TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 358, Number 1, Pages 285–314
S 0002-9947(05)03646-9
Article electronically published on February 18, 2005

RANDOM FRACTAL STRINGS: THEIR ZETA FUNCTIONS,COMPLEX DIMENSIONS AND SPECTRAL ASYMPTOTICS
B. M. HAMBLY AND MICHEL L. LAPIDUS

Abstract. In this paper a string is a sequence of positive non-increasing real
numbers which sums to one. For our purposes a fractal string is a string
formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensionsof the string, the poles of an associated zeta function, it is possible to obtain
detailed information about the behaviour of the asymptotic properties of thestring. We consider random versions of fractal strings. We show that by usinga random recursive self-similar construction, it is possible to obtain similar
results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex
dimensions can only lie on the real line. The results allow us to discuss thegeometric and spectral asymptotics of one-dimensional domains with random
fractal boundary.

1. Introduction
In [14] there is an extensive analysis of fractal strings. These are sequences
extracted from the construction of fractal subsets of R and are used as models forother inﬁnite series such as the primes. The focus of [14], Chapter 2, was on some
simple strings arising from iterated function systems on the real line and, in thatsetting, by considering an associated zeta function, it is possible to give explicit
formulas for various quantities associated with the resulting self-similar string. The
explicit formulas are determined by the complex dimensions of the string, deﬁned
as the poles of the zeta function.
In this work we will see how some aspects of this theory carry over to the settingof random fractal strings. There are many natural random fractals which generate
strings, and we will discuss three. First, a natural generalization of the iteratedfunction system is the random recursive fractal and it generates a corresponding
random string. Second, the zero set of Brownian motion provides a fractal set of
dimension 1/2 and it too can be used to generate a string. We extend this to strings
arising from stable subordinators. We also consider some strings that have larger
Received by the editors October 21, 2003 and, in revised form, February 17, 2004.
2000 Mathematics Subject Classiﬁcation. Primary 28A80, 60D05; Secondary 11M41, 58J50,
60J80.
Key words and phrases. Random recursive fractal, general branching process, fractal string,zeta function, complex dimensions, stable subordinator, tubular neighbourhoods, spectral
asymptotics.
The second author was supported in part by the U.S. National Science Foundation under grants
DMS-9623002, DMS-0070497.
c2005
American Mathematical Society

285

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B. M. HAMBLY AND MICHEL L. LAPIDUS

random ﬂuctuations, which shows that there is little that can be said in such cases
using the techniques developed here.
One reason for the development of the study of fractal strings was that they provide a suﬃciently simple set for the study of the second order term in the asymptotics of the eigenvalue counting function of the Laplace operator on a boundeddomain. This problem, of hearing the shape of a fractal drum, was considered ina number of papers. See for example [2], [4], [11], [5], [12], [13]. In the case of
two-dimensional domains with an irregular boundary there is a range of possible
behaviour for the second order term in the asymptotics of the eigenvalue counting
function. The problem is made simpler if we reduce to one dimension and consider
a fractal string as a set with fractal boundary. The explicit formulas that follow
from our results on the complex dimensions allow us to obtain precise results on
the eigenvalue asymptotics for our random fractal strings.
For example, if we consider the fractal string generated as the complement of
the zero set of Brownian motion, then the fractal boundary problem associated tothe string has the following property. If we let N (λ) denote the eigenvalue counting
function (the number of eigenvalues of the Dirichlet Laplace operator on the string
whose value is less than λ), then we have the following asymptotic formula; for any
 > 0, almost surely we haveN (λ) = λ + Lζr (1/2)λ1/2 + o(λ1/4+ ),where L is the local time at 0 of Brownian motion run for time 1 and ζr is the
Riemann zeta function. The error term could be reduced if we could establish
meromorphic continuation of the associated zeta function to the left of the region{z ∈ C : Re(z) > 1/4}.
An outline of the paper is as follows. We begin by stating some of the results
concerning the behaviour of self-similar strings. This is followed by a discussionof the general branching process, a random process which describes the underlying
tree structure in a random recursive fractal. We will establish a rate of convergence
theorem for such processes; this is required to prove meromorphic continuationof the zeta function of our random recursive strings. In Section 4 we discuss therandom recursive strings. In particular, we show that the poles (of the meromorphic
continuation) of their zeta functions are almost surely the complex solutions of anatural ‘Moran–type’ expectation equation. In Section 5 we consider the stablestrings and show that their zeta functions can be meromorphically continued andwill not have any complex dimensions with non-zero imaginary part. We conclude
the discussion of meromorphic continuation by exhibiting in Section 6 a particular
type of scale irregular random Cantor set which has large ﬂuctuations, and ourtechniques do not allow us to prove meromorphic continuation. Finally in Section 7
we discuss some properties of the string that can be deduced from the complex
dimensions. These include the asymptotic behaviour of tubular neighbourhoods ofthe boundary and the asymptotics of the eigenvalue counting function.
2. Self-similar fractal strings and their zeta functions
Fractal strings were introduced in [12], and in [14] a theory of complex dimensionsof fractal strings was developed and its connections with number theory explored.
We introduce the terminology and several results. Our aim will be to consider

RANDOM FRACTAL STRINGS

287

the zeta functions of our strings to determine their complex dimensions and the
behaviour of the zeta function at its poles.
In short, a fractal string is a bounded open subset of the real line, U ⊂ R, withboundary F = ∂U , a totally disconnected (compact) subset of R with measure 0.
Frequently, a fractal string U is identiﬁed with the sequence L = {Li }∞i=1 of lengthsof the connected components (largest open sub-intervals) of U , written in nonincreasing order according to multiplicity. To avoid trivial situations it is always
it canassumed that the sequence is inﬁnite and Li → 0. Without loss of generality
L=
1.
be assumed that U ⊂ [0, 1] and that L has total length one: |U | = ∞i
i=1
We now focus on self-similar strings; we ﬁrst recall their construction before
discussing some of the results of [14] concerning their complex dimensions. We
start with the interval [0,1]. A self-similar set can be constructed from a family ofsimilitudes {ψi : i = 1, . . . , N }, where ψi : [0, 1] → [0, 1], and we write ri (0 < ri <1) for the Lipschitz constant of ψi . We assume that our maps satisfy the open set
condition (OSC) in that there is an open set O ⊂ [0, 1] such that ψi (O) ∩ ψj (O) = ∅N
and O ⊃ i=1 ψi (O). The self-similar set F is the unique ﬁxed point in the set ofsubsets of [0,1] such thatNψi (F ).
F =i=1

It is also obtained as the limit of repeated application of the maps. If we let I1 ={1, . . . , N } and set In = I1n for the sequences of length n we can use this to index
the subsets of the fractal. That is, if i = (i1 , . . . , in ) we write ψi = ψi1 ◦ψi2 ◦· · ·◦ψin
and
∞ 

ψi ([0, 1]).
F =n=1 i∈In

Note that the subsets of the fractal are indexed by an N -ary tree.
The set F is a subset of [0,1], and we note that there are subintervals Gj ⊂[0, 1], j = 1, . . . , M , such that Gj ∩ Gj  = ∅ for j = j  ∈ {1, . . . , M } and[0, 1] =

M


Gj ∪

j=1

N


ψi ([0, 1]).

i=1The maps which form these intervals are written ϕi , so that ϕj ([0, 1]) = Gj . The
lengths of the removed intervals Gj are denoted by lj . It is these removed intervals
that we use to form our string. From our recursive construction we see that at each
iteration of the maps a new collection of subsets Gij = ψi ◦ ϕj ([0, 1]) is formed withnlength L̃ij = k=1 rik lj . We now place the collection of all lengths in non-increasingorder to form our string,L = (L1 , L2 , . . . ), Li ≥ Li+1 ,∞Li = 1.

i=1

A simple example is the middle third Cantor string in which M = 1, N = 2, l =1/3, r1 = r2 = 1/3.
The geometric zeta function of the string is given byζ(s) =∞
i=1

Lsi , s ∈ C.

