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 given below.

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.