ON THE EXCEPTIONAL SET IN A CONJECTURE OF
LITTLEWOOD
LUKAS GEYER
1. Introduction and Main Result
0
2|f (z)|
denote the spherFor a meromorphic function f let f # (z) = 1+|f
(z)|2
−1
ical derivative of f , and let Sing(f ) denote the set of singular values,
i.e. the set of all critical and asymptotic values of f . Let S be the class
of entire functions with finitely many singular values, and B the class
of entire functions with a bounded set of finite singular values. For
Borel sets A ⊆ C we write #A for the cardinality of a set A, and |A|
for its (two-dimensional) Lebesgue measure. If A and B are Borel sets
with |B| ∈ (0, ∞) we define dens(A, B) = |A∩B|
as the density of A in
|B|
B. We denote the unit disk by D, the disk of radius r centered at 0 by
Dr , and the annulus r < |z| < s by Ar,s .
If R is a rational function of degree n, an application of the CauchySchwarz inequality yields
Z Z
1/2 Z Z
1/2
ZZ
#
#
2
R (z)dx dy ≤
R (z) dx dx
dx dy
(1)
D
D
D
√
≤ (4πn)1/2 π 1/2 = 2π n,
since R covers the sphere n times, and the area of the sphere is 4π.
For general rational functions this is asymptotically best possible, but
Littlewood conjectured that for polynomials this estimate could be
improved. More precisely, he conjectured the following, which was
later proved by Lewis and Wu.
Theorem 1 (Lewis, Wu [3]). There exist absolute constants C and
α > 0 with
ZZ
(2)
P # (z)dx dy ≤ Cn1/2−α
D
for all n and all polynomials P of degree n.
Remark. In fact, Lewis and Wu showed that one can choose α = 2−264 ,
and Eremenko proved in [2] that one cannot choose α arbitrarily close
to 1/2.
1
2
LUKAS GEYER
Littlewood showed that this result had a curious implication for the
value distribution of entire functions of finite order. Roughly speaking,
most values are taken in a very small subset of the plane.
Corollary 2 (Littlewood[4]). Let f be an entire function of finite order ρ ∈ (0, ∞), and let β ∈ (0, α), where α is the constant of Theorem 1. Then there exists a constant C1 and an open set S ⊂ C
with dens(S, Ar,2r ) ≤ C1 r−2ρβ for all r > 0, such that for almost all
w ∈ C and all > 0, there exists a constant C2 such that the set
Ew = f −1 (w) \ S satisfies #(Ew ∩ Dr ) ≤ C2 rρ−(α−β)ρ+ for all r > 1.
We call Ew the set of exceptional preimages. Since we expect to have
roughly rρ preimages in Dr for a function of order ρ and a typical point
w, the estimate on the cardinality shows that most preimages of w lie in
S, which has a geometrically decreasing Lebesgue density. Obviously
meromorphic functions of finite order do not have this property, as
shown by the Weierstrass ℘-function.
It is conceivable that the estimate on the cardinality of the set of
exceptional preimages can be improved considerably in the classes S
and B. In particular, it is an open question whether in the class S one
can always choose S so that Ew is a finite set for all w. This would
be especially interesting in the theory of dynamics of entire functions,
as it would imply rigidity of the growth order ρ under quasiconformal
deformations. In this note we show that in the class B the exceptional
set can indeed contain ≥ c log r points, as made precise in the following
theorem.
log 2
, ∞) there exists a function
Theorem 3. For almost every ρ ∈ ( log
3
f ∈ B of order ρ and a set W of positive measure such that for every
w ∈ W the set Ew of Corollary 2 satisfies
#(Ew ∩ Dr )
ρ
(3)
lim inf
≥
.
r→∞
log r
log 2
Furthermore, for every > 0 there exists a function of growth order
ρ ∈ (1/2, 1/2 + ) satisfying (3).
Remark. It is conceivable that our technique of proof may be pushed
to yield the result for all ρ > 1/2, but it will never produce examples
of growth order ρ ≤ 1/2.
2. Proof
The entire functions we use will be Poincaré functions of quadratic
polynomials at repelling fixed points. We obtain the exceptional preimages as preimages of a rotation domain under the Poincaré function. In
ON THE EXCEPTIONAL SET IN A CONJECTURE OF LITTLEWOOD
3
2
order to get almost all ρ > log
, we use explicit polynomials with fixed
log 3
Siegel disks; in order to obtain ρ arbitrarily close to 1/2, we choose perturbations of the Chebyshev polynomial T (z) = z 2 − 2 with periodic
Siegel disks.
Let P (z) = λw+w2 with λ = e2πiγ . By a classical result of Siegel, for
almost all γ ∈ R the function P can be linearized near 0, i.e. there exists
an analytic linearizing map h(z) = z+O(z 2 ) near 0 such that P (h(z)) =
h(λz). The power series of h has a finite radius of convergence R, and
h maps DR conformally onto the Siegel disk V of P centered at 0. The
polynomial P has another finite fixed point at z0 = 1−λ with multiplier
µ := P 0 (z0 ) = 2 − λ. Since |µ| > 1, there exists a local linearizing
function f (z) = z0 + z + O(z 2 ) with P (f (z)) = f (µz). In this case the
functional equation allows to extend f to an entire function of growth
log 2
, the Poincaré function of P at z0 . We now fix such a
order ρ = log
|µ|
function f associated to a polynomial P with a Siegel disk centered at
0. We will show that f ∈ B and that the exceptional set satisfies the
asymptotic estimate (3). Since |µ| = |2 − λ| attains almost every value
in the interval (1, 3), this proves the theorem.
