The Hausdorff dimension of the set of dissipative points for a Cantor-like model set for singly cusped parabolic dynamics

0.18 MB·English

by  J. Schmeling

THE HAUSDORFF DIMENSION OF THE SET OF
DISSIPATIVE POINTS FOR A CANTOR-LIKE MODEL SET
FOR SINGLY CUSPED PARABOLIC DYNAMICS
JO¨RG SCHMELINGANDBERND O. STRATMANN
8
0
0
Abstract. In this paper we introduce and study a certain intricate
2
Cantor-likeset C contained in unit interval. Ourmain result is toshow

n thatthesetCitself,aswellasthesetofdissipativepointswithinC,both
a have Hausdorff dimension equal to 1. The proof uses the transience of
J
a certain non-symmetricCauchy-typerandom walk.

5
2
1. Introduction
]
S
InthispaperweestimatetheHausdorffdimensionofthesetC ofdissipative
D ∞
points within a certain Cantor-like subset C of the unit interval [0,1) ⊂ R.
.
h There are two ways to define these sets. The first is via the interation of a
t
a certain interval map Φ, where C represents the set of points with an infinite
m forward orbit, and C is equal to the basin of attraction of the set of critical
∞
[ points of Φ (see Remark 1.1 at the end of this introduction). The second

construction is purely in terms of fractal geometry. Let us remark that our
1
v motivation for considering the sets C and C∞ stems from the investigations
2 in [10] of the geometry of limit sets of Kleinian groups with singly cusped
6
parabolic dynamics. For the purposes of this paper this link is irrelevant
9
3 and therefore we omit the details. However, intuition coming from Kleinian
. groups has historically played a very important role in in the development
1
0 of Real and Complex Dynamics, and this paper can be seen as adding to
8 this tradition.
0
Let us begin with by giving the slightly intricate, but more down-to-earth
:
v fractal geometric construction of the sets C and C . For this we have to
∞
i
X define certain families of fundamental intervals by induction as follows. We
r start with the unit interval [0,1), and then partition the left half of [0,1)
a
into the infinitely many intervals
k k+1
3 3 3
I := 0, , and I := l−2, l−2 , for k ∈N.
1 π2 k+1 π2 π2
" !
(cid:20) (cid:19) l=1 l=1
X X
The family of these first-level intervals will be denoted by C . Note that
1
the right half 1,1 of the unit interval, which is clearly not captured by
2
C , should be interpreted as ‘the hole at the first level’. The second step
1
(cid:2) (cid:1)
is to partition each element I ∈ C as follows. By starting from the left
k1 1
2000 Mathematics Subject Classification. Primary: 60D05, 14L35, 51N30.
Keywordsandphrases. Fractalgeometry,Hausdorffdimension,Cauchyrandomwalks,
Kleinian groups.
1
2 TheHausdorffdimensionofthesetofdissipativepoints
endpointofI ,wepartitionthelefthalfofI intoinfinitelymanymutually
k1 k1
adjacent intervals
I ,··· ,I ,··· ,
k1k1+1 k1k1+l
where the diameters of these intervals are given by
3 |I |
|I |= k1 , for l ∈ N.
k1k1+l π2 l2
Similarly, by starting from the right endpoint of I we insert into the right
k1
half of I the (k +1) mutually adjacent intervals
k1 1
I ,I ,··· ,I ,
k1k1 k1k1−1 k10
with diameters given by
3 |Ik1| for l = 0
|Ik1k1−l|= ( π23(2|kI1k)12| for l ∈ {1,...,k }.
π2 l2 1
The family of these second-level intervals will be denoted by C . Note that
2
in this way we have perforated each I ∈ C such that there is a ‘hole’ in
k1 1
I with diameter of order |I |/k .
k1 k1 1
We then proceed by induction as follows. Suppose that for n ≥ 2 the n-th
level interval I has been constructed. The (n + 1)-th level intervals
k1...kn
arising from I are then obtained as follows. There are two cases to
k1...kn
consider. Thefirstcase is that k = 0, and herethe partition only continues
n
inthelefthalfofI . Moreprecisely, inthiscasewestartfromtheleft
k1...kn−10
endpoint of I and partition the left half of I into infinitely
k1...kn−10 k1...kn−10
many mutually adjacent intervals
I ,··· ,I ,··· ,
k1···kn−101 k1···kn−10l
with diameters given by
3 |I |
I = k1···kn , for l ∈N.
