DOI: 10.1017/S0143385702000810 Printed in the United Kingdom
A zero–infinity law for well-approximable
points in Julia sets
RICHARD HILL† and SANJU L. VELANI‡§
† Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
(e-mail: [email protected])
‡ Department of Mathematics, Queen Mary, University of London, Mile End Road, London E1 4NS, UK
(e-mail: [email protected])
(Received 20 May 2001 and accepted in revised form 20 December 2001)
In memory of Noel Baker
Abstract. LetT : J → J be an expanding rational map of the Riemann sphere acting on its Julia set J and f : J → R denote a H¨older continuous function satisfying
f (x) > log|T(x)|for allx inJ. Then for any pointz0inJ define the setDz0(f )of
‘well-approximable’ points to be the set of points inJwhich lie in the Euclidean ball
B
y,exp
−
n−1
i=0 f (Tiy)
for infinitely many pairs(y, n)satisfyingTn(y)=z0. In our 1997 paper, we calculated
the Hausdorff dimension ofDz0(f ). In the present paper, we shall show that the Hausdorff
measureHs of this set is either zero or infinite. This is in line with the general philosophy that all ‘naturally’ occurring sets of well-approximable points should have zero or infinite Hausdorff measure.
1. Introduction
In [4], we formulated the following general problem. Consider a metric spaceJ equipped with a Borel probability measurem. IfT : J → J is measure preserving and ergodic, we know by the ergodic theorem that for any ballBof positivem-measure, the subset
{z∈J :Tn(z)∈Bfor infinitely manyn∈N}
ofJ has fullm-measure. This means that the trajectories ofm-almost all points will go through the ballBinfinitely often. A natural question to ask is what happens if the ballB
shrinks with time. More precisely, if at timenwe have a ballB(z0,rad(n))centred at a pointz0∈J of radius rad(n)(rad(n)→0 asn→ ∞), then what kind of properties does
the setWof pointszhave, whose imagesTn(z)are inB(z0,rad(n))for infinitely manyn? These points can be thought of as trajectories which hit a shrinking target infinitely often and are called ‘well approximable’ with respect to the function ‘rad’, in analogy to those in the classical theory of Diophantine approximation.
In [4], we considered a special case of the above general ‘shrinking target’ problem in whichT is an expanding rational map of the Riemann sphereC=C∪ {∞}andJ =J (T )
is its Julia set. By the definition of expanding, there exists a constantλ >1 and an integer
p≥1 such that
|(Tp)(z)| ≥λ for allz∈J,
whereTis the derivative ofT. For such maps, it is known (see [8]) thatJ is not the whole ofCand we may and will assume that∞ ∈J. Thus we can think ofJ as a metric space with the usual metric onC. Specifically, for anyτ >0 andz0∈Jwe considered the sets
Wz•
0(τ ):= {z∈J :T
n(z)∈B(z0,|(Tn)(z)|−τ)
for infinitely manyn∈N}
and
Wz0(τ ):= {zinJ :T
n(z)∈B(z
0, e−nτ)for infinitely manyn∈N},
which we referred to as the ‘local’ and ‘global’ well-approximable sets, respectively. Here the backward orbit of a selected pointz0 inJ corresponds to the rationals in the classical setW (τ ) := {x ∈ R : |x−p/q| ≤ q−τ for infinitely many rationalsp/q}of well approximable numbers. For Julia sets associated with rational maps, we proved [4] the following analogue of the Jarn´ık–Besicovitch Theorem in the classical theory of metric Diophantine approximation.
THEOREM1. For the local set one has dimWz•0(τ )=δ/(1+τ ), whereδis the Hausdorff dimension ofJ.
In [4], we also obtained a partial result on the dimension of the global set Wz0(τ ).
However, the breakthrough in calculating the dimension of this set came by considering the following generalization. Letf : J → R≥0 denote a H¨older continuous function satisfying
f (z) >log|T(z)| for allz∈J,
and writefn(x)for thenth ergodic sum, that is
fn(x):= n−1
i=0
f (Tix).