288

B. M. HAMBLY AND MICHEL L. LAPIDUS

The complex dimensions of the string are deﬁned to be the poles of the meromorphic continuation of the geometric zeta function. A result in [14] is to provethat the geometric zeta functions can be analytically continued to the left of therightmost pole, which occurs at the dimension of the string and to calculate theresidues of the complex dimensions of the string. These give rise to so-called explicit formulas giving more detailed information about this object. In particular,
these explicit formulas are expressed in terms of the (visible) complex dimensions
and provide a precise description of the oscillatory behaviour in the underlying
geometry, dynamics or spectrum.
We will ﬁnd it convenient to think (as in [14]) of the string as a measure. The
sequence (Li )∞i=1 of ordered lengths deﬁnes a measure on [1, ∞) asη(dx) =∞
δL−1 (dx),ii=1where δx denotes the Dirac measure concentrated at x. We will write the counting
function as x∞
η(dy) =
I{L−1 ≤x} ,(2.1)
η(x) =i0

i=1

where IA denotes the indicator function of the set A. Using the expression for thestring as a measure we can write the zeta function as a Mellin transform of themeasure∞ζ(s) =x−s η(dx).

0

We conclude with a summary of results.
Theorem 2.1. For a self-similar string, the zeta function is given byM s
j=1 ljζ(s) =
N s .
1 − i=1 ri
The set of complex dimensions is contained in
C = {s ∈ C :N
ris = 1}.

i=1The zeta function can be meromorphically continued to the left of the rightmost pole(in fact to all of C).
Remark 2.2. (1) The only complex dimension located on the real axis is equal todf , the fractal dimension (Minkowski, upper box dimension) of F . It is equal to thesimilarity dimension of F , that is, the unique real solution to the ‘Moran equation’N s
i=1 ri = 1, and also coincides with the abscissa of convergence of the Dirichlet
series deﬁning ζ(s). Moreover, df ∈ (0, 1) and is a simple pole of ζ(s).
(2) In general the complex dimensions are contained in C, as there may becancellations between the zeros of the numerator and those of the denominator of
ζ(s), as shown in [15]. Of course, in the case of a single gap, M = 1, then the setof complex dimensions precisely coincides with C. (An analogous statement holds
for the lattice case of Theorem 2.3 below.)RANDOM FRACTAL STRINGS

289

There is a dichotomy in the structure of the complex dimensions. A self-similar
string is non-lattice if the numbers log ri , i = 1, . . . , N , are rationally independent,
and lattice otherwise. In the lattice case, there exists r ∈ (0, 1), called the ‘multiplicative generator’ of the string, and positive integers without common divisors,
ki , i = 1, . . . , N , such that ri = r ki . We will write T = − log r. In the lattice casethe string has a line of complex dimensions at each zero of the denominator of thezeta function. In the non-lattice case the real parts of the complex dimensions aredense to the left of the rightmost pole of the zeta function. The non-lattice casecan be approximated by a sequence of lattice strings whose periods diverge.
Theorem 2.3 ([14], Theorem 2.13). In the lattice case ζ(s) is a rational function ofr s = e−T s . There are ﬁnitely many poles ω0 := df , ω1 , . . . , ωq of the zeta function
N ki(obtained as roots of the polynomial equation= 1, with z = r s ) and
i=1 z
there is an oscillatory period p = 2π/T , such that the set of complex dimensions iscontained in√
C = {ωu + −1 np : n ∈ Z, u = 0, . . . , q}.
In the non-lattice case, df is the unique pole of ζ(s) on the line Re(s) = df . The
complex dimensions of the string can be approximated by the complex dimensions ofa sequence of lattice strings with larger and larger oscillatory period. In particular,
there are countably many complex dimensions (a subsequence of which tends to,but does not touch, the line Re(s) = df ) and they are all located in a horizontallybounded strip.
One sees that in the lattice case, the complex dimensions are periodically distributed on ﬁnitely many lines, whereas in the non-lattice case, they exhibit a‘quasi-periodic behaviour’; the latter is studied in more detail in [14] (see also [15]).
In the ﬁnal section we will discuss the application of these results to the geometry
and spectral properties of the string. By essentially inverting the zeta function we
can recover asymptotic properties of various counting functions associated with thestring.
3. The general branching process
We will see that random fractal strings display a wider variety of behaviour
than their deterministic counterpart. In order to obtain theorems on the complex
dimensions analogous to the deterministic case we will need some results from thetheory of general or Crump-Mode-Jagers (CMJ) branching processes.
The self-similar set F can be described by the sequence space I = ∞n=0 In ,which we can think of as a tree. In a similar way (to be described below) we thinkof random recursive fractals as indexed by random trees; see [9]. The sample pathsof classical Galton–Watson branching processes therefore provide a description of arandom recursive fractal. For a tree which contains more structure we can consider
the sample paths
n of the general branching process or branching random walk.
Let In = k=0 Nk and I = k Ik , where I0 is the empty set. The concatenationof sequences i and j ∈ I will be written ij. For an inﬁnite sequence i ∈ I\In , denoteby i|n the sequence truncated to length n and by i[n] the n-th element of i. We
write j ≤ i if i = jk for some k, and denote by |i| the length of the sequence i.
The general branching process will have a sample space deﬁned by a random
subset T of I. An element i ∈ Nn represents an n-th generation individual in the290

B. M. HAMBLY AND MICHEL L. LAPIDUS

branching process. If i has j children, then these are i1, i2, . . . , ij. A tree T is asubset of the space I such that:i) ∅ ∈ T is the root of the tree;ii) i ∈ T implies i|k ∈ T for all k < |i|;iii) if ij ∈ T for some j ∈ N, then i1, i2, . . . , i(j − 1) ∈ T .
Level n of the tree will be denoted Tn = T ∩ Nn . We have a one-to-one correspondence between trees and feasible realisations of the set of individuals in a branchingprocess.
For the individual indexed by i we have a description Ui = (ξi , λi , φi ). The Ui
are independent and identically distributed and consist of three components. The
reproduction process ξ : R+ → Z+ (where R+ = [0, ∞), Z+ = 0 ∪ N) is a point
process describing the oﬀspring produced at each time; λ ∈ R+ is the life span, andφ is a random characteristic, a product-measurable non-negative random processthat assigns a score to an individual. Note that we make no assumption about thejoint distribution of the components of U .
For t ≤ λi , ξi (t) is the number of children born to i in time t. We write Ni = ξi (λi )for the total number of children born to i. Let T ⊂ I be the individuals in a CMJ
process. We take ∅ as the initial ancestor, and then for each individual i ∈ T
include its children i1, i2, . . . , iNi ∈ T . Clearly T is a random tree. Let the birthtime of ∅ be σ∅ = 0, and let the birth time for ij be σij = σi + ti (j), whereti (j) = inf{t : ξi (t) ≥ j}. Any distribution for U∅ induces a probability space(Ω, B, P), where Ω is the space of all trees T and their associated {Ui : i ∈ T }.
Let Z(t) be the population alive at time t, given Z(0) = 1. By considering theoﬀspring of the initial individual ∅, we have a decomposition of the process asξ∅ (t)

Z(t) = I{λ∅ >t} +(3.1)

Zi (t − σi ),i=1where each Zi is an independent copy of the branching process with initial ancestor
the individual with address i. The process counted with random characteristic φ isthen

φi (t − σi ),Z φ (t) =i∈T

where the φi are i.i.d., and may depend upon the whole process started from i.
We note that Z φ (t) = Z(t) if we use the random characteristic φi (t) = I{0≤t<λi } .
Let Is denote the set of sequences corresponding to individuals to be born after
time s whose mothers were born before s. We can decompose the process withcharacteristic, at any time s < t, as φφi (t − σi ) +Zi (t − σi ).
(3.2)
Z φ (t) =
i∈Is

i∈Is

We make the following assumptions (writing E for the expectation under P).
Assumption 3.1. (A1) Eξ∅ (0) = 0, and there exists a unique Malthusian parameter α ∈ (0, ∞) such that
E

N∅

i=1

−ασi

e

=1

and

E

N∅

i=1

σi e−ασi < ∞.