The fact that f ∈ B is well-known, it follows from the fact that P
has a connected Julia set, and thus a bounded post-critical set. As a
Poincaré function, f has no finite asymptotic values, and its critical
values are the points in forward orbits of finite critical points of P .
Since P is a quadratic polynomial with a Siegel disk, the finite critical
point and its forward orbit are contained in the Julia set of P , which
is a compact subset of the plane.
It remains to show (3). Let α > 0 be the constant from Theorem 1,
β ∈ (0, α), and S ⊂ C given by Corollary 2. We write Ck for constants
depending only on α, β, and the growth order ρ.
Let W := h(DR/2 ) be a sub-Siegel disk. The Koebe Distortion Theorem implies that there exists an absolute constant M such that
dens(g(A), g(W ))
1
(4)
≤
≤M
M
dens(A, W )
for all conformal maps g : V → C and all Borel sets A ⊆ C of positive
measure.
Since there are no critical points in the Fatou set of P , the Siegel disk
V contains no singular values of f , and since V is simply connected,
f maps every component of f −1 (V ) conformally onto V . Let U0 be
one such component, and let Uk = µk U . Then f (Uk ) = f (µk U0 ) =
P k (V ) = V , so (Uk ) is a sequence of components of f −1 (V ). Let
Wk = f −1 (W ) ∩ Uk and Sk = S ∩ Wk . By Corollary 2, dens(S, Wk ) =
dens(Sk , Wk ) ≤ C3 |µ|−2ρβk for all k. Applying (4) to the branch of f −1
4
LUKAS GEYER
mapping
≤ C4 |µ|−2ρβk . Setting E :=
T∞ S V to Uk yields dens(f (Sk ), W ) P
∞
−2ρβk
for every n,
n=1
k=n f (Sk ), we get dens (E, W ) ≤
k=n C4 |µ|
so dens(E, W ) = 0, and hence |E| = 0. This implies that almost every
w ∈ W satisfies g −1 (w) ∩ Sk = ∅ for all but finitely many indices k.
Thus Ew contains µk zw for some zw 6= 0 and all k ≥ 0, and hence
(5)
lim inf
k→∞
#(Ew ∩ Dr )
1
ρ
≥
=
.
log r
log |µ|
log 2
In order to produce examples of growth order arbitrarily close to
1/2, we need to modify the construction slightly. Instead of using explicit polynomials with Siegel fixed points, we use perturbations of the
Chebyshev polynomial T (z) = z 2 − 2 which have cycles of Siegel disks.
Existence of these polynomials is well-known, but for the convenience
of the reader we give a sketch of the proof.
The intermediate value theorem shows that there exists a decreasing
sequence of real numbers (an ) with an → −2 such that Qn (z) = z 2 + an
has a super-attracting periodic point, i.e., it satisfies Qqnn (0) = 0 for
some qn ≥ 1. Perturbing Qn and using the implicit function theorem,
we get a sequence of polynomials Rn (z) = z 2 + bn with −2 < bn < an
having a parabolic periodic point zn with multiplier −1, i.e., Rnqn (zn ) =
zn and (Rnqn )0 (zn ) = −1. (Essentially this is the well-known fact that
there are Feigenbaum bifurcations arbitrarily close to the Chebyshev
polynomial in the quadratic family.) By the same result of Siegel that
we used in the first part of the proof, there are numbers γ arbitrarily
close to 1/2 such that any analytic function F (z) = e2πiγ z + O(z 2 ) is
linearizable. Since the multiplier is a non-constant analytic function of
the parameter near bn , there exists cn ∈ C with |cn − bn | < n1 such that
Pn (z) = z 2 + cn has a periodic Siegel disk of period qn . In this way
we have constructed a sequence of polynomials Pn (z) = z 2 + cn with
periodic Siegel disks and cn → −2.
The repelling fixed point z = 2 of T (z) = z 2 − 2 varies analytically
with the parameter, so Pn has a repelling fixed point zn with zn → −2
and µn = Pn0 (zn ) → 4 for n → ∞. It follows from classical results
in complex dynamics that the Julia set of Pn is connected, and this
implies |µn | < 4 (see [1]). Let fn denote the Poincaré function of Pn at
the fixed point zn . Since Pn has connected Julia set, we get fn ∈ B with
log 2
→ 1/2. It remains to show that fn satisfies
growth order ρn = log
µn
(3), and this follows along the same lines as the first part.
Let n be fixed and, and let U1 , . . . Uq−1 be the cycle of Siegel disks
of Pn containing periodic points ζ1 , . . . , ζq−1 . Let hk : DRk → Uk be
the linearizing map
of P q in Uk , normalized as hk (z) = ζk + z + O(z 2 ).
Sq−1
Now we let W = k=1 hk (DRk /2 ). Then Pn (W ) = W , and W is a finite
ON THE EXCEPTIONAL SET IN A CONJECTURE OF LITTLEWOOD
5
union of sub-Siegel disks, so we also get (4) for all maps g which are
conformal in any Uj , where the constants do not depend on j. Applying
this to branches of fn−1 exactly as in the first part of the proof yields
the desired estimate (3).
References
[1] X. Buff, On the Bieberbach Conjecture and holomorphic dynamics, Proc. AMS
131 (3) (2002), 755–759.
[2] A. E. Eremenko, Lower estimate in Littlewood’s conjecture on the mean spherical derivative of a polynomial and iteration theory, Proc. AMS 112(3) (1991),
713–715.
[3] J. L. Lewis and J.-M. Wu, On conjectures of Arakelyan and Littlewood, J.
d’Analyse Math. 50 (1988), 259–283.
[4] J. E. Littlewood, On some conjectural inequalities, with applications to the
theory of integral functions, J. London Math. Soc. 27 (1952), 387–393.