Then define the setDz0(f )of well-approximable points to be the set of points inJwhich
lie in the ball
B(y,exp(−fn(y)))
THEOREM2. The setDz0(f )has Hausdorff dimensionα, whereαis the unique positive
number satisfying the pressure equation
P (T ,−α·f )=0.
HereP (–,–)denotes the topological pressure.
It is easy to verify (see §1.3 in [5]) that the ‘local’ set corresponds to Dz0(f )with
f (z) = (1+τ )log|T(z)| whilst the ‘global’ set corresponds toDz0(f )withf (z) =
log|T(z)| +τ. This solves the problem of calculating dimWz0(τ ). The theorem may
be viewed as an extension of the Bowen–Manning–McCluskey formula, which states that
P (T ,−δlog|T|) = 0. In [5], we also demonstrated an unexpected link between the dimension results forDz0(f )and the dimension of exceptional sets arising from points
with ‘badly behaved’ ergodic averages. In short, given an ergodic measurem on J, these are pointsz inJ at which the ergodic average limn→∞n−1fn(z) of f does not
tend to the expected limit Jf dm. In turn, Falconer [3] has recently shown a rather elegant connection between the dimension results forDz0(f )and the multifractal spectrum
associated with the dynamical systemT :J →J. Also, the dimension result for the local set has recently been extended to parabolic rational maps [7]. This concludes our brief overview of recent developments and various connections. Returning to the main theme, it follows from the definition ofs-dimensional Hausdorff measureHsthat
Hs(D
z0(f ))=
0 ifs > α ∞ ifs < α.
However, ifs=αthenHs(D
z0(f ))may be zero or infinite, or may satisfy
0<Hs(Dz0(f )) <∞.
In this paper, we shall prove that the latter is impossible.
THEOREM3. Let α be Hausdorff dimension of Dz0(f ). Then the α-dimensional
Hausdorff measure ofDz0(f )is either zero or infinity.
Note that if one setsf = log|T|, the set Dz0(f )has fullδ-dimensional Hausdorff
measure inJ, whereδ denotes the Hausdorff dimension ofJ. However, for expanding rational maps,J has finite positiveδ-dimensional Hausdorff measure and so clearly the theorem does not extend to this case. In fact, the conditionf >log|T|everywhere onJ
guarantees thatHδ(D
z0(f ))=0—see the Appendix.
In the language of geometric measure theory, Theorem 3 simply states that the sets
Dz0(f )are nots-sets. This is a known fact for the analogous, classical sets of
2. Conformal measures
We shall writeHs for thes-dimensional Hausdorff measure. Note that thes-dimensional Hausdorff measure iss-conformal; that is, ifT is injective on some setX⊂J then
Hs(T X)=
X
|T(x)|sdHs(x).
Recall that a functionf : J → Ris said to be H¨older continuous if and only if there is a constantC satisfying the following condition. For any ballB inJ and any natural numbernsuch thatTnis injective onB, one has for allx, yinB∩J
|fn(x)−fn(y)| ≤C.
The constantCwill be referred to as a distortion constant for the functionf. We need the following powerful result from complex analysis (see [6]).
K ¨OBE DISTORTION THEOREM. Let ⊂ C be a topological disc with boundary containing at least two points and letV ⊂ be compact. Then there exists a constant
C( , V )such that for any univalent holomorphic function U : → Cthe following inequality is satisfied,
sup
x,y∈V
|U(x)|
|U(y)| ≤C( , V ).
This implies that the function log|T|onJis H¨older continuous.
A more general class of conformal measures was introduced by Denker and Urba´nski in [1] and numerous other papers. Iff :J →Ris H¨older continuous, then a measureνonJ
is said to bef-conformal if and only if the following holds. For all measurable subsetsX
ofJon whichT is injective:
ν(T X)=
X
exp(f (x)) dν(x). (1)
In this notation,Hs isslog|T|-conformal. Denker and Urba´nski have proved (see [1]) the following theorem.