RANDOM FRACTAL STRINGS(A2) For Xα =

291

N∅

−ασi
,
i=1 e

EXα log+ Xα < ∞.
(A3) There exists
 ∞on [0, ∞) a non-increasing, positive integrable function g suchthat E(supt g(t)−1 t e−αu ξ(du)) < ∞.
We will assume that our characteristics are bounded.
We note that if mφt = Ee−αt Ztφ , then it satisﬁes a renewal equation
 t
mφt = Ee−αt φ(t) +mt−s µα (ds),0twhere µα (t) = E 0 e−αs ξ(ds). This is the distribution function of a probability
measure by our choice of α. We will refer to a measure as lattice if its support lies
on a lattice with period T . If the measure is not lattice we call it non-lattice. By
the renewal theorem we have that as t → ∞, if the measure µα is non-lattice,∞E(φ(t))e−αt dt
φφ.
mt → m∞ = 0 ∞ −αtte µ(dt)
0
In the lattice case we obtain limits down an appropriate subsequence,∞−αjT
j=0 E(φ(jT ))eφ∞, as n → ∞.
mnT →te−αt µ(dt)0In either case we will write the limit as mφ∞ .
Let
e−ασi .
Wt =i∈It

It is shown in [18] that this process is a martingale and, as it is positive, converges
to a limit W . We observe that this can be decomposed as
e−ασi Wi ,
(3.3)
W =
i∈Is

where Wi denotes the limit random variable started from the individual with addressi (and we write W = W∅ ).
Under Assumption 3.1 we have the following.
Theorem 3.2 ([18, 7]). Under (A1), (A3), for any characteristic φ there is arandom variable W such that:(1) If µα is non-lattice, then e−αt Z φ (t) → mφ∞ W almost surely as t → ∞.
(2) If µα is lattice, then e−αnT Z φ (nT ) → mφ∞ W almost surely as n → ∞.
Moreover, EW > 0 if and only if (A2) holds.
Note that in the lattice case we have a ﬁnite constant c1 such thatlim sup e−αt Zt ≤ c1 W.
t→∞

We will now give two assumptions which each provide a restriction on the branching process.
Assumption 3.3. The probability space Ω is ﬁnite. There are only a ﬁnite numberof possible life histories U .
Assumption 3.4. The reproduction process is bounded in that there exist ﬁnite
constants c2 and Λ1 , Λ2 such that ξi (λi ) ≤ c2 and Λ1 ≤ λi ≤ Λ2 for all i ∈ T .

292

B. M. HAMBLY AND MICHEL L. LAPIDUS

Clearly Assumption 3.3 will imply Assumption 3.4 and we will therefore make thesecond assumption unless we require a stronger statement. Both these assumptions
imply that the random variable W has moments of all orders (see [3]).
We state here a rate of convergence theorem for the renewal theorem. This
will enable us to prove a rate of convergence theorem for the general branchingprocess which will be required to prove the meromorphic continuation of our random
recursive strings.
We state the result in a more general form. Let f, g be functions satisfying therenewal equation for a probability measure ν, t
f (t − s)ν(ds).
(3.4)
f (t) = g(t) +0∞
We will write the ﬁrst moment as ν̄ = 0 sν(ds).
Statements of the rate of convergence in the renewal theorem have been given
in a few places. In order to ﬁnd this theorem we refer the reader to [20]; otherapproaches in the lattice and non-lattice cases can be found in [16], [10]. We call ameasure ν strongly non-lattice if
 ∞lim inf |1 −eiθx ν(dx)| > 0.
|θ|→∞

−∞

Lemma 3.5. Let f satisfy the renewal equation (3.4) with g a directly Riemannintegrable
 ∞ function and ν a probability measure which has exponential moments, inthat 0 eδs ν(ds) < ∞ for some δ > 0. Then there exists a constant ρ > 0 suchthat(1) In the strongly non-lattice case
1 tg(s)ds| = 0.
lim eρt |f (t) −t→∞ν̄ 0(2) In the lattice case, where ν lies on a lattice with period T ,lim eρt |f (t) − G(t)| = 0,t→∞where G is the T -periodic function given by∞T G(t) =
g(t + jT ).
ν̄ j=−∞
Corollary 3.6. Let g be such that g(t) ≤ e−δt and let ν be a lattice
 ∞or strongly
non-lattice sub-probability measure in that ν is a measure such that 0 ν(ds) < 1,∞then there exists a constant ρ > 0 such that 0 eρs ν(ds) = 1 and there exists a
constant c3 such thatf (t) ≤ c3 e−ρ t ,for any ρ < min(δ, ρ).
Remark 3.7. (1) We do not have an analogous result for the general non-latticecase.
(2) There are other conditions under which such results can be obtained. In thecase of a ﬁnite measure (where the strong non-lattice case is the non-lattice case)
we refer to [10], where the precise value of ρ is given.
Assumption 3.8. The general branching process mean measure is either lattice orstrongly non-lattice.

RANDOM FRACTAL STRINGS

293

Under this assumption, we will let ρ be the rate of convergence in the renewal
theorem for the mean of the general branching process,ρ := sup{β : lim eβt |mφt − mφ∞ | = 0}.
t→∞This constant could depend upon φ but as we will only use one particular φ in what
follows, we omit the dependence.
Theorem 3.9. Under Assumptions 3.1, 3.4 and 3.8, for any 0 < κ < min(α, ρ)/2,
then
lim e−(α−κ)t |Ztφ − W EZtφ | = 0, P-a.s.
t→∞

The proof will follow from a series of lemmas. We begin with the following
renewal-type theorem.
Lemma 3.10. Under the assumptions of Theorem 3.9, if a(t) is bounded over ﬁnite
intervals and satisﬁes the equationa(t) = EN∅e−2ασi a(t − σi ) + b(t), ∀t > Λ2 ,i=1

with 0 < |a(0)| < ∞, where |b(t)| ≤ c1 e−γt for all t ≥ 0 for some constant c1 , thenthere exists a constant c2 such that for all γ  < min(α, γ),|a(t)| ≤ c2 e−γ t , ∀t ≥ 0.
Proof. Fix a γ  < min(α, γ). As a(t) is ﬁnite initially and is bounded over ﬁniteintervals we can set c2 = supt∈[0,t0 ] eγ t |a(t)| < ∞, where t0 will be chosen later.
Then for t ∈ [t0 , t0 + Λ1 ] we have|a(t)| ≤ E

N∅




e−2ασi c2 e−γ (t−σi ) + |b(t)|i=1
≤ c2 e−γ t

E

N∅






e−(2α−γ )σi + c1 e−(γ−γ )t.

i=1

N∅ −(2α−γ  )σie< 1 and hence for γ  < γ
Hence, for γ  < α, we know that C = E i=1−(γ−γ  )t0
we can choose t0 such that C + c1 e
≤ 1. Thus|a(t)| ≤ c2 e−γ t , t ∈ [0, t0 + Λ1 ].
Proceeding in the same way we can extend the estimate to [t0 + nΛ1 , t0 + (n + 1)Λ1 ]for all integers n, giving the result.
Remark 3.11. Even though the equation satisﬁed by a(t) is a renewal equation, we
have chosen to prove this result directly, as it is all we require for the proof of ourmain theorem.
Now we turn to a variance calculation that will establish Theorem 3.9. Let
βiφ (t) = e−αt Ziφ (t), so that mφt = Eβtφ , and note that Xi (t) = βiφ (t) − mφt Wi hasmean 0. Let V (t) = E(X 2 (t)) = E(β φ (t) − mφt W )2 . This quantity is bounded overﬁnite time intervals as, under Assumption 3.4, all the moments of W and β φ exist.

