Discrete Holomorphic Local Dynamical Systems

304 Dierk Schleicher
logarithmic singularity if there is an Ur so that f : Ur → Dχ(a,r)\{a} is a universal
covering. In particular, if f is a transcendental entire function, then ∞ is always a
direct asymptotic value (see Theorem 1.14 below). For entire functions of bounded
type (see Section 3), ∞ is always a logarithmic singularity.
Recall from Definition 1.1 that we defined singular value to be a critical value,
an asymptotic value, or a limit point thereof.
Theorem 1.13 (Singular Values).
Any a ∈ C that is not a singular value has a neighborhoodU so that f : f −1 (U) →U
is an unbranched covering (i.e., a is a regular value for all tracts).
Proof. Choose a neighborhoodU of a, small enough so that it is disjoint from S( f );
it thus contains no critical or asymptotic value. If U has an unbounded preimage
component, then it is not hard to show that U contains an asymptotic value (successively
subdivide U so as to obtain a nested sequence of open sets with diameters
tending to 0 but with unbounded preimages). Therefore, a has a simply connected
neighborhood for which all preimage components are bounded. If V is such a preimage
component, then f : V → U is a branched covering, and if U contains no critical
value, then f : V → U is a conformal isomorphism. The claim follows. ⊓⊔
The set of direct asymptotic values is always countable [He57] (but the number
of associated asymptotic tracts need not be). There are entire functions for which
every a ∈ C is an asymptotic value [Gr18b]. On the other hand, according to the
Gross Star Theorem [Gr18a], every entire function f has the property that for every
a ∈ C and for every b ∈ C with f (b)=a, and for almost every direction, the ray at
a in this direction can be lifted under f to a curve starting at b.
The following theorem of Iversen is important.
Theorem 1.14 (Omitted Values are Asymptotic Values).
If some a ∈ C is assumed only finitely often by some transcendental entire function
f , then a is a direct asymptotic value of f . In particular, for every entire function, ∞
is always a direct asymptotic value.
Proof. For any r > 0, any bounded component of f −1 (Dχ(r,a)) contains a point z
with f (z) =a. Since by Picard’s Theorem A.4, at most one point in C is assumed
only finitely often, it follows that for every r > 0, the set f −1 (Dχ(r,a)) cannot consist
of bounded components only. Therefore, for each r > 0, there is at least one
unbounded component, and thus at least one asymptotic tract over the asymptotic
value a; for such a tract, a is a direct singularity. Since the point ∞ is omitted for
each entire function, it is a direct asymptotic value. ⊓⊔
The following definition is of great importance in function theory:
Definition 1.15 (Order of Growth).
The order of an entire function f is
ord f := limsup
z→∞
loglog| f |
.
log|z|