THEOREM4. Let T,f andαbe as in Theorem 2. Then there is a unique non-atomic
α·f-conformal probability measure onJ.
From now on we shall refer to the uniqueα·f-conformal probability measure asν. We recall the following fact concerning conformal measures.
LEMMA1. LetTnbe injective on a ballBinJ. Then one has
ν(TnB)exp(αfn(B))ν(B)
wherefn(B)is the value offnat any point ofBand the implied constants are independent
ofBandn. Similarly one has for any measurable subsetA⊂B
Hα(TnA) |(Tn)(B)|αHα(A).
Next, we state a useful formula for theν measure of an arbitrary ball. For any ballB
inJ, we shall writen0(B)for the largest natural numbernfor whichTnis injective onB.
Using the fact thatT is expanding one may show (see, for example, Lemma 4 of [5]) that there is anN ∈Nsuch that for any ballBinJ,
J ⊂Tn0(B)+NB.
This fact together with the previous lemma gives the following lemma.
LEMMA2. For any ballBone hasν(B)exp(−αfn0(B)(B)).
3. Proof of Theorem 3
Define the H¨older continuous functiong:J →R>0by
g(x):=f (x)−log|T(x)|.
For means of calculation we shall introduce the following sets forC >0:
E(C):= {x∈J :Tn(x)∈B(z0, Cexp(−gn(x)))for infinitely manyn∈N}
wheregn is thenth ergodic sum ofg. It follows from the K¨obe Distortion Theorem that
there is a constantC >1 such that
E(C−1)⊂Dz0(f )⊂E(C).
Therefore, in order to prove Theorem 3 it is sufficient to show that eitherE(C)has zero Hausdorff measure for allC, or thatE(C)has infinite Hausdorff measure for allC.
LEMMA3. Forx ∈J, one hasx ∈E(C)if and only ifT (x)∈E(Ce−g(x)).
Proof. Letx∈E(C). This means thatTn(x)∈B(z0, Cexp(−gn(x)))for infinitely many
natural numbersn. This is equivalent toTn−1(T x) ∈B(z0, Cexp(−gn−1(T x)−g(x)))
for infinitely many natural numbersn. Replacing n−1 bynin this we obtainT (x) ∈
E(Cexp(−g(x))). ✷
LEMMA4. Forx ∈J one hasx∈E(C)if and only ifTn(x)∈E(exp(−gn(x))C).
Proof. This follows from Lemma 3 by induction onn. ✷
LEMMA5. For any a > 0 we have Hα(E(aC)) Hα(E(C)), where the implied constants depend ona, but not onC.
Proof. Suppose a < 1. Then clearly Hα(E(aC)) ≤ Hα(E(C)). On the other hand, by the conformality of Hausdorff measure (see Lemma 1) and the fact that T
is expanding, we have that Hα(Tm(E(C))) Hα(E(C))—here the implied constant depends on m. As g(z) > 0 everywhere on the compact set J, there is an ) > 0 such that g(z) > ). Thus gm(z) > m), which together with Lemma 4 implies that
Tm(E(C))⊂E(Cexp(−m))). Hence
Hα(E(C))Hα(E(C
exp(−m)))),
and choosingmsuch that exp(−m))≤ a ≤ exp(−(m−1)))completes the proof when
a < 1. For the case whena > 1, start by consideringHα(Tm(E(aC)))and repeat the
LEMMA6. For any ballB=B(z, r)centred onJ we have Hα(B∩E(C)) |(Tn0(B))(z)|−αHα(E(exp(−g
n0(B)(z))C)).
The implied constants are independent ofBandC. Proof. Letn0=n0(B(z, r)). By Lemma 1 we have
Hα(B∩E(C)) |(Tn0)(z)|−αHα(Tn0(B∩E(C))).
