I spent time working with cubic mappings. With two critical points, there are Julia sets that consist of infinitely many disconnected components that are still locally connected, corresponding two one critical point escaping and one falling into one of the basins of attraction.
Of course that was about 15 years ago, and I haven't really kept up. Too much time keeping up with software.

On 12/16/06, Jacques Carette < carette@mcmaster.ca> wrote:
Steve Downey wrote:
> No fair. Although I've a B.S. in Mathematics, I spent most of my time
> in complex analytic dynamical systems, rather than hanging with the
> algebraists.  Except for a bit where I did some graph theory.

Never thought I'd run into a fellow dynamicist on haskell-cafe!  I did
my PhD on the links between the geometry and dynamics of Julia sets (co
supervised by the (late) Adrien Douady and John Hubbard).  What
'flavour' of this stuff are you interested in?

Jacques

PS: now I have converted to being a computer scientist, which is why I
hang out on haskell-cafe!