294

B. M. HAMBLY AND MICHEL L. LAPIDUS

Lemma 3.12. Under Assumptions 3.1, 3.4 and 3.8 for any γ < min(α, ρ), there
exists a constant c3 such that
V (t) ≤ c3 e−2γt , t > 0.
Proof. We will consider s = 0+ in (3.3), so that we are just splitting the population
based on the ﬁrst individual and then i = i for i = 1, . . . , N∅ . Thus, for t > Λ2 ,
N∅


=

Xt

φ
e−ασi βt−σ
−
i

N∅


i=1

i=1


N∅

=

e−ασi mφt Wi

e−ασi Xt−σi +

N∅


i=1

e−ασi (mφt−σi − mφt )Wi

i=1

and, by conditioning on the ﬁrst event, using independence between branches andthe fact that EXt = 0, we have that
V (t) =
=

EXt2
E(

N∅


e−ασi Xt−σi )2 + 2E

i=1

N∅


e−ασi Xt−σi

i=1


N∅

+E(

N∅


e−ασi (mφt−σi − mφt )Wii=1e−ασi (mφt−σi − mφt )Wi )2

i=1


N∅

=

E

e−2ασi V (t − σi ) + g(t),

i=1

where
(3.5)
g(t) = EN∅e−2ασi (mφt−σi − mφt )E(Xt−σi Wi ) + E(

i=1

N∅


e−ασi (mφt−σi − mφt )Wi )2 .

i=1

We need to control g(t) and for this we need to compute f (t) = E(W X(t)) using
Lemma 3.10. By splitting at the ﬁrst individual, for t > Λ2 ,f (t) = EWN∅e−2ασi X(t − σi ) + EW

i=1

N∅


(mφt − mφt−σi )e−ασi Wi .

i=1

This expression can be written, using conditioning, as
f (t) =EN∅e−ασi Wi

i=1


N∅

(3.6)

=

E

N∅


e−2ασi (X(t − σi ) − (mφt − mφt−σi )Wi )i=1e−2ασi f (t − σi ) + h(t),

i=1

where
h(t) = E

N∅ N∅



(mφt−σi − mφt )EWi Wj .

i=1 j=1

By Lemma 3.5 we have constants C1 , C2 such that|mφt − mφt−σi | ≤ |mφt − mφ∞ | + |mφ∞ − mφt−σi | ≤ C1 e−ρt + C2 e−ρ(t−σi ) .

RANDOM FRACTAL STRINGS

295

Observing that σi ≤ Λ2 we have|mφt − mφt−σi | ≤ C3 e−ρt .

(3.7)
Now, using (3.7), we have|h(t)| ≤ E

N∅ N∅



|mφt−σi − mφt |EWi Wj

i=1 j=1≤ C3 e−ρt (EN∅ (N∅ − 1) + EN∅ EW 2 )≤ C4 e−ρt .
As f is bounded over ﬁnite intervals, we can now apply Lemma 3.10 to (3.6) toobtain the bound that |f (t)| ≤ C5 e−ρ t for any ρ < min(α, ρ). Putting this into
the expression for g(t) we have|g(t)| ≤EN∅e−2ασi |mφt−σi − mφt ||f (t − σi )| + E(

i=1

≤
≤

−ρt

C3 e

e−ασi |mφt−σi − mφt |Wi )2

i=1
−ρ t

C5 e

−(ρ+ρ )t

C6 e

N∅


−2ρt

E(W )
2

+ C3 e.

Thus, as V is bounded over ﬁnite intervals, we can apply Lemma 3.10 once again
to obtain the existence of the constant c3 such thatV (t) ≤ c3 e−2ρ t ,for any ρ < min(α, ρ) as required.

Proof of Theorem 3.9. The almost sure control on Xt is then accomplished via
Chebychev’s inequality and our variance calculation as, for any γ < min(α, ρ),P (Xt > θ) ≤1
V (t) ≤ c6 θ −2 e−γt .
θ2Hence, for  > 0, we can set θ = e−(γ−)t/2 to obtain, by Borel–Cantelli, that there
is a constant C such thatXt ≤ Ce−(γ−)t/2 , P-a.s.
As this holds for arbitrary , we have the desired result.



4. Random recursive strings
We now turn to our examples of random fractal strings. The ﬁrst class is generated by random recursive fractals in one dimension.
Let A be a ﬁnite set. For each a ∈ A we have a set of Na similitudes ψ a = {ψia :i = 1, . . . , Na }, where ψia : [0, 1] → [0, 1] is a map with contraction factor ria andψ a satisﬁes the open set condition (with the same open set) for each a ∈ A. The
fractal Fa is the unique compact set satisfying
Fa =

Na

i=1

ψia (Fa ).

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B. M. HAMBLY AND MICHEL L. LAPIDUS

The string is formed from the sequence of removed subintervals, and for each a we
can writeMNaaϕaj ([0, 1]) ∪ψia ([0, 1]),[0, 1] =
j=1
a

i=1

= {ϕaj : j = 1, . . . , Ma } are similitudes and ϕaj has contraction

where the maps ϕfactor lja .
We now wish to combine these into a random recursive fractal string using aprobability measure on the set A. The informal description is that we start withthe unit interval and choose a set of similitudes (ψ a , ϕa ) according to this probability
measure and apply it to the unit interval. We retain the removed lengths as partof the string and then repeat this procedure independently for each copy of theunit interval given by the ψia . The collection of removed lengths, in non-increasingorder, is the random fractal string.
More formally we proceed as follows. In order to construct a random recursive
string we need an address space. We use the same address space we used to label
the individuals in the general branching process, the set of integer sequences I. Let
Ui = (Ni , Mi , r1 (i), . . . , rN (i), l1 (i), . . . , lM (i)), i ∈ I,
be independent and identically distributed (N, N, (0, 1)N )-valued random variables.
We deﬁne a probability measure Φn on (0, 1)n and take {pn,m }n,m∈N to be a probability distribution on N × N. Then the probability measure for U is given byP(U ∈ (n, m, Sn )) = pn,m Φn+m (Sn+m ), Sn+m ⊂ (0, 1)n+m , n ∈ N.
A random tree T is a subset of I such that(i) ∅ ∈ T is the root of the tree;(ii) i ∈ T implies i|k ∈ T for all k < |i|;(iii) i ∈ T implies i1, i2, . . . , iN (i) ∈ T , and i(N (i) + 1), i(N (i) + 2), . . . ∈ T .
We will write (Ω, B, P) for the natural probability space associated with these trees.
That is, a sample point ω ∈ Ω will denote a random tree T and the associated{Ui : i ∈ T }. The σ-algebras are deﬁned as
Bn = σ(Ui ; i ∈ Tn−1 ),

B=

∞


Bn ,

n=1

and the probability measure, P, is as deﬁned previously. If we project this measure onto its ﬁrst coordinate it is the oﬀspring distribution for a Galton–Watson
branching process.
We now regard our random variables Ui with probability measure P as taking
values in the set A of possible families of maps by taking Na as the number of maps
in the family ψ a and Ma as the number of removed intervals needed for the string.
The {ria , i = 1, . . . , N } and {lja , j = 1, . . . , M } are the Lipschitz constants of therespective maps. We note that we must have 1 ≤ Ma ≤ Na + 1. From now on wedrop the sub or superscript a and regard the N, M, ri , lj as random variables.
The address of each branch in the tree is now used to specify our random Cantor
set. LetUU

Ci = ψi ([0, 1]) := ψi|1i|1 (· · · (ψi|ni|n ([0, 1]))).