k1···kn−10l π2 l2
In the second case(cid:12)we have th(cid:12)at k ∈ N, and here we start from the left
(cid:12) (cid:12) n
endpoint of I and partition the left half of I into infinitely many
k1...kn k1...kn
mutually adjacent intervals
I ,··· ,I ,··· .
k1···knkn+1 k1···knkn+l
The diameters of these intervals are
3 |I |
|I | = k1···kn , for l ∈ N. (1)
k1···knkn+l π2 l2
Similarly, by starting from the right endpoint of I we insert into the
k1...kn
right half of I the (k +1) mutually adjacent intervals
k1...kn n
I ,I ,··· ,I ,
k1···knkn k1···knkn−1 k1···kn0
with diameters given by
3 |Ik1···kn| for l = 0
|I |= π2 (2kn)2 (2)
k1···knkn−l ( 3 |Ik1···kn| for l ∈ {1,...,k }.
π2 l2 n
J.SchmelingandB.O.Stratmann 3
The so obtained set of intervals of the (n+1)-th level will be denoted by
C . That is,
n+1
C = {I : k ∈ N,k ∈N for i ∈ N, and if k = 0 then k 6= 0}.
n k1...kn 1 i+1 0 i i+1
Again, note that by this we have perforated I such that in the first
k1...kn
case ‘the hole’ is precisely the right half of I , whereas in the second
k1...kn
case the diameter of the hole is of order |I |/k . Also, let us emphasize
k1...kn n
that by construction, the state 0 necessarily has to renew itself. That is,
the generation following the interval I is given by {I :
k1···kn−10 k1···kn−10kn+1
k ∈N}. Moreover, note that the system can only be stationary at states
n+1
k ∈ N, which means that if I is a given interval of some level n then
n k1...kn
I exists if and only if k 6= 0. Finally, note that we always assume
k1...knkn n
that the intervals I are half open, namely closed to the left and open
k1...kn
to the right.
With this inductive construction of the generating intervals I at hand,
k1...kn
the Cantor-like set C is defined by
C := I.
n\∈NI∈[Cn
Next, we define the set of dissipative points in C. For this we require the
following canonical coding of the elements in C. A finite or infinite sequence
(k ,k ,...) is called admissable if I ∈ C , for all n ∈ N. Clearly, the
1 2 k1···kn n
diameter of I tends to zero as n tends to ∞, for every fixed infinite
k1···kn
admissable sequence (kn)n∈N, and therefore,
∞
I is a singleton.
k1···kn
n=1
\
In particular, each x ∈ C is coded uniquely by an infinite admissable se-
quence, and this gives rise to the bijection
∞
ρ: Σ → C,(k ,k ,...) 7→ I ,
1 2 k1···kn
n=1
\
where Σ refers to the set of all admissable sequences. Using this coding, the
set C ⊂ C of dissipative points is then given by
∞
C := x∈ C : x= ρ(k ,k ,...) and lim k = ∞ .
∞ 1 2 n
n→∞
n o
Thefollowing theoremgives themainresultofthispaper. Here, dim refers
H
to the Hausdorff dimension.
Main Theorem. For C and C as defined above, we have
∞
dim (C )= dim (C) = 1.
H ∞ H
Remark 1.1. As already mentioned at the beginning of the introduction,
the sets C and C can alternatively be defined in terms of a certain interval
∞
map
Φ : I → [0,1).
k
Ik[∈C1
4 TheHausdorffdimensionofthesetofdissipativepoints
Namely, the map Φ is given piecewise by Φ| := φ , for each k ∈ N, where
Ik k
the maps φ : I → I are piecewise linear in the following sense.
k Ikl∈C2 kl k
For each k ∈ N and l ∈ N , the map φ | is a linear and
S 0 k Ikl
∞ I for l 6= 0
m=l−1 km
φ | (I ) =
k Ikl kl  S
I \I for l = 0.
 k k0
One then immediately verifies that the set C is equal to the set of points

which have an infinite forward orbit under Φ. Moreover, the centre c of I
k k
is the critical point of φ , for each k ∈ N, and thus Crit(Φ):= {c : k ∈ N}
k k
represents the countable set of critical points of Φ. With ω (x) referring
Φ
to the ω-limit set of an element x ∈ [0,1/2) with respect to Φ (that is,
ω (x) denotes the set of accumulation points of {Φn(x) :n ∈ N}), and with
Φ
B(Crit(Φ)) := {x : ω (x) ⊂ Crit(Φ)} denoting the basin of attraction of
Φ
Crit(Φ) under Φ, we then have
C = B(Crit(Φ)).
