Discrete Holomorphic Local Dynamical Systems

326 Dierk Schleicher
One way to interpret this definition is as follows. We write g =(ϕ ◦ f ◦ ϕ −1 ) ◦
(ϕ ◦ ψ −1 ),sog is a quasiconformally conjugate function to f , postcomposed with
another quasiconformal homeomorphism. In analogy, every quadratic polynomial
is conjugate to one of the form z 2 + c, so any two quadratic polynomial differ from
each other by conjugation, postcomposed with an automorphism of C (here, there
are few enough marked points so that for the postcomposition, it suffices to use
complex automorphisms).
Theorem 6.2. Let f ,g be two entire functions of bounded type that are quasiconformally
equivalent near ∞. Then there exist R > 0 and a quasiconformal
homeomorphism ϑ : C → C so that ϑ ◦ f = g ◦ ϑ on
AR := {z ∈ C: | f ◦n (z)| > R for all n ≥ 1} .
Furthermore ϑ has zero dilatation on {z ∈ AR : | f ◦n (z)|→∞}.
In particular, quasiconformally equivalent entire functions of bounded type are
quasiconformally conjugate on their escaping sets. This result may be viewed as
an analog to Schröder’s theorem (that any two polynomials of equal degree are
conformally conjugate in a neighborhood of ∞); it is due to Rempe [Re], together
with the following corollary.
Corollary 6.3 (No Invariant Line Fields).
Entire functions of bounded type do not support measurable invariant line fields on
their sets of escaping points.
A lot of work has been done on the space of the simplest entire functions: that
is the space of exponential functions. It plays an important role as a prototypical
example, in a similar way as quadratic polynomials play an important prototypical
role for polynomials. By now, there is a good body of results on the parameter space
of exponential functions, in analogy to the Mandelbrot set. Some of the results go
as follows:
Theorem 6.4 (Exponential Parameter Space).
The parameter space of exponential maps z ↦→ λ e z has the following properties:
1. there is a unique hyperbolic component W of period 1; it is conformally
parametrized by a conformal isomorphism μ : D ∗ → W, μ ↦→ μ exp(−μ),
so that the map E λ with λ = μ exp(−μ) has an attracting fixed point with
multiplier μ;
2. for every period n ≥ 2, there are countably many hyperbolic components of
period n; on each component, the multiplier map μ : W → D ∗ is a universal
covering;
3. for every hyperbolic component W of period ≥ 2, there is a preferred conformal
isomorphism Φ : W → H − with μ = exp◦Φ (where H − is the left half plane);
4. there is an explicit canonical classification of hyperbolic components and
hyperbolic parameters;