RANDOM FRACTAL STRINGS

297

A random recursive fractal can then be deﬁned by
Fω =

∞




Ci .

n=1 i∈Tn (ω)

We now deﬁne a speciﬁc general branching process Z with the property that attime t the individuals alive in the population each correspond to a set in the fractalof roughly size e−t . Let the reproduction process and life span be given by(ξ, λ) =Nδ− log ri ,i=1

max − log ri

1≤i≤N

,

where U  = (N, r1 , . . . , rN ) is chosen according to the appropriate marginal distribution of P. The birth time σi of an individual with address (ancestry) i can be
written as|i|σi = − log ri , where ri =ri[k] (i|k−1 ).
k=1

Let (Ω∅ , B∅ , P) be the probability space generated by U∅ . The branching process
assumptions of Section 3 can be reexpressed for the fractals as
Assumption 4.1. (A1) P(r1 (∅) = 1) = 0, and there exists a unique α ∈ (0, ∞)such that
N (∅)
N (∅)


α
E
ri (∅) = 1 and E| log ri (∅)|ri (∅)α < ∞.
i=1

i=1

(A2) Ω∅ is ﬁnite.
(A3) The ri are bounded away from 0.
(A4) The measure µ, with distribution function
 x N
(∅)
y −α δ− log ri (∅) dy,µ(x) = E0i=1

is either lattice or strongly non-lattice.
Remark 4.2. (1) Under Assumption 4.1(A2), Φn is a ﬁnite atomic measure on(0, 1)n .
(2) Under both Assumption 4.1(A2) and (A3) there exist r∗ , r ∗ such that 0 <r∗ ≤ ri (∅, ω) ≤ r ∗ < 1 for all i = 1, . . . , N (∅, ω), ω ∈ Ω∅ and N (∅, ω) ≤ N̄ < ∞ a.s.,
for some scalar N̄ .
(3) Assumption 4.1(A4) gives the lattice/non-lattice dichotomy for our strings,
that is, it is lattice if log ria are rationally related for all i = 1, . . . , Na , a ∈ A. In
particular we see that if the probability measure has a density on A, we cannot
have a lattice case.
The Hausdorﬀ dimension of the set F ω can be found by applying the results of[6], [17] and, as we have assumed the OSC, it is given by the Malthusian parameterof the CMJ process,N (∅) α 
r
(∅)
=
1
, for P-a.e. ω ∈ Ω.
(4.1)
df (F ω ) = inf α ≥ 0 : E
i=1 iNote that for these fractals the other natural notions of dimension, Minkowski,
upper box, etc. all coincide with the Hausdorﬀ dimension.

298

B. M. HAMBLY AND MICHEL L. LAPIDUS

Now we describe the construction of the random fractal string. To begin weconstruct the sample path of this process as our random tree. Associate with eachvertex of the tree the set of removed lengths (lj : j = 1, . . . , M ) of the type ofoﬀspring generated from that vertex.
The lengths that compose the random fractalrstring are then given by Lij = n−1
j=1 ij lj at vertex i = (i1 , . . . , in ). We then write
them in length order to obtain the string L = (Li )∞i=1 .
As in Section 2 we think of the string as a measure, and for the random string
it will be a random measure. Our general branching process can be used to encode
this random measure η by use of a suitable characteristic. We set
φi (t) =Mi

I{− log lj ≤t} .

j=1

That is, the lengths of the removed pieces of string are a function of the individual
representing that application of the similitude. We also only count the removed part
if it is suﬃciently large. We can check that with this choice our general branchingprocess has encoded the random measure η.
By the deﬁnition of the general branching process and the counting function ηin (2.1), we haveφ(t − σi )
ztφ =
i∈I∞

=

Mi
 

I{− log lj ≤t−σi }i∈I∞ j=1

=

=

Mi
 

I{L−1 ≤et }ij

i∈I∞ j=1∞


I{L−1 ≤et } = η(et ).
i

i=1

We note that, as all lengths are less than one, the branching process begins at
t = 0 when Z(0) = 1 and z0φ = 0.
We can then use our Theorem 3.2, concerning random characteristics, to determine the growth rate of the string.
Corollary 4.3. Under Assumption 4.1 there is a mean one random variable W
such that in the non-lattice caselim e−αt η(et ) = mηα (∞)W, P-a.s.

t→∞

and in the lattice caselim e−αnT η(enT ) = mηα (∞)W, P-a.s.

n→∞

By using the fact that the quantity mηα (∞) is the limit of the normalized mean
growth rate we can write, in the non-lattice case,η(et )= W, P-a.s.
t→∞ Eη(et )lim

We have a similar expression in the lattice case down a subsequence. The randommeasure η will satisfy a distributional equality in that it can be expressed as a sum

RANDOM FRACTAL STRINGS

299

of scaled copies of itself,
η(dx) =Mδl−1 +

N


j

j=1

η(ri dx).

i=1

The zeta function of the random string is a random function deﬁned as the Mellintransform of the measure η, ∞ζ(s) =x−s η(dx).
0

The complex dimensions of the random fractal string are the poles of this function.
In order to compute them we consider the mean zeta function which is given by ∞x−s Eη(dx).
Eζ(s) =
0

Lemma 4.4. The mean geometric zeta function can be expressed ass
E M
j=1 ljEζ(s) =.
s
1 − E( Ni=1 ri )If Assumption 4.1(A2) holds, the family size is ﬁnite, then the mean zeta function
can be meromorphically continued to C. If Assumption 4.1(A3) holds, then themean zeta function can be meromorphically continued up to the line Re(s) = 0. In
either case, Eζ(s) is given by the above expression for s in the indicated region.
Proof. We can use the branching structure of the random set to ﬁnd a closed form
expression for this mean zeta function. Observe that by construction we have thefollowing equality in distribution:(4.2)

ζ(s) =

M

j=1

ljs +

N


ris ζi (s),i=1where the ζi are i.i.d. copies of the original zeta function and the random variables
lj , ri are determined by the particular choice of a ∈ A corresponding to the ﬁrst
individual.
If we take expectations in (4.2) and condition on the ﬁrst map,Eζ(s) = EM
j=1

ljs + E(N

ris Eζi (s)).

i=1

As the ζi are just copies of ζ we can rearrange to getM
E j=1 ljs
(4.3)
Eζ(s) =.
N
1 − E( i=1 ris )NThe poles of this function lie in the set of s ∈ C such that E( i=1 ris ) = 1.
Under Assumption 4.1(A2) the probability space is ﬁnite and we have the following explicit formula for the mean zeta function as written in (4.3):
Ma a

a∈A
j=1 ljEζ(s) =

Na a s .
1 − a∈A i=1 (ri ) pa
As the sums are ﬁnite Eζ(s) can be meromorphically continued to the whole of thecomplex plane.

