A most interesting treatize of iteration of polynomials.
I've just started reading, no results, but it promises much.
http://www.math.cornell.edu/~hubbard/Ite...ubics1.pdf
http://www.math.cornell.edu/~hubbard/Ite...ubics2.pdf
The iteration of cubic polynomial
by
BODIL BRANNER (The Technical University of Denmark and Cornell University, Ithaca, NY, U.S.A.) and
JOHN H. HUBBARD (Corner University, Ithaca, NY, U.S.A.)
( 1988 )
Part I: The global topology of parameter
Table of contents
Introduction ................................ 143
CHAPTER I. Univalent functions in complex analytic dynamics ...... 147
1. Attraction to infinity ....................... 147
2. Parametrizing the space of polynomials ............. 150
3. Compactness of the connectedness locus ............ 153
4. The mapping q~e is close to the identity ............. 156
5. The high level sets of H are spheres ............... 160
CHAPTER II. Wringing the complex structure ............... 165
6. Beltrami forms invariant under a polynomial .......... 166
7. Analytic dependence on parameters ............... 169
8. Stretching and wringing the complex structure ......... 171
9. Continuity on the structural stability set ............. 177
10. Continuity for cubic polynomials ................. 179
CHAPTER III. The global topology of parameter space .......... 183
11. Fibrations ............................. 184
12. The structure of the fiber, part 1 ................. 185
13. The structure of the fiber, part 2 ................. 193
14. The global topology of parameter space ............. 197
References ................................. 205
Also the webpage of John Hubbard has a lot of material whose titles (and cooperations) look promising...
Gottfried