Ph.D. Projects. Ph.D. Projects.

For me, Complex Dynamics principally means iterating rational functions, that is, studying the behaviour of zn, where

zn = f(zn),
where f is a rational function. f could, of course, be a polynomial, even a quadratic polynomial of the form
f(z) = z2+c.
The parameter space of quadratic polynomials must be one of the most studied in all of Dynamics. Many undergraduate courses involve some study o real quadratic polynomials, especially of the well-known chaotic map
f(x) = x2-4.
A very detailed structure is known for the space of quadratic polynomials, although there are still some important open problems concerning this family of an analytical nature. Some good sources for the basic theory of Complex Dynamics and for the theory of quadratic polynomials are in an outstanding resource for Complex dynamics:

The SUNY Stony Brook IMS preprint archive.

Two excellent expository accounts are the following.

J. Milnor: Dynamics in one variable: introductory lectures.

J. Milnor: Periodic Orbits, External Rays and the Mandelbrot Set: an expository account.

In electronic form, the archive is available from 1990, and is especially strong in Complex Dynamics, with other interesting areas of dynamics also well represented. Although, of course, the preprints available in the archive are by people who have had a connection with IMS at Stony Brook over that time, some important papers in the subject are available in the series. For more suggestions of preprints in the series which might be more closely related to the material on these webpages, click here.

My own interest is in quadratic rational maps, where detailed combinatorial structure is only now starting to emerge. The structure for quadratic polynomial space is interesting enough, but it turns out that there are very significant extra twists to the structure when one considers quadratic rational maps.

As an example of what can occur the reader might like to consider the following family

fa(x) = 1- a+1
x
 + a
x2
 .
We restrict to real parameter a and real variable x, so considering fa as a function on the real line. The corresponding family of complex functions wth complex parameter is very interesting but this will suffice for the moment.

If you want to read more, click here.

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On 6 Nov 2005, 20:05.