300

B. M. HAMBLY AND MICHEL L. LAPIDUS

Under Assumption 4.1(A3) the expectations in (4.3) can be computed for Re(s)> 0 and hence Eζ(s) can be continued to this region.
The expression for the poles is exactly the expression for the dimension of theboundary of the fractal string and hence we see that the zeta function is welldeﬁned for Re(s) > df . We now wish to show that the random zeta function canbe meromorphically continued to the left of this line.
The approach we take is based on the Euler–Maclaurin series approach to proving meromorphic continuation for the usual Riemann zeta function. From thediscussion prior to Lemma 4.4 we can write ∞x−s V (dx), P-a.s.,(4.4)
ζ(s) = W Eζ(s) +0where W is the limiting random variable for the normed limit of the distributionfunction for the measure η and V (dx) = (η − W E(η))(dx) is a random signed
measure. In order to extend our zeta function we need to be able to compute thesecond term for suitable s ∈ C and for this purpose we look at the behaviour of x
V (x) =V (dy) = η(x) − W E(η(x)).
0

The ﬂuctuations in this function follow from those of the corresponding generalbranching process derived in Section 3.
We can now use this result to establish our main theorem.
Theorem 4.5. Under Assumptions 4.1(A1), (A3), (A4), there is a number τ with
0 < τ < min(α, ρ)/2 such that the random zeta function ζ(s) can be meromorphically extended to the right half-plane D := {s : Re(s) > α − τ }, where α is given in
Assumption 4.1(A1) above. The poles of the random zeta function lie in the set ofcomplex solutions z ∈ D to the equationE(Nriz ) = 1.

i=1

The residues at the poles z (when the latter are simple) are given by
W Res(E(ζ(s)); s = z),where Eζ(s) is the mean zeta function given in Lemma 4.4.
Proof. With our Theorem 3.9 we immediately deduce the meromorphic
 ∞ extension
from (4.4). This follows if we can prove the existence of the integral 0 x−s V (dx).
By construction it is a question of determining the behaviour of V at ∞ and, by
Theorem 3.9, there is a random constant C and a 0 < τ < min(α, ρ)/2 (where ρis the rate of convergence in the renewal theorem for the branching process mean)∞such that V (y) ≤ Cy α−τ for large y. Hence 0 x−s V (dx) is well deﬁned forRe(s) > α − τ .
We immediately see thatM
 ∞E j=1 ljs
x−s V (dx),ζ(s) = W
N s +
1 − E( i=1 ri )0

RANDOM FRACTAL STRINGS

301

so that the poles of ζ(s) in D occur at the same place as those of E(ζ). To computethe residues we again use the formula (4.4).
Res(ζ(s); s = z)=lim (s − z)ζ(s)s→z= W Res(E(ζ(s)); s = z) + lim (s − z)
s→z

 ∞

x−s V (dx)0

= W Res(E(ζ(s)); s = z),

as desired.

Remark 4.6. (1) Note that in general there may be cancellations between the zerosN ssof the numerator E( M
j=1 lj ) and those of the denominator 1 − E(
i=1 ri ) of themean zeta function. In the deterministic case examples of such cancellations canbe found in [15]. However if M = 1, the poles of the random zeta function ζ(s) areN
precisely located at the complex solutions of E( i=1 riz ) = 1 (such that Re(z) >α − τ ).
(2) In the deterministic case, the poles may have multiplicity greater than one[14], Section 2.2.3; the corresponding Laurent series of the zeta function can, however, be used instead. In the random case, the above proof shows that the Laurent
series of ζ(s) at a pole z can be expressed in terms of that of Eζ(s) at z.
5. Stable strings
In this section we will discuss the case where the string is generated by the excursions of a stable subordinator. The main example to bear in mind is the Brownian
string. One-dimensional Brownian motion run for a unit time can be decomposed
into a countable family of excursions away from the origin. The local time of Brownian motion at 0 can be deﬁned as the density of the occupation measure of theprocess at 0. We can then regard the Brownian path as a Poisson process of excursions indexed by the local time process. Our string will be constructed as theseries of ranked excursion lengths, and as the time at the origin has 0 Lebesgue
measure, it will have the property of total unit length. Our aim will be to consider
stable strings, establish the asymptotic behaviour of the string, show that its zetafunction can be meromorphically continued to the left of its rightmost pole andobtain an expression for the zeta function at the poles. Our main conclusion is thatfor such strings the complex dimensions are purely real.
5.1. Stable subordinators. The ranked lengths of the excursions of stable subordinators are known to have a Poisson–Dirichlet, PD(α, 0) distribution for 0 < α < 1.
This type of distribution initially arose when deﬁning the relative frequencies of various species in a large population. There are a number of ways of deﬁning such
distributions and we refer to [19] for an extensive survey. For our purposes werecall the deﬁnition through subordination and the construction through a familyof independent identically distributed random variables.
Let τs be a stable subordinator. This is an increasing process with stationary
independent increments which can be characterised via its Laplace transform
E[exp(−λτs )] = exp(−sCΓ(1 − α)λα ),where C is a constant and Γ is the Gamma function. Let V1 (T ) ≥ V2 (T ) ≥ . . .
denote the ranked lengths of components of the set [0, T ]\Z, where Z is the closure

302

B. M. HAMBLY AND MICHEL L. LAPIDUS

of the range of {τs ; s ≥ 0}. A basic proposition is that for every s > 0,
V1 (τs ) V2 (τs )
,
,...
τs
τs
has the PD(α, 0) distribution, and also for ﬁxed times t > 0,
V1 (t) V2 (t)
,
,...
t
t
has the PD(α, 0) distribution.
∞
From now on we consider the case t = 1 and write (V1 , V2 , . . . ), where i=1 Vi =1, for a random sequence with the PD(α, 0) distribution. An observation which canbe found in [19] is that the random variables Rn = Vn+1 /Vn have beta distributionβ(nα, 1) and are mutually independent. For completeness we recall that the β(u, v)
distribution has density function
fu,v (x) =Γ(u + v) u−1x(1 − x)v−1 , x ∈ [0, 1].
Γ(u)Γ(v)Thus, given V1 we can recursively construct the relative values of the lengths in thestring and then determine V1 by normalization.
We also note here a ﬁrst limit result and an important representation.
Lemma 5.1 ([19], Proposition 10). If (Vn ) has the PD(α, 0) distribution, then
(1) The limit L = limn→∞ nVnα exists almost surely.
(2) Let Xn = LVn−α . Then the points (Xn ) are the points of a Poisson randommeasure with Lebesgue measure as its intensity.
nWe note that the second part of the lemma means that we can write Xn =i=1 ξi , where the ξi are independent random variables with the exponential distribution with mean 1.
Once we have deﬁned the ranked excursions we wish to consider the associatedzeta function
∞

Vns .
ζα (s) =n=1

The key tool will again be the random measure η on [1, ∞) determined from thestring by∞η(dx) =
δVn−1 (dx).
n=1

5.2. The mean zeta function. The ﬁrst step in our analysis is to consider themean behaviour of the zeta function. We write Pα,θ for the general PD(α, θ) distribution and Eα,θ for the expectation with respect to this distribution. When θ = 0
we will write Pα and Eα for the distribution and its expectation, respectively.
We begin with a useful formula from [19] derived from size biased sampling fromthe distribution.
Lemma 5.2 ([19], Corollary 3). For the general PD(α, θ) distribution we have 1∞
Γ(θ + 1)(5.1)
Eα,θf (Vn ) =(1 − u)α+θ−1 u−α−1 f (u)du.
Γ(θ
+
α)Γ(1
−
α)
0
n=1

RANDOM FRACTAL STRINGS

303

From this we can easily compute the expected value of the zeta function directly,∞Eα ζα (s) = Eα,0
Vns .
n=1

Setting f (x) = xs in (5.1) and integrating we obtainΓ(s − α).
Γ(1 − α)Γ(s)The Gamma function can be analytically continued to the whole complex plane. It
has no zeros and has simple poles that lie on the real axis at 0, −1, −2, . . . . As themean zeta function is a ratio of Gamma functions, we know that the poles lie on
the real line at the values α, α − 1, α − 2, . . . and we have the following lemma.

