THE THEORY OF POLYNOMIAL-LIKE MAPPINGS
-- THE IMPORTANCE OF QUADRATIC POLYNOMIALS
Nuria Fagella
Dep. de Matematica Aplicada i Analisi
Universitat de Barcelona
Gran Via 585
08007 Barcelona
Spain
e-mail: fagella@maia.ub.es
ABSTRACT
In the field of complex dynamics and, in particular, iteration
of functions of one complex variable, the topic that has by far
been object of the most attention is the iteration of the family
of quadratic polynomials $Q_c := z^2+c$. In this paper we aim to
answer the question of why this very particular family of polynomials
is important for the understanding of iteration of general
complex functions, by means of the theory of polynomial-like
mappings of Douady and Hubbard.
This theory explains how the understanding of polynomials is not
only interesting {\em per s\'e }, but helps understand a much wider
class of functions namely those that locally behave as polynomials do.
Most of the definitions and results in this paper may be found in
the work of Douady and Hubbard {\em ``On the Dynamics of Polynomial-like
Mappings''} \cite{dh3}.Our goal is to state their most important results
as well as to give several examples that illustrate them.
This is the third paper in the ``Complex Dynamics'' series of EWM 95.
We assume that the reader is familiar with the basic definitions and
theorems concerning the dynamics of quadratic polynomials which are
the topic of the first article by Bodil Branner.
(The whole proceedings may be downloaded from
http://www.math.helsinki.fi/EWM/meetings/proceedings/madrid.html
)
REFERENCE
Proceedings of the 7th EWM meeting, Madrid 1995
Editors Bodil Branner and Nuria Fagella