Discrete Holomorphic Local Dynamical Systems

Dynamics of Entire Functions 301
an r > 0 with Dr(0) ⊂ h(D). Butthencn > r/2 for all large n. Thisimpliesthatno
subsequence of the cn tends to 0, and hence that inf{cn} > 0.
There is thus a c > 0, and there are c ′ n > c with ∂D c ′ n (0) ⊂ hn(D) for all large n.
This implies
∂D c ′ nM(rn/2; f )(0) ⊂ f (Drn (0)) ⊂ f (Dn) ⊂ Dn
and thus, by the definition of rn,
cM(rn/2; f ) < rn .
Therefore, M(rn/2; f )/rn is bounded.
Define an entire function g via g(z) := f (z)/z − f ′ (0) (the isolated singularity at
z = 0 is removable because f (0) =0, and thus g(0)=0). It has the property that
M(rn/2;g) is bounded; since rn → ∞, this implies that g is bounded on C and hence
constant. But g(0)=0, hence f (z)=zf ′ (0) for all z,so f is a polynomial of degree
at most 1 as claimed.
We have now shown that, if f is not a polynomial of degree 0 or 1, then J :=
J( f ) �= /0. If |J| > 1, then it follows from Picard’s Theorem A.4 that J is infinite and
unbounded: as soon as J contains at least two points a,a ′ , every neighborhood of ∞
contains infinitely many preimages of a or a ′ .
We finally have to consider the case that |J| = 1, say J = {a}. In this case, f (a)=
a, and f must have another fixed point, say p, andW := C \{a} equals the Fatou
set. We must have | f ′ (p)|≤1, and | f ′ (p)| = 1 would lead to the same contradiction
as above. This implies that | f ′ (p)| < 1. But any loop in W starting and ending at p
would converge uniformly to p, and by the maximum modulus principle this would
imply that W was simply connected, a contradiction. This shows that |J| > 1inall
cases, hence that J is always an infinite set (we will show below in Theorem 1.7 that
J is always uncountable). ⊓⊔
For an entire function f ,anexceptional point is a point z ∈ C with finite backwards
orbit. There can be at most one exceptional point: in fact, any finite set of
exceptional points must have cardinality at most 1 (or the complement to the union
of their backwards orbits would be forward invariant and hence contained in the
Fatou set, by Montel’s Theorem A.2, so the Julia set would be finite). If an entire
function f has an exceptional point p, it is either an omitted value or a fixed point
(such as the point 0 for e z or ze z ); in such cases, f restricts to a holomorphic self-map
of the infinite cylinder C \{p}.
Theorem 1.7 (Topological Properties of the Julia Set).
For every entire function f (other than polynomials of degree 0 or 1), the Julia set
is the smallest closed backward invariant set with at least 2 points. The Julia set
is contained in the backwards orbit of any non-exceptional point in C, it has no
isolated points, and it is locally uncountable.
Proof. Suppose z ∈ C has a neighborhood U so that f ◦n (U) avoids a. Thenitalso
avoids all points in f −1 (a).Unlessa is an exceptional point, Montel’s Theorem A.2
implies that z is in the Fatou set. The backwards orbit of any non-exceptional point
in C thus accumulates at each point in the Julia set. Any closed backwards invariant