∞
Similar types of interval maps have been studied for instance in [4] and [11]
in connection with ‘wildCantor sets’ andthesearch for Juliasets of positive
Lebesgue measure. It seems worthwhile to point out that the ‘Martingale
Argument’ of Keller (see [4], Section 4.1), which gives a criterion for the
basin of attraction of a critical point to be of positive Lebesgue measure, is
notapplicabletothemapΦandhencedoesnotallowtodrawanyconclusion
for the Lebesgue measure of C . Nevertheless, recent studies in the theory
∞
of Kleinian groups (cf. [1] [3] [5], and also [6]) have confirmed the Ahlfors
Conjecture, and applying these to our situation here strongly suggests that
C and C are both of 1-dimensional Lebesgue measure equal to 0. However,
∞
currently it is still a conjecture that the Lebesgue measure of C and C is
∞
equal to zero, and it would be desirable to have an elementary proof of this
conjecture.
Acknowledgement: We like to thank the EPSRC for supporting a one
week visit of the first author to the School of Mathematics at the University
of St. Andrews. During this visit we began with the work towards this
paper. Also, we like to thank the Erwin Schrdinger Institute, Vienna for
supportingthe workshopErgodic Theory - LimitTheorems andDimensions
(Vienna,17-21December2007)duringwhichwemadethefinishingtouches
to this paper. Finally, we are grateful to Tatyana Turova for very helpful
discussions concerning the random walk argument which we employ here.
2. Proof of the Main Theorem
Since C ⊂ C ⊂[0,1), we have
∞
dim (C ) ≤ dim (C) ≤ 1.
H ∞ H
Therefore, our strategy will beto construct a family of probability measures
µ on C , for 1/2 < α < 1, such that the Hausdorff dimension dim (µ )
α ∞ H α
of the measure µ tends to 1 for α tending to 1. Clearly, this will then be
α
sufficient for the proof of the Main Theorem.
J.SchmelingandB.O.Stratmann 5
2.1. The family of measures µ .
α
Let 1/2 < α ≤ 1 be fixed. We then define a set function µ on the intervals
α
I byinductioninthefollowingway. DefineI := [0,1)andsetI := I
k1···kn 0 0k k
for all k ∈ N. Then let
µ (I ):= 1,
α 0
anddefineµ I foreachfiniteadmissablesequence(k ,··· ,k )
α k1···knkn+1 1 n+1
as follows. With ζ(s):= ∞ m−s referring to the Riemann zeta function,
(cid:0) (cid:1) m=1
we define for k 6= k ,
n n+1
P
µα(Ik1···kn) 1 + 1 fork 6= 0
µ I := 2ζ(2α) |kn+1−kn|2α (kn+1+kn)2α n+1
α k1···knkn+1  (cid:16) µα(Ik1···kn) 1(cid:17)fork = 0.
(cid:0) (cid:1)  2ζ(2α) kn2α n+1

Also, for k = k let
n n+1
µ (I ) 1
µ (I ) := α k1···kn .
α k1···knkn 2ζ(2α) (2k )2α
n
On the first sight, this definition of the set function µ might appear to be
α
slightlyartificial. However, inthenextsectionwewillseethatthisdefinition
reflects the transition probabilities of a certain (transient) random walk on
N , and therefore is rather canonical. Before we come to this, let us first
0
state the following consistency property for µ . This property can also be
α
deduced using the random walk of Section 2.2. Nevertheless, the following
gives an elementary proof of this consistency property.
Lemma 2.1. For each finite admissable sequence (k ,··· ,k ), we have
1 n
µ (I ) = µ I .
α k1···kn α k1···knkn+1
knX+1≥0 (cid:0) (cid:1)
Proof. For k =0, we have
n
∞
µ I 2
µ I = α k1···kn−10 = µ I .
