Discrete Holomorphic Local Dynamical Systems

328 Dierk Schleicher
indifferent orbit with multiplier e iy ; if not, then the corresponding parameter would
be missing from the classification.
(7) Parameter rays in the space of exponential maps are constructed and classified
in [FS09]. There are additional parameters for which the singular value escapes;
these are landing points of certain parameter rays and are classified in [FRS08].
(8) Itisshownin[BBS08] that parameter rays in exponential parameter space
have Hausdorff dimension 1, and in [Qiu94] that all escaping parameters have Hausdorff
dimension 2. By [FRS08], every escaping parameter is either on a parameter
ray or a landing point of one of them.
(9) Contrary to one of the principal conjectures on the Mandelbrot set, and many
combinatorial similarities between the parameter spaces of exponential maps and
quadratic polynomials, the exponential bifurcation locus is not locally connected
at any point on any parameter ray: in fact, any parameter ray is approximated by
other parameter rays on both sides, and between any pair of parameter rays there
are are hyperbolic components. This destroys local connectivity of the exponential
bifurcation locus [RSch08].
(10) For rational maps, there is a fundamental theorem of Thurston [DH93] that
characterizes rational maps with finite critical orbit and that is at the basis of most
classification theorems in rational dynamics. Unfortunately, there is no analogous
result for transcendental maps. Currently, the only extension of Thurston’s theorem
to the case of transcendental maps is [HSS09] on exponential maps with finite singular
orbits (see also [Se09] for work in progress in this direction); the resulting
classification of postsingularly finite exponential maps is in [LSV08]. This classification
had been expected for a long time [DGH].
(11) The fundamental study of the Mandelbrot starts with Douady and Hubbard’s
result about connectivity of the Mandelbrot set, or equivalently of its boundary,
which is the bifurcation locus in the space of quadratic polynomials. The corresponding
result about exponential maps is that the exponential bifurcation locus is
connected. Unlike in the polynomial case, where connectivity of the bifurcation locus
of quadratic polynomials is the starting point for much of the theory of the Mandelbrot
set, this result comes at the end of a detailed study of exponential parameter
space. It was shown in [RSch09].
Just like for the whole field of entire dynamics, it is impossible to do justice to
the large body of knowledge that has been established from many different points
of view even on exponential parameter space. Among further existing work, we
would like to mention [GKS04, UZ07, Ye94]. There is also a significant amount of
work on other one-dimensional parameter spaces of explicit entire functions; we
only mention the work by Fagella, partly with coauthors, on the families z ↦→ λ ze z
[Fa99] and on z ↦→ λ z m e z (with m ≥ 2) [FaGa07]. Even though these are not entire
functions, we would also like to mention work on the family of tangent maps by
[KK97].
Conjecture 6.5 (Exponential Parameter Space).
• Hyperbolicity is dense in the space of exponential maps.
• Fibers in exponential parameter space are trivial.