This may be a bit of a weird question but here goes...
My project for next year is on Complex Dynamics which includes looking at Mandelbrot and Julia sets. Its going to be a mixture of theoretical and computer work but probably slightly leaning towards theoretical.

Here's the official project description;

This project is concerned with the behavior of iterations of an analytic function of a complex variable. Even for quite simple functions, such as quadratic polynomials, the study of such iterations can lead to very intricate and spectacular patterns, such as the well-known `Julia sets' and `Mandelbrot set'. These sets have been the subject of active research over the last 25 years, combining computational work with deep theoretical results.
The object of the project would be first to explore the theoretical aspects, which mainly concern the study of fixed points and periodic points of analytic functions, which are central to understanding the dynamics. This can then be applied to explore the quadratic and possibly other families of functions. This can be done analytically or computationally, or (preferably) using a combination of the two, by exploring the dynamics computationally and interpreting the results theoretically. The emphasis of the project could range from entirely theoretical to mainly computational.

My Q is this... Is there an area of complex dynamics that hasn't been that well researched? I would like to try throw in some original work as well as just following the project guidelines. I know this is quite a vague question but I hope some of the more senior members might have an idea of what I'm getting at.

Cheers

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I'm sure there are many areas that have not been well researched. I doubt even more general polynomials than x^n+c have been well researched, and certainly not all rational functions. x^n+c is very easy because it has only one critical point. Even so it is an interesting study. For example can you work out how to find: all such c such that the Julia set for x^n+c is connected but has empty interior - these points are dense in the boundary of Mandelbrot sets (e.g. find all values value of c such that after 12 iterations starting from 0 x^n+c is periodic with period 7). I was quite pleased when I discovered how to find these points because it lets you draw the most interesting Julia sets without guessing. It is possible to find these points by finding the roots of certain polynomials, but because the polynomials have enormous degree it can be tricky.
I am very interested in Kleinian groups - which are the limit sets of discrete groups of Moebius transformations. There has been an enormous amount of research into these in recent years because of the connection with hyperbolic manifold theory, but most of it is very hard to follow unless you are a specialist. But just drawing limit sets is quite easy. Yet nobody can really tell if the coordinates of a proposed 2-generator Kleinian group (specified as the traces of 3 matrices or something equivalent) actually belong to a discrete group or not. There is a lot of research, but almost all on groups with a parabolic commutator, but groups with elliptic commutators and generators can have beautiful fractals. For example I have found out how to create 2-generator Kleinian groups which have the symmetries of any triangle group.
Going back to a more purely dynamic point of view, Julia sets of polynomials in trigonometric functions are interesting. For example the Mandelbrot set for sin(z*pi)^4+c contains small copies of the Mandelbrot set for z^2+c and that for z^4+c - or at least parts that look very like that, and it might be interesting to look at what happens for more complicated powers like z^12. Similar things happen for certain rational functions: in fact the classic z^2 Mandelbrot set occurs in the Mandelbrot set for numerous other functions.
Also, there must be many interesting topics connected with measures- Julia sets have an invariant measure associated with them, but this must be very hard to calculate, even for Julia sets of quadratic polynomials - this is maybe important because while you can draw good pictures of IFS fractals for monoids of affine transformations using random iteration, you cannot for Julia sets of polynomials because the density of the measure varies too much and you naver get to the more interesting parts of the limit set - finding the measure would help compensate for this.