Discrete Holomorphic Local Dynamical Systems

298 Dierk Schleicher
In a brief appendix, we state a few important theorems from complex analysis
that we use throughout.
The research field of complex dynamics is large and active, and many people
are working on it from many different points of view. We have selected some topics
that we find particularly interesting, and acknowledge that there are a number
of other active and interesting topics that deserve no less attention. We mention in
particular results on measure theory, including the thermodynamic formalism (see
Urbański [Ur03] for a recent survey). Further important omitted areas that we should
mention are Siegel disks and their boundaries (see e.g., Rempe [Re04]), questions
of linearizations and small cycles (see e.g. Geyer [Ge01]), the construction of entire
maps with specific geometric or dynamic properties (here various results of
Bergweiler and Eremenko should be mentioned), the relation of transcendental dynamics
to Nevanlinna theory, and Thurston theory for transcendental maps (see
Selinger [Se09]).
We tried to give many references to the literature, but are acutely aware that the
literature is vast, and we apologize to those whose work we failed to mention. Many
further references can be found in the 1993 survey article of Bergweiler [Be93] on
the dynamics of meromorphic functions.
The illustration on the first page shows the Julia set of a hyperbolic exponential
map with an attracting periodic point of period 26. The Fatou set is in white. We
thank Lasse Rempe for having provided this picture.
ACKNOWLEDGEMENTS. I would like to thank my friends and colleagues for
many interesting and helpful discussions we have had on the field, and in particular
on drafts of this manuscript. I would like to especially mention Walter Bergweiler,Jan
Cannizzo, Yauhen Mikulich, Lasse Rempe, Phil Rippon, Nikita Selinger,
and Gwyneth Stallard. And of course I would like to thank the CIME foundation
for having made possible the summer school in Cosenza, and Graziano Gentili and
Giorgio Patrizio for having made this such a memorable event!
1 Fatou and Julia Set of Entire Functions
Throughout this text, f will always denote a transcendental entire function
f : C → C. The dynamics is to a large extent determined by the singular values, so
we start with their definition.
Definition 1.1 (Singular Value).
A critical value is a point w = f (z) with f ′ (z) =0; the point z is a critical point.
An asymptotic value is a point w ∈ C such that there exists a curve γ : [0,∞) → C
so that γ(t) → ∞ and f (γ(t)) → w as t → ∞. Thesetofsingular values of f is the
closed set
S( f ) := {critical and asymptotic values} .
This definition is not completely standard: some authors do not take the closure in
this definition.