α k1···kn−10kn+1 2ζ(2α) l2α α k1···kn−10
kknnX++11≥6=00 (cid:0) (cid:1) (cid:0) (cid:1) Xl=1 (cid:0) (cid:1)
6 TheHausdorffdimensionofthesetofdissipativepoints
If k 6= 0, then we compute
n
µ I
α k1···knkn+1
knX+1≥0 (cid:0) (cid:1)
1 1 1
= + µ (I )+
2ζ(2α) |k −k |2α (k +k )2α α k1···kn
knX+1>0 (cid:18) n+1 n n+1 n (cid:19)
kn+16=kn
1 1
+ µ (I )+ µ (I )
2ζ(2α)k2α α k1···kn 2ζ(2α)(2k )2α α k1···kn
n n
µ (I ) 1 1
= α k1···kn  + +
2ζ(2α) |k −k |2α (k +k )2α
n+1 n n+1 n
kknnX++116=>k0n (cid:16) 
1 1 
+ +
k2α (2k )2α
n n
(cid:17)
µ (I ) 1 1
= α k1···kn + +
2ζ(2α) (k −k )2α (k −k )2α
n+1 n n n+1
(cid:16)kn+X1>kn 0<kXn+1<kn
1 1 1 1
+ + + +
(k +k )2α (k +k )2α k2α (2k )2α
n+1 n n+1 n n n
0<kXn+1<kn kn+X1>kn (cid:17)
∞
µ (I ) 1
= α k1···kn ·2 = µ (I ).
2ζ(2α) l2α α k1···kn
l=1
X
(cid:3)
The following is an immediate consequence of the previous lemma.
Corollary 2.2. The measure µ is a probability measure on C.
α
2.2. The associated random walk.
In this section we show that the measure µ can be interpreted in terms
α
of a certain random walk. In particular, this will give that µ has the
α
Markov property. For this, let the random variables X˜α be defined by the
n
probability (with respect to µ ) being in the interval I given that
α k1···knkn+1
in the previous step the process has been in the interval I . That is,
k1···kn
the random variables X˜α is given as follows.
n
• For k > 0 such that k 6=k , let
n+1 n+1 n
µ I
P X˜α = I |X˜α = I := α k1···knkn+1
n+1 k1···knkn+1 n k1···kn µ (I )
(cid:0)α k1···kn (cid:1)
(cid:16) (cid:17)
1 1 1
= + .
2ζ(2α) |k −k |2α (k +k )2α
(cid:18) n+1 n n+1 n (cid:19)
• For k > 0 such that k =k , let
n+1 n+1 n
µ (I ) 1 1
P X˜α = I |X˜α = I := α k1···knkn = .
n+1 k1···knkn n k1···kn µ (I ) 2ζ(2α)(2k )2α
α k1···kn n
(cid:16) (cid:17)
• If k = 0, then
n+1
µ (I ) 1 1
P X˜α = I |X˜α = I := α k1···kn0 = .
n+1 k1···kn0 n k1···kn µ (I ) 2ζ(2α)k2α
α k1···kn n
(cid:16) (cid:17)
J.SchmelingandB.O.Stratmann 7
Clearly,theseconditionalprobabilitiesdonotdependonk ,··· ,k . Hence,
1 n−1
we can define an associated random walk Xα on N by the following tran-
n 0
sition probabilities.
• For l,m ∈ N , let
0
1 1 + 1 for l 6= 0,l 6= m
2ζ(2α) |m−l|2α (m+l)2α
P(Xα =l|Xα = m) :=  (cid:16) 2ζ(12α)(2m1)2α(cid:17) for l 6= 0,l = m
n+1 n  1 1 for l = 0,m 6= 0
 2ζ(2α)m2α
0 for l = m = 0.


The random walk Xα is very closely connected to our original geometric
n
setting, since it allows to recover the measure µ as follows.
α
P(Xα = k ,··· ,Xα = k )
1 1 n n
= P(Xα = k |Xα = k )·P(Xα = k |Xα = k )·
n n n−1 n−1 n−1 n−1 n−2 n−2
... ·P(Xα = k |Xα = 0)
1 1 0
µ (I ) µ I µ (I )
= α k1···kn α k1···kn−1 ··· α k1 = µ (I ).
µ I µ I µ ([0,1)) α k1···kn
α k1···kn−1 α(cid:0) k1···kn−2(cid:1) α
The aim now(cid:0)is to sho(cid:1)w th(cid:0)at the ra(cid:1)ndom walk Xα is transient. This will
n
then allow us to deduce that µ is non-trivial on C .