(5.2)

Eα ζα (s) =Lemma 5.3. The mean zeta function can be meromorphically continued to thewhole complex plane and is given there by (5.2).
We will also be interested in the mean growth of our measure η. This can becomputed from (5.1) as
 x
η(dy)
Eα η(x) = Eα
1

= Eα

∞


I{Vn ≥1/x}

n=1(x − 1)αΓ(α + 1)Γ(1 − α)sin πα=(x − 1)α ,(5.3)
πα
where the last result follows from the fundamental identities for the Gamma function.
=

5.3. The zeta function for the string. Finally we consider the actual zeta function associated with the string. The ﬁrst stage is to prove that it can be meromorphically continued to the left of the line Re(s) = α and to do this we will use thefollowing.
Theorem 5.4. For all 0 < α < 1 we have
πα
η(x)
=
L, Pα -a.s.
limx→∞ Eα (η(x))
sin παProof. The key to the result, and to some of the later ones, is that we can obtain
limit information about η from the asymptotics of Vn .
We can write∞η(x) =
I{Vn−1 ≤x}
n=1


(5.4)

=

Thus
x−α η(x) =

0
m

∞


x < V1−1−1
Vm−1 ≤ x < Vm+1.
x−α nI{Vn−1 ≤x<V −1 }
n+1

n=1

304

and

B. M. HAMBLY AND MICHEL L. LAPIDUS

∞


α
nVn+1
I{Vn−1 ≤x<V −1 } < x−α η(x) ≤

∞


n+1

nVnα I{Vn−1 ≤x<V −1 } .
n+1

n=1
n=1
Now, as Vn−1 → ∞ as n → ∞, we see that taking limits as x → ∞ is equivalent totaking limits as n → ∞ and gives, by Lemma 5.1,
∞


lim

x→∞

nVnα I{Vn−1 ≤x<V −1 } = lim nVnα = L, Pα -a.s.
n→∞

n+1

n=1

For the left-hand side the change to Vn+1 does not alter the limit and hence,
as upper and lower bounds have the same limit, we have that Pα -almost surely
limx→∞ x−α η(x) = L. We now obtain the stated result by substituting in thefunction Eα η(x) from (5.3).
In order to establish the meromorphic continuation we require a rate of convergence in the above limit theorem. In order to do this we will use the second partof Lemma 5.1. As we can write LVn−α as a sum of independent and identically
distributed random variables, we can use the independence to control the growthrate of LVn−α − n. We need two preliminary results.
Lemma 5.5. For  > 0
lim n−(1+)/2 LVn−α − n = 0, Pα -a.s.

n→∞Proof. We begin by observing thatn

LVn−α − n =(ξi − 1),i=1where the ξi are independent exponential random variables with mean 1. We use
a standard large deviation approach
Pα (

n
n


(ξi − 1) > x) ≤ Pα (exp(θξi ) > eθ(x+n) )i=1

i=1−θ(x+n)≤ e

Eα (eθX )n= e−θ(x+n) (1 − θ)−n .
As this holds for all θ < 1 we can optimize the bound by setting θ = x/(x + n) to
get
(5.5)

Pα (

n

1 x2
2 x
x
(ξi − 1) > x) ≤ e−x (1 + )n ≤ e− 2 n (1− 3 n ) .
ni=1

We can follow the same line of argument to show that
(5.6)

Pα (

n

1 x2
(1 − ξi ) > x) ≤ e− 2 n .
i=1Hence, if x < 12 n, then in (5.5) we have 1 − 23 nx ≥ 23 and combining this with (5.6),
we arrive at
Pα (|

n

i=1

1 x2

(1 − ξi )| > x) ≤ 2e− 3 n .

RANDOM FRACTAL STRINGS

305

Thus for x = δn(1+)/2 , we have summability of the probabilities and hence theﬁrst Borel–Cantelli lemma shows that
|LVn−α − n|≤ δ, Pα -a.s.
lim sup1n→∞n 2 (1+)As this holds for every δ > 0, the limit exists as required.

We are now ready to state our convergence rate result. For convenience we writeL̃ = (πα/ sin πα)L.
Theorem 5.6. For any  > 0, η(x)
lim xα(1−)/2 − L̃ = 0, Pα -a.s.
x→∞Eα η(x)

Proof. The proof is an application of Lemma 5.5. It is enough to show that, forany δ > 0,
lim sup xα(1−)/2 x−α η(x) − L ≤ δ.
x→∞

As in the proof of the almost sure convergence in Theorem 5.4 we can writexα(1−)/2 x−α η(x) − L = x−α(1+)/2 |η(x) − Lxα |= x−α(1+)/2 |n − Lxα | I{Vn−1 ≤x<V −1 }n+1α(1+)/2α≤ Vn+1max(|n − LVn−α |, |n − LVn+1|)I{Vn−1 ≤x<V −1 } .
n+1To use Lemma 5.5 we note that
|LVn−α − n|
|LVn−α − n|α
= lim suplim (nVn+1)(1+)/2 .
lim sup −α(1+)/2(1+)/2
n→∞
n
n→∞ V
n→∞
n+1
Thus, applying the result we established in Lemma 5.5, and using the almost sureconvergence of nVnα → L as n → ∞, we have the result.

Therefore we have shown that for any  > 0, Pα -almost surely, for a constantC > 0 we have
V (x) = η(x) − L̃Eα (η(x)) ≤ Cxα(1+)/2 .
Hence we can use the Euler–Maclaurin approach, as outlined in the proof of Theorem 4.5 in the random recursive fractal case, to show that the zeta function can bemeromorphically continued beyond the line Re(s) = α to the set {s : Re(s) > α/2}.
From the properties of the Gamma function we see that the complex dimensions
for these strings are purely real and we can state our ﬁnal theorem for this section.
(Note that the Brownian string corresponds to the case α = 1/2.)
Theorem 5.7. The geometric zeta function for the stable string is deﬁned meromorphically for all s : Re(s) > α/2 and has a simple pole at the value s = α andno other visible poles. The value of the residue at the pole is αL.
Remark 5.8. The stable bridge is a stable process with the property that it isconditioned to return to 0 at time 1. It is known [19] that such a process has law
PD(α, α) and hence its mean zeta function can be calculated using (5.1) and has
a meromorphic continuation to the whole complex plane. It also has a random
recursive structure as indicated in [7], as we can decompose the Cantor set formedby the visits to zero at the ﬁrst and last zeros either side of 1/2. Thus we canalso regard this string as a random recursive string but from an uncountable family