By Lemma 4 we have
Tn0(B∩E(C))⊂E(exp(−g
n0(z)+a)C),
whereais a distortion constant forg. Therefore,
Hα(B∩E(C)) |(Tn0)(z)|−αHα(E(exp(−g
n0(z)+a)C)).
By Lemma 5 this implies
Hα(B∩E(C)) |(Tn0)(z)|−αHα(E(exp(−g
n0(z))C)).
On the other hand, there is a constantNsuch thatTn0+N(B)⊃J. This implies
Hα(B∩E(C)) |(Tn0+N)(z)|−αHα(Tn0+N(B∩E(C)))
|(Tn0)(z)|−αHα(E(exp(−g
n0+N(z)−a)C)),
and again by Lemma 5 we have
Hα(B∩E(C)) |(Tn0)(z)|−αHα(E(exp(−g
n0(z))C)). ✷
To prove the next lemma, we will use a carefully chosen cover of the Julia setJ which we now describe. LetI be the set of pairs (y, n) ∈ J ×N such that Tn(y) = z0. Suppose we havec1, C > 0 and setI (C, c1) = {(y, n) ∈ I : |C −loggn(y)| < c1}.
It is known that ifc1is sufficiently large (depending only onJandf) then for anyC >0 the following is a cover ofJ:
C(C)= {B(y, c1|(Tn)(y)|−1):(y, n)∈I (C, c1)}.
We shall fixc1sufficiently large so thatC(C)is a cover. There is a constantNdepending
only onJ,f andc1such that no point ofJis in more thanNof the balls in the coverC(C). These statements are contained in Lemma 8 of [5]. The upshot of this is that for any measureµonJand any measurable subsetA⊆J, we have
µ(A)
(y,n)∈I (C,c1)
µ(A∩B(y, c1|(Tn)(y)|−1)). (2)
The implied constants are independent ofµ andC (in fact, the implied constants are 1 andN).
Proof. By Lemma 6 it is sufficient to prove the lemma in the caseB=J. By (2) we have
Hα(E(C))
(y,n)∈I (C,c1)
Hα(B(y, c1|(Tn)(y)|−1)∩E(C)).
For the moment we shall concentrate on one of the ballsB =B(y, c1|(Tn)(y)|−1)in the coverC(C). Note thatn0(B)=n+O(1)(see Lemma 5 of [5]). Therefore, by Lemma 6 we have
Hα(B∩E(C)) |(Tn)(y)|−αHα(E(Ce−gn(y))).
Now by Lemma 5 as exp(gn(y))C, we have
Hα(E(C)∩B) |(Tn)(y)|−αHα(E(
1)).
Again since exp(gn(y))C, by Lemma 2 we have
|(Tn)(y)|−αCαexp(−αfn(y))Cαν(B).
Summing this over the ballsBin our coverC(C)we obtain
Hα(E(C))CαHα(E(
1))
(y,n)∈I (C,c1)
ν(B(y, c1|(Tn)(y)|−1)).
Now, using (2) in the opposite direction we have
Hα(E(C))CαHα(E(
1))ν(J )=CαHα(E(1)).
This proves the lemma. ✷
For a ballBinJwe shall use the notation
m(B):=Hα(B∩E(1)).
LEMMA8. For any ballBinJwe havem(B)Hα(E(1))ν(B).
Proof. GivenB =B(z, r), letn0=n0(B(z, r)). By Lemma 6 withC=1 and Lemma 7 withB=J, we have
m(B) |(Tn0)(z)|−αHα(E(exp(−g
n0(z))))
exp(−αfn0(z))H
α(E(1)).
The lemma now follows on applying Lemma 2. ✷
During the proof of the next result we will require the following fact (see Corollary 1
of [5]):
y:Tn(y)=z 0
exp(−αfn(y))1.
PROPOSITION1. ν(E(1))=0.
