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    Ph.D. Projects.
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For me, Complex 
Dynamics principally means iterating rational functions, that is, 
studying the behaviour of $z_{n}$, where
$$z_{n}=f(z_{n}),$$
where $f$ is a rational function. $f$ could, of course, be a 
polynomial, even a quadratic polynomial of the form 
$$f(z)=z^{2}+c.$$
The parameter space of quadratic polynomials must be one of the most 
studied in all of Dnamics. Many undergraduate courses involve some 
study o real quadratic polynomials, especially of the well-known 
chaotic map
$$f(x)=x^{2}-4.$$
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

$$f_{a}(x)=1-{a+1\over x}+{a\over x^{2}}.$$
We restrict to real parameter $a$ and real variable $x$, so 
considering $f_{a}$ as a function on the real line $\mathbb R$. 
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, follow the link.
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