Discrete Holomorphic Local Dynamical Systems

296 Dierk Schleicher
made possible detailed studies of particular maps; most often, quadratic polynomials
as the simplest non-trivial holomorphic mappings were studied. While a number
of deep questions on quadratic polynomials remain, interest expanded to specific
(usually complex one- or two- dimensional) families of holomorphic maps, such
as quadratic rational maps, cubic polynomials, or other families in which critical
orbit relations reduced the space to simple families of maps: for instance, families
of polynomials of degrees d ≥ 2 with a single critical point of higher multiplicity.
Only in recent years has the progress achieved so far allowed people to shift interest
towards higher-dimensional families of iterated maps, such as general polynomials
of degree d ≥ 2. The study of iterated rational maps, as opposed to polynomials,
seems much more difficult, mainly because of lack of a good partition to obtain
a good encoding for symbolic dynamics: the superattracting fixed point at ∞, and
the dynamic rays emanating from it, are important ingredients for deep studies of
polynomials that are not available for general rational maps. A notable exception
are rational maps that arise from Newton maps of polynomials: for these, it seems
that good partitions for symbolic dynamics are indeed possible.
Transcendental iteration theory has been much less visible for a long time, even
though its study goes back to Fatou (we will even treat a question of Euler in
Section 4), and a solid body of knowledge has been developed by Baker and coauthors,
and later also by Eremenko and Lyubich and by Devaney and coauthors, for
more than four decades. Complex dynamics is known for employing methods from
many different fields of mathematics, including geometry, complex analysis, algebra
and even number theory. Transcendental dynamics unites two different general
directions of research: there is a substantial body of knowledge coming from value
distribution theory that often yields very general results on large classes of iterated
transcendental functions; among the key contributors to this direction of research are
Baker, Bergweiler, Eremenko, Rippon, and Stallard. The other direction of research
sees transcendental functions as limits of rational functions and employs methods
adapted from polynomial or rational iteration; here one usually obtains results on
more specific maps or families of maps, most often the prototypical families of
exponential or cosine maps; this direction of research was initiated by Devaney
and coauthors. Others, like Lyubich and Rempe, have worked from both points
of view.
In recent years, transcendental iteration theory has substantially gained interest.
There have recently been international conferences specifically on transcendental
iteration theory, and at more general conferences transcendental dynamics is obtaining
more visibility.
In this survey article, we try to introduce the reader several aspects of transcendental
dynamics. It is based on lecture notes of the CIME summer school held in
Cosenza/Italy in summer 2008, but substantially expanded. The topic of that course,
“dynamics of entire functions”, also became the title of this article. We thus focus
almost entirely on entire functions: their dynamical theory is much simpler than the
theory of general meromorphic transcendental functions, much in the same way that
polynomial iteration theory is much simpler (and more successful) than rational iteration
theory. However, we believe that some of the successes of the polynomial