusion $f(\overline{A})\subset\overline{f(A)}$ follows by continuity of $f$. Suppose
$\{x_n\}$ converges to $y$ then $\lim_{n\to\infty} f(x_n)=f(\lim_{n\to\infty}x_n)=f(y)\in \overline{f(A)}$.

The remaining inclusion $f(\overline{A})\supset\overline{f(A)}$ is obtained as follows: suppose there exists a sequence $\{x_n\}\subset A$ such that $\lim_{n\to \infty} f(x_n)=C\in
\overline{f(A)}$. Then by compactness of X, there exists a converging subsequence $\{x_{n_k}\}$: so $\lim_{k\to\infty} x_{n_k}=y \in \overline{A}$ and $f(y)=C$ since $f(x_n)$ and $f(x_{n_k})$ converge to the same value. Thus for all $C\in \overline{f(A)}$ there exists a $y\in \overline{A}$ such that $f(y)=C$.

On the uniform attraction property of attractors

Please see the class notes for Friday 18 Oct for a more elaborate discussion of the uniform attraction property of attractors. While what I presented in class was true, it was certainly not self-evident. I hope the notes are now entirely clear. (see pages 21-22 of the 181019 class notes).

How to prepare for the class test of 25 Oct

The test will concern the following material:

Class notes of 11, 14 and 18 October concerning topological dynamics (pages 7 -21; so NOT the material from page 21 of the class notes about attractors, topological conjugacy and symbolic dynamics)

• definition of chaos (following Devaney): density of periodic orbits, topological transitivity, sensitive dependence - concepts and applications to simple examples;
• observation that sensitive dependence follows from density of periodic orbits and topological transitivity if the state space is not a periodic orbit 
• topological mixing and relationships to other topological dynamical properties
• invariant sets and $\omega$-limit sets: properties 
• Problem sheet #1: questions 1, 2, 3, 7, 8, 9

Relevant lecture notes (see Blackboard):

• [HK] chap 7; section 7.1 (not prop 7.1.5 and lem 7.1.6; not sections 7.1.4 and 7.1.5); section 7.2 (not sections 7.2.1 and 7.2.4)
• [MR] Dynamical systems lecture notes 2017: chapter 3 sections 1 and 2

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...

Problem sheet #1 correction question 2 (update 21/10)

It was kindly pointed out to me that in question 2 the sensitivity constant cannot always be choses any number smaller rather than the diameter. But it can always be chosen to be anything smaller than 1/2 the diameter. I have amended the question on the problem sheet and model answer, accordingly. [I took the erroneous question from [HK] exercise 7.2.3.] While correct, half the diameter is not always optimal. Namely, if X is compact, the sensitivity constant can be chosen to be any number smaller than $\inf_{x\in X}(\sup_{y\in X} d(x,y))$. I included the argument with the model answer. This quantity can be equal to the diameter; eg on $S^1$.

Class tests

As announced earlier, there will be 3 short class tests during this term for this course. Each class test will weigh 3.33% to your exam result. Class test will be aimed to assess your understanding of the material rather than your memory, so please invest your time accordingly.

The class tests will be on the following days

• Friday 25 October 2019, 
• Friday 22 November 2019
• Friday 6 December 2019

12:00-12:30 in HXLY 342 (so at the start of the scheduled lecture). Please make sure to be on time.

The precise material for the first test will be announced on Friday 18 October (roughly the lecture material and Problem sheet #1). 

First problem sheet

The first problem sheet can now be found from the link in the right-hand-side margin. The lecture on monday 21 Oct will be dedicated to this problem sheet and selected problems will be discussed. Model answers will also be published on monday.

The problem sheet is provided to help you understand and work through the relevant lecture material. It is important to make some effort and thereafter reflect critically on your effort with the model answers.

Some background reading

As is inevitably the case, the mathematics we employ during this course will use some basic mathematical concepts (from analysis and metric spaces and their topology, mainly) that not all students are equally aware of or comfortable with. While I will do my very best to keep the course as self-contained as possible, and will always aim to emphasize examples that illustrate any abstract facts, students may benefit from some a compact background reference as the one given in Appendix A of [HK]. It can be found also at blackboard, for your convenience.

The most important notions are metric, open and closed sets, the existence of limits, and the contraction principle (Banach fixed point or contraction mapping theorem). While the course (and exam) are of course not about these abstract notions and objects in principle, of course we need some of such to be able to make statements and prove them.

[HK] Chapter 7

The lectures on topological dynamics, starting Friday 11/10, we follow in the first instance loosely Chapter 7 of Hasselblatt and Katok [HK], also available on Blackboard. 

Decimal expansion of rational numbers

Just to correct my answer to a question in the lecture of Monday 7/10: any real number is rational if and only if its decimal expansion is ultimately periodic, see for instance https://en.wikipedia.org/wiki/Repeating_decimal. So indeed, it follows that the set of eventually periodic points of the map $x\to 10 x$ mod 1 on [0,1) is exactly equal to the set of rational points, which is obviously dense in [0,1).

Suggested reading

The course material will be made available as a combination of purpose lecture notes and selected parts of other texts. It will always be made entirely clear which material is relevant to tests and the exam.

Some of my favourite texts are:

[HK] Boris Hasselblatt and Anatole Katok. A first course in Dynamics. 2003.

- This is very readable as an introduction. It guides the reader's understanding of many important concepts in dynamical systems.

[BS] Michael Brin and Garrett Stuck. Introduction to Dynamical Systems 2002.

- This is a very good book, but in parts it may be written too compactly for a first introduction.


Other good texts: 

John Guckenheimer and Philip Holmes. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. 1983. (somewhat dated but inspiring in scope and context)

Anatole Katok and Boris Hasselblatt. Introduction to the Modern Theory of Dynamical Systems.1995. (impressive reference text; not so appropriate for as a first text)

Clark Robinson. Dynamical Systems. Stability, Symbolic Dynamics and Chaos. 1995. (advanced textbook)
On blackboard there are some additional suggestions for background reading. 

New schedule of lectures

Due to the size of the student group and room availability issues, the lecture schedule has been changed to the following:

Monday 12-1pm in HXLY 140 for weeks 2 and 11, and in HXLY 342 for weeks 3-10

Friday 12-2pm in HXLY 342 for weeks 2-7 and 9-10 and in HXLY 658 in week 8 (latter one tbc). 

I hope this does not lead to too many clashes. Unfortunately, despite a lot of effort from the administration, this was the best solution to the scheduling problem.

Some illustrations of dynamical systems

The page https://brain.cc.kogakuin.ac.jp/~kanamaru/Chaos/e/ contains some interesting illustrations of dynamical systems, some of which we will show in class.

Welcome

DYNAMICAL SYSTEMS

Prof Jeroen S.W. Lamb

Autumn 2019

Lectures: monday 12:00-13:00 & friday 13:00-15:00 in Huxley 642
Office hour: monday 11:00-12:00 in Huxley 638

The aim of this course is to provide an introduction to basic concepts of dynamical systems, also popularly known as Chaos Theory, both from a topological and probabilistic point of view.

This course is particularly recommended for students intending to take the following optional courses: Dynamics of Games, Bifurcation Theory, Advanced Dynamical Systems and Random Dynamical Systems and Ergodic Theory.

Lecture notes and selected book chapters that cover the relevant material will be posted in due course.

Some of the lectures will be devoted to discussion of exercises, details tba. There will also be three class tests, each counting for 3.33% of your total exam mark. Dates of these will be announced asap in due course.