Discrete Holomorphic Local Dynamical Systems

Dynamics of Entire Functions 329
Remark 6.6. Fibers in exponential parameter space are defined in analogy as for the
Mandelbrot set (see [RSch08] for the exponential case and [Sch04b] fortheMandelbrot
case). The second conjecture says that all non-hyperbolic exponential maps
are combinatorially rigid (their landing patterns of periodic dynamic rays differ); it
is the analog to the famous conjecture that the Mandelbrot set is locally connected.
The second conjecture implies the first [RSch08].
We conclude this section with a question of Euler [E1777] that was raised in 1777,
and its generalization to complex numbers.
Question 6.7 (Euler Question: Iterated Exponentiation).
For which real a > 0 does the sequence a, aa , aaa, aaaa ,...,havealimit?
This question can be rephrased, by setting λ := loga, as follows: for which λ ∈ R
does the sequence x0 := 0, xn+1 = λ e xn , have a limit? In this form, it makes sense
for λ ∈ C ∗ . The answer is surprisingly complicated. Set E λ (z) := λ e z .Tobegin
with, we note that if the sequence xn converges to some b ∈ C but is not eventually
constant, then b must be an attracting or parabolic fixed point (clearly, no repelling
fixed point and no center of a Siegel disk can be a limit point of an orbit unless the
latter is eventually constant; and it is not hard to show that a Cremer point cannot be
the limit of the unique singular orbit).
The map E λ has an attracting fixed point if and only if λ = μe −μ with μ ∈ D ∗
(since 0 is the only singular value of E λ , it follows that in these cases, the orbit of
0 must converge to the attracting fixed point, by Theorem 2.3). Similarly, E λ has a
rationally indifferent fixed point if and only if λ = μe −μ and μ is a root of unity.
This takes care of all cases where the orbit of 0 converges to a finite limit point
in C without eventually being constant. (The analogous question for periodic limit
points leads to the classification of hyperbolic components and their boundaries in
exponential parameter space, and thus items (4)and(6) in Theorem 6.4.)
The description of parameters λ in which the orbit of 0 is eventually fixed (or
eventually periodic) involves a classification of postsingularly finite exponential
maps, and this is settled by item (10) in Theorem 6.4. Finally, the case of parameters
λ in which the orbit of 0 converges to ∞ is item (7) in that theorem.
7 Newton Maps of Entire Functions
If f is an entire holomorphic function, then its associated Newton map Nf :=
id − f / f ′ is a meromorphic function that naturally “wants to be” iterated. While
the iteration of general meromorphic functions falls outside of the scope of this
manuscript, there are a number of results specifically on Newton maps of entire
functions. In rational iteration theory, polynomials are the easiest maps to work on,
and their Newton maps have useful properties that make them easier to investigate
than general rational maps. Since we believe that the situation should be similar for
in the transcendental world, this section is included.