How to study for the exam

I have now posted some detailed notes on Blackboard, on how to best prepare for the exam. In this note I detail in particular material that you need not study in detail and I provide a description of the material that will feature in the exam.

Master's exam question material

As earlier flagged, the material of the last lecture forms the basis for the final question (5) of the exam, which is not to be answered by M3 students. The relevant handwritten notes for this lecture have now been posted on Blackboard under the name "Symbolic Dynamics 2D".

Solutions to exercises of chapter 5.

I have now posted the answers to the chapter 5 exercises on Blackboard. I had inadvertently forgotten to do so earlier. Thanks to one of you for pointing this out. The model solutions are now on blackboard.

Final lecture 11/12 about Master's exam question

The last lecture of the course will be entirely dedicated to the material that covers the Master's exam question (the 5th exam question, for MSci and MSc students only). This material will concern some examples of chaotic dynamics in two dimensions: horseshoes and hyperbolic total automorphisms.

Relevant handouts will be posted in due course.

Exercise class 7/12 about Chapter 5 exercises

The lecture of Fri 7/12 will be an exercise class (moderated by Andrew Clarke) about the exercises of chapter 5 in the course notes by Dr Rasmussen.

There is an opportunity to get feedback on your own work on these exercises (not for credit) if you hand in your work by wed 5/12, 5pm in my mailbox in the staff mailroom on Huxley 6th floor.

Exercise class 30/11 about the exercises on the "symbolic dynamics" notes

The lecture of Fri 30/11 will be an exercise class about the exercises of chapter four on my notes on symbolic dynamics.

There is an opportunity to get feedback on your own work on these exercises (not for credit) if you hand in your work by wed 28/11, 5pm in my mailbox in the staff mailroom on Huxley 6th floor.

Model answers for these exercises have now been posted on blackboard (if you do not have access but would like to get them, drop me an e-mail at jswlamb@ic.ac.uk).

Dynamics of (sub)shifts

In the lectures we skipped section 3 of chapter 4 of Martin Rasmussen's notes (on Chaos in 2 dimensions: the horseshoe mapping) and instead discussed one-dimensional dynamics
(semi-)conjugate to subshifts associated to topological Markov Chains. You find relevant notes (by me) on Blackboard, as well as part of Chapter 7 from Hasselblatt and Katok (broadly dealing with the same matter). 

Exercise class 16/11 about Chapter 4 exercises

The lecture of Fri 16/11 will be an exercise class (moderated by Andrew Clarke) about the exercises of chapter four in the course notes by Dr Rasmussen.

There is an opportunity to get feedback on your own work on these exercises (not for credit) if you hand in your work by wed 14/11, 5pm in my mailbox in the staff mailroom on Huxley 6th floor.

Model answers for these exercises have now been posted on blackboard (if you do not have access but would like to get them, drop me an e-mail at jswlamb@ic.ac.uk).

Exercise class 2/11 about Chapter 3 exercises

The lecture of Fri 2/11 will be an exercise class (moderated by Andrew Clarke) about the exercises of chapter three (p42) in the course notes by Dr Rasmussen.

There is an opportunity to get feedback on your own work on these exercises (not for credit) if you hand in your work by wed 31/10, 5pm in my mailbox in the staff mailroom on Huxley 6th floor.

Model answers for these exercises have now been posted on blackboard (if you do not have access but would like to get them, drop me an e-mail at jswlamb@ic.ac.uk).

Exercise class 19/10 about Chapter 2 exercises

The lecture of Fri 19/10 will be an exercise class (moderated by Andrew Clarke) about the exercises of chapter two (p22-23) in the course notes by Dr Rasmussen.

There is an opportunity to get feedback on your own work on these exercises (not for credit) if you hand in your work by wed 5pm in my mailbox in the staff mailroom on Huxley 6th floor.

Model answers for these exercises have now been posted on blackboard (if you do not have access but would like to get them, drop me an e-mail at jswlamb@ic.ac.uk).

Distribution of orbits of the doubling map.

Charles had a good remark after the lecture: I told that the orbits of the map x(n+1)=2x(n) mod 1has a uniform distribution on the unit circle, but of course it is easy to see that the point x=0 is a fixed point, so it does not hold for all points (in particular it does not hold for any point with rational initial condition. So let me clarify: my comment holds for almost all initial conditions (a set of initial conditions of full arc length or Lebesgue measure). The exceptional points are dense, but have (as a set) zero length. One can interpret this as saying that if one choses an initial condition randomly, there is zero chance that its orbit does not converge to a uniform distribution.

My comments about this example are only tasters for what is to come later and the explanation of all of the above will be central to the course. 

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

M345PA23

DYNAMICAL SYSTEMS

Prof Jeroen S.W. Lamb

Autumn 2018

Lectures: tuesday 11:00-13:00 & friday 12:00-13:00 in Huxley 139
Office hour: friday 13:00-14:00 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 Dynamics of Games (MxPA48), Bifurcation Theory (MxPA24), Advanced Dynamical Systems (MxPA38) or Random Dynamical Systems and Ergodic Theory (MxPA40).

The course broadly follows lecture notes by Dr Martin Rasmussen, who taught this course the last few years, which are available to download from the course page at Blackboard (please contact me by e-mail if you do not have access so I can send you the notes by e-mail). Further guidance and eventual additional notes will be posted here in due course.

Some of the friday lecture hours will be devoted to discussion of the exercises, details tba.