Proof. ForC sufficiently large the setE(1)is contained in the limsup of the setsA(n)
defined by
A(n):=
y:Tn(y)=z 0
Therefore, in view of the Borel–Cantelli Lemma it is sufficient to show thatnν(A(n))
converges. By Lemma 1 we have
ν(A(n))
y:Tn(y)=z 0
exp(−αfn(y))ν(B(z0, C2exp(−gn(y)))).
Asg(z) > 0 everywhere on the compact setJ, there is an) > 0 such thatg(z) > ). Thusgn(z) > n). This together with the above fact implies that
ν(A(n))ν(B(z0, C2exp(−n))))
y:Tn(y)=z 0
exp(−αfn(y))ν(B(z0, C2exp(−n)))).
Next, note that there is aρ >0 such thatT is injective on any ballBcentred onJof radius
r < ρ. Therefore,
n0(B(z, r))≥ log(ρ/r)
logT −log(r),
whereT is the supremum norm of T on theδ-neighbourhood of J. Therefore, by Lemma 2 and the fact thatT is expanding, we have
logν(B(z, r)) −n0(B(z, r))log(r).
We now have
logν(A(n))log(C2exp(−n))) −n.
By the ratio test it follows thatν(A(n))converges. ✷
Proof of Theorem 3. Choose) >0. In view of the above proposition, there is a cover{Bi}
ofE(1)such that
i
ν(Bi) < ).
This implies by Lemma 8 that
i
m(Bi))Hα(E(1)).
Now{Bi}is a cover ofE(1), so we also have that
Hα(E(
1))≤
i
m(Bi).
Therefore,
Hα(E(
1)))Hα(E(1)).
Since) >0 is arbitrary, this implies thatHα(E(1))is either zero or infinite. By Lemma 5 we have Hα(E(C)) = Hα(E(1)) for all C > 0 which completes the proof of the
theorem. ✷
A. Appendix
LEMMAA.1.
(i) Iff =log|T|, then
0<Hδ(Dz0(f )) <∞.
(ii) Iff >log|T|everywhere onJ, then Hδ(D
z0(f ))=0.
The proof of the lemma makes use of certain well-known facts which we now summarize. For an expanding rational mapT, theδ-conformal measureν supported on
J is equivalent to δ-dimensional Hausdorff measure Hδ and soHδ(J ) is positive and finite. Furthermore,νhas a unique (and hence ergodic) equivalentT-invariant probability measure,µ which is the unique equilibrium state forT and−δlog|T|. Thus, for any measurable subsetXofJ we have that
µ(X)Hδ(X). (A.1)
Also
µ(B(x, r))rδ (A.2)
for any ballBwith centrexinJ and radiusr < r0. For further details see [4, 8].
Proof. We work with theT-invariant probability measureµ. Recall, that there is a constant
C >1 such that
E(C−1)⊂Dz0(f )⊂E(C).
Iff =log|T|, thenE(C−1)consists of pointsx inJ whose forward orbitTn(x)lands in the fixed ballB(z0, C−1)infinitely often. Either by Poincar´e recurrence or the ergodic theorem,µ(E(C−1)) = 1. Thusµ(Dz0(f )) = 1, which together with (A.1) proves the
first part of the lemma.
If f > log|T| everywhere on J, then as already mentioned above gn(x) > n)
everywhere onJ. Thus,E(C)⊂E∗(C):=lim supn→∞E∗n(C)where
En∗(C):= {x ∈J :Tn(x)∈B(z0, Cexp(−n)))}.
By (A.2) and the fact thatµisT–invariant, we have
µ(En∗(C))=µ(B(z0, Cexp(−n))))(Cexp(−n)))δ.
Hence
∞
n=1
µ(En∗(C))
∞
n=1
exp(−n)δ) <∞,
and so, by the Borel–Cantelli Lemmaµ(E∗(C))=0. Thusµ(E(C))=0, which together
with (A.1) completes the proof of the lemma. ✷
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