α ∞
Theorem 2.3. For each 1/2 < α < 1, the random walk Xα on N is
n 0
transient. That is, we have P-almost surely,
lim Xα = ∞.
n
n→∞
Proof. Let 1 < β < 2 be fixed, and consider the Cauchy-type random walk
Yβ on Z, given by the transition probabilities
n
1 1
P(Yβ = m+l|Yβ = m) := for n∈ N,m ∈ Z and l ∈ Z\{0}.
n+1 n 2ζ(β)|l|β
β β
Itiswell knownthatY is symmetric, andthatY is transientif andonly if
n n
β < 2. LetY˜β denotetherandomwalkwhicharisesfromYβ inthefollowing
n n
β
way. Let l ,··· ,l be the sequence of jumps of Y after the first n steps.
1 n n
With r referring to the first time at which Yβ crosses 0, we set Y˜β := |Yβ|.
1 k r1 r1
Subsequently, we apply the jumps |l | up to |l |, where r refers to the
r1+1 r2 2
β
next time Y crosses 0. After that, we apply the jumps |l | up to |l |,
n r2+1 r3
β
wherer denotes thenexttimeat whichthe processY firstcrosses 0again.
3 k
β
More precisely, with r referring to the i-th time the process Y crosses 0,
i k
we let Y˜β := |Yβ|, and between each two consecutive crossings r and r
ri ri i i+1
we define the jumps of Y˜β to be |l |,...,|l |. Note that the random
ri ri+1 ri+1
β
walk we have just described is equal to the random walk |Y |. Also, note
n
β
that since Y is a symmetric random walk, the above modification of the
n
sample path does not alter its probability. Therefore, it immediately follows
that the transience of Yβ implies that Y˜β is transient. Using the fact that
n n
by symmetry of Yβ we have P(Yβ = m) = P(Yβ = −m), we now compute
n n n
8 TheHausdorffdimensionofthesetofdissipativepoints
the transition probabilities of the random walk Y˜β on N as follows. For
n 0
l,m ∈ N such that l 6= 0, and m 6= l, we have
0
P(Y˜β = l|Y˜β = m)= P(|Yβ |= l||Yβ|= m)
n+1 n n+1 n
P(|Yβ | = l, Yβ = m or Yβ = −m)
= n+1 n n
P(Yβ = m or Yβ = −m)
n n
P(|Yβ | = l, Yβ = m) P(|Yβ | =l, Yβ = −m)
= n+1 n + n+1 n
2P(Yβ = m) 2P(Yβ = −m)
n n
= P(|Ynβ+1| = l, Ynβ = m) = 1 1 + 1 .
P(Yβ = m) 2ζ(β) |m−l|β (m+l)β
n (cid:18) (cid:19)
Similarly, we obtain for l = m 6= 0,
1 1
P(Y˜β = m|Y˜β = m)= ,
n+1 n 2ζ(β)(2m)β
and for l = 0 and m 6= l,
1 1
P(Y˜β =0|Y˜β = m)= .
n+1 n 2ζ(β)mβ
Finally, note that we immediately have
P(Y˜β = 0|Y˜β = 0) = 0.
n+1 n
This shows that the transition probabilities of Y˜β coincide with the ones
n
of Xβ/2. Therefore, since Y˜β is transient, it follows that Xβ/2 is transient.
n n n
This finishes the proof of the theorem. (cid:3)
As already announced before, Theorem 2.3 has the following important im-
plication.
Corollary 2.4. For every 1/2 < α < 1, we have
µ (C )= 1.
α ∞
Remark 2.5. Note that the proof of Theorem 2.3 relies heavily on the fact
that 1/2 < α < 1. Namely, for instance for α = 1 the associated random
walk is recurrent, and consequently the measure µ vanishes on C .
α ∞
2.3. Approximating the essential support of µ .
α
In order to prepare our estimate of the lower pointwise dimension of µ , we
α
need a further approximation of the essential support of this measure. We
will see that µ -almost surely the diameters of the coding intervals of an
α
element of C do not shrink too fast. For this we define, for γ ∈ R,
∞
|k −k |
Cγ := x∈ C : x = ρ(k ,k ,···) such that limsup n+1 n ≤ 1 .
∞ ∞ 1 2 nγ
n→∞
(cid:26) (cid:27)
Lemma 2.6. For each 1/2 <α < 1 and γ > 1/(2α−1), we have
µ (Cγ )= 1.
