Holomorphic Dynamics and Hyperbolic Geometry (LTCC Feb-March 2013)


Mondays 10.50-12.50, 18th February to 18th March (inclusive) at the LMS Headquarters, De Morgan House in Russell Square (use the staircase to the basement just along from the main entrance).


There is an extensive list of books at the end of the Notes. The two books by Beardon are very readable and provide a good introduction to two themes of this course - rational maps and hyperbolic geometry. Milnor's book is excellent both as an introduction and for providing an insight into some more advanced topics. Mumford, Series and Wright is great to look at - and to read. Al Marden's book goes deeper into recent advances in Kleinian groups and hyperbolic 3-manifolds. Michele Audin's book tells the fascinating story of the work of Julia and Fatou during and after the First World War. Jim Anderson's book covers the geometry and trigonometry of the hyperbolic plane.

To draw computer pictures of Julia sets and the Mandelbrot set, for Mac users I recommend FractalStream - which is both fast and easy to use (you can download the package and a user guide from the web). Also easy to use are the Java applets Bob Devaney has made available on his Boston University website. I do not know of any programs for plotting limit sets of Kleinian groups which are quite as easy to use as either of these, but there is a Kleinian groups program on Curt McMullen's webpage and there may well be others elsewhere.

There is a helpful list of software for holomorphic dynamics on the Stony Brook Dynamics website www.math.sunysb.edu/dynamics/programs/ . One program that looks particularly useful is Wolf Jung's 'Mandel', which you can find at www.mndynamics.com/indexp.html . There are also complex dynamics programs on the webpages of Arnaud Cheritat, Curt McMullen and Mitsuhiro Shishikura.

The DVD "Dimensions" by Etienne Ghys and his colleagues is available for free download at www.dimensions-math.org . It contains some introductory material on complex maps - with excellent graphics - and also some very visual material on geometry and topology in higher dimensions.

Lecture Notes

Detailed lecture notes will be handed out each week. They will also appear here a few days before the lecture if I finish them in time - and otherwise shortly after the lecture. They will be corrected from time to time: the final version at the end of the course should be the most accurate.

Week 1 Notes (18th Feb)

Week 2 Notes (25th Feb)

Week 3 Notes (4th March)

Week 4 Notes (11th March)

Week 5 Notes (18th March)

Complete Set of Notes


Week 1 Exercises Solutions to Week 1 Exercises

Week 2 Exercises Solutions to Week 2 Exercises

Week 3 Exercises Solutions to Week 3 Exercises

Week 4 Exercises Solution to Week 4 Exercise 1

Assessment Exercises Solutions to Assessment Exercises

Solutions to the assessment exercises should be sent to me (e.g. by e-mail) by the end of March. A full solution to any one of questions 2,3 or 4 on the Week 4 sheet is an acceptable alternative to a set of solutions to the 'Assessment Exercises'.

Last updated: 8th April 2013.
Prof Shaun Bullett.