α ∞
J.SchmelingandB.O.Stratmann 9
Proof. The proof is an easy consequence of the Borel-Cantelli lemma. In-
deed, first note that for β =2α and k ∈ N we have
P(|Yβ −Yβ | ≥ k)≥ P(|Y˜β −Y˜β |≥ k).
n n−1 n n−1
The latter is an immediate consequence of the fact that the random walk
Yβ has the same distribution as Y˜β, but without reflections at 0. Hence, it
n n
β
suffices to prove the lemma for the symmetric random walk Y . For this we
n
define
pγ := P(|Yβ −Yβ |≥ nγ).
n n n−1
We then have, for each n ∈ N and with c(β) > 0 referring to some universal
constant,
∞
1 1
pγ = 2 ≤ c(β) .
n kβ nγ(β−1)
k=nγ
X
Since the series ∞ pγ converges for γ > 1/(β − 1), the Borel-Cantelli
n=1 n
Lemmaimplies thatP-almostsurelythereareatmostfinitely manynwhich
P
satisfy the inequality
|Yβ −Yβ | ≥ nγ.
n n−1
This shows that for µ -almost every x= ρ(k ,k ,...)∈ C we have
α 1 2
|k −k |
n+1 n
limsup ≤ 1.
nγ
n→∞
(cid:3)
2.4. The lower pointwise dimension on fundamental intervals.
The main result of this section will be the following estimate for the lower
pointwise dimension of the measure µ restricted to the fundamental inter-
α
vals I .
k1...kn
Proposition 2.7. For each ǫ > 0 there exists 1/2 < α < 1 and γ >
1/(2α−1) such that for every x= ρ(k ,k ,...)∈ Cγ ,
1 2 ∞
logµ (I )
liminf α k1···kn ≥ α−ǫ.
n→∞ log|Ik1···kn|
Furthermore, in here we have that α tends to 1 for ǫ tending to 0.
Proof. Since we are interested in the asymptotic behaviour of I for
k1...kn
γ
points x = ρ(k ,k ,...) ∈ C , Corollary 2.4 and Lemma 2.6 imply that we
1 2 ∞
can assume without loss of generality that k > 0 and |k −k |≤ nγ, for
n n+1 n
all n ∈ N. Furthermore, for ease of exposition we only consider sequences
which do not contain repetitions. That is, we assume that k 6= k , for
n n+1
all n ∈ N. The case with repetitions can be dealt with in similar way and is
10 TheHausdorffdimensionofthesetofdissipativepoints
left to the reader. Using the definition of µ , we then have
α
logµ (I )
α k1···knkn+1
log|I |
k1···knkn+1
log 1 + 1 µ (I )
= −log(2ζ(2α)) + |kn+1−kn|2α (kn+kn+1)2α α k1···kn
log|I | h(cid:16) log|I | (cid:17) i
k1···knkn+1 k1···knkn+1
log 2 µ (I )
≥ −log(2ζ(2α)) + |kn+1−kn|2α α k1···kn
log|I | h log|I | i
k1···knkn+1 k1···knkn+1
log 2·2αζ(2)α|Ik1···kn+1|αµ (I )
= −log(2ζ(2α)) + |Ik1···kn|2α α k1···kn
log|I | h log|I | i
k1···knkn+1 k1···knkn+1
= log 2αζ(ζ2(α2))α + log(|Ik1···kn+1|α) + log µ|αIk(I1k··1·k··n·k|nα)
log|I | log(I ) log|I
k1···knkn+1 k1···kn+1 k1···knkn+1|
log 2αζ(2)α log µα(Ik1···kn)
= α+ ζ(2α) + |Ik1···kn|α .
log|I | log|I |
k1···knkn+1 k1···knkn+1
Since
n
1
|I | < , (3)
k1···kn+1 2ζ(2)
(cid:18) (cid:19)
it follows for each κ > 0 and for all n sufficiently large,
2αζ(2)α
log
ζ(2α)
> −κ. (4)
log|I |
k1···knkn+1
Clearly, we even have that the limit of the latter expression is equal to 0.
This settles the second term in the final line in the above calculation. The
third term is more subtle, and for this we proceed as follows. Using (1)
and (2), we derive with the convention k ≡ 0,
0
n−1
1 1
|I |α = .
k1···kn 2αζ(2)α|k −k |2α
i=0(cid:18) i+1 i (cid:19)
Y
Similarly, using the recursive definition of µ , we obtain
α
n−1
1 1 1
µ (I ) = + .
α k1···kn 2ζ(2α) |k −k |2α (k +k )2α
i=0(cid:18) (cid:20) i+1 i i+1 i (cid:21)(cid:19)
Y