306

B. M. HAMBLY AND MICHEL L. LAPIDUS

which does not have scaling ratios bounded away from 0 (as the law of the ﬁrst
return time before 1/2 has support on the interval [0,1/2]). As yet we have notheorem that enables us to prove meromorphic continuation. If this can be done,
we can obtain the same results as for the stable string. The poles of the zetafunction occur at α, α − 1, . . . and hence there are no non-real complex dimensions.
The only change is that the residue at s = α is diﬀerent. This demonstrates thatthe class of random recursive strings includes strings that are quite diﬀerent fromdeterministic self-similar strings.
6. A homogeneous random string
We discuss a spatially homogeneous string arising from a random Cantor set.
This shows that there are random strings for which the techniques we have employed here to prove meromorphic continuation break down. This example is a
simpliﬁcation of the construction of the random recursive fractals. In this case
we do not choose a new family of similitudes for each interval, instead we choose
the same family for all sets of a given size. Random fractals of this type havebeen discussed in [1]. This has the eﬀect of introducing much larger ﬂuctuationsin the string and for this reason we are not able to control the extra term in the
Euler–Maclaurin approach.
We begin by deﬁning a scale irregular random Cantor set. We let ψ a = {ψ1a , ψ2a }denote the family of two similitudes given for any 0 < a < 1/2 byψ1 (x) = ax, ψ2 (x) = 1 − a(1 − x).
Thus the eﬀect of applying the map ψ a to the unit interval is to produce twointervals of length a with a gap of length 1−2a between them. For a given sequencea = {ai }∞i=1 , where 0 < ai < 1/2 for all i, we can deﬁne a scale irregular Cantor set
as
∞


Ca =
ψia11 ◦ · · · ◦ ψiann ([0, 1]).
n=1 i∈{1,2}nNote that if we write σ(a) = (a2 , a3 . . . ) for the shift map on one-sided sequences,then
Ca = ψ a1 (Cσ(a) ).
At this stage we make no assumption about the sequence. We can deﬁne homogeneous random Cantor sets by generating the sequence a using a probability
measure.
Theorem 6.1. The Hausdorﬀ dimension and lower box counting dimension of thescale irregular Cantor set Ca are given bynlog i=1 ai.
dlb = dimH = lim infn→∞n log 2The packing dimension and upper box counting (Minkowski) dimensions arenlog i=1 ai
dub = dimP = lim sup.
n log 2n→∞Proof. This is the same as the case of scale irregular Sierpinski gaskets as given in[1].
RANDOM FRACTAL STRINGS307

Our next step is to describe the scale irregular string that is determined bythis scale irregular Cantor set. From this point we assume for convenience thatan < 1/3 for all n. This assumption ensures that the set of lengths Ln = 1 − 2an isstrictly decreasing with n. By construction the string is the set of lengths Ln , withmultiplicity 2n , given byn−1

L1 = 1 − 2a1 , Ln =ai (1 − 2an ).
i=1

By construction of the zeta function,
ζa (s) =
=

∞

n=1
∞

n=1

=

∞


2n Lsnn−1

asi (1 − 2an )s

n

2

i=1

exp n log 2 + s

n=1

n−1


log ai + log (1 − 2an ),i=1which will be ﬁnite provided the real number s is such thatn log 2 + s

n−1log ai + log (1 − 2an )

< 0.

i=1

Thus we havenlog i=1 ai= dub .
ζa (s) < ∞, for s > lim supn log 2n→∞

The question we address now is the meromorphic continuation. Let us make somemore assumptions about our model. Assume that there is a probability distribution
P on (0,1/3) and that {Ai } is a sequence of independent and identically distributed
random variables with law P. In this case we see that−E(log A1 ), P-a.s.
df = dimH (Ca ) = dlb = dub = dimP (Ca ) =log 2
We consider η(x), the number of intervals with length greater than 1/x. By the
structure of our sequence we have the identityηa (x) = I{L1 ≥1/x} + 2ησ(a) (A1 x).
As the sequence is i.i.d. we haveEηa (x) = P(L1 ≥ 1/x) + 2Eηa (A1 x).
This can be written as a renewal equation if we set m(x) = e−γx Eηa (ex ), g(x) =e−γx P(L1 ≥ e−x ) and choose γ such that P̃(dx) = 2e−γx P(− log A1 ∈ dx) is a
probability measure, ∞m(x) = g(x) +m(x − y)P̃(dy).
0As g(x) is directly Riemannintegrable, we can apply the renewal theorem to obtain
∞∞m(x) → 0 g(x)dx/ 0 2xe−γx P(log A1 ∈ dx) as x → ∞, giving the growth rate

308

B. M. HAMBLY AND MICHEL L. LAPIDUS

of the counting function. Thus we have γ deﬁned to be the solution to 2EAγ1 = 1
and as x → ∞,Eηa (x) ∼ cxγ .
Our next task is to show that η(x) does not grow like its mean and hence wecannot hope to control the zeta function with the mean zeta function as we havedone before. We note that the mean zeta function is straightforward to compute asζa (s) = Ls1 + 2As1 ζσ(a) (s),and henceE((1 − 2A1 )s ).
1 − E(2As1 )Thus the poles of the zeta function occur at a subset of the complex solutions to
2EAs1 = 1, whose maximal real solution is not df . We now apply our previous
technique and attempt to control the zeta function by using a random variable W
and signed measure V = η − W Eη, such that ∞ζa (s) = W Eζa (s) +x−s V (dz).
Eζa (s) =0

By construction of the string we havenη(x) = 2n − 1 if {n−1

Ai (1 − 2An+1 ) < 1/x ≤i=1Taking the subsequence xn = (by the ergodic theorem

n−1

Ai (1 − 2An )}.
i=1

i=1 Ai (1 − 2An ))−1

we have η(xn ) = 2n − 1 and1
1− log Ai + log(1 − 2An )) → exp(−E(log A1 )),
n i=1
n
n−1

x1/n
= exp(n
as n → ∞. Now

η(xn ) = xnn log 2/ log xn − 1 ∼ xnlog 2/−E(log A1 ) ,as n → ∞. We note that this almost sure growth rate log 2/ − E(log A1 ) is diﬀerent
from γ, the mean growth rate. We can make the oscillation precise by using thelaw of the iterated logarithm (LIL) for i.i.d.
n random variables to control the size ofthe ﬂuctuation in the convergence of n1 i=1 log Ai .
Note that by the LIL, for all  > 0,lim sup
n→∞

1

n

|
1/2+

n


(log Ai − E(log Ai ))| = 0.

i=1Thus, for any c > 0, there exists N < ∞ such that xn ≤ exp(−nE(log Ai )+cn1/2− )for all n ≥ N , P-a.s. Also there will be a subsequence nk such that1/2+xn ≥ exp(−nk E(log Ai ) + cnk), ∀k.

Thus we cannot control the growth of the counting function η(x) by its mean growthrate, or even by its almost sure growth rate, as the ﬂuctuations are too large. Hence
our Euler–Maclaurin approach to meromorphic continuation does not work and adiﬀerent approach would be required to prove results about the complex dimensionsof this random fractal string.

RANDOM FRACTAL STRINGS

309

7. Tubular neighbourhoods and eigenvalue asymptotics
We conclude with an application of our results on the complex dimensions of
the string to two problems. First we discuss the geometric problem of determining
the volume of a tubular neighbourhood of the string. Second we consider the asymptotic growth of the eigenvalue counting function for the Laplace operator on abounded domain. In this case we regard a fractal string as a one-dimensional domain with a fractal boundary consisting of the Cantor set which is the complementof the string. We will restrict our attention to random recursive fractals satisfying
Assumption 3.3, and to stable strings.
We will extend the results of [14] concerning explicit formulas for the expansionof the zeta function from the deterministic to the random case. We begin with
two hypotheses on the growth of the zeta function. First we need two deﬁnitions.
Let r : R → [−∞, α] be a bounded Lipschitz continuous function and deﬁne thescreen S to be the curve S : t → r(t) + it for t ∈ R and the window W = {s ∈ C :Re(s) ≥ r(Im(s))}. We will write Dη (W) for the set of complex dimensions of thestring η which lie in the window W; these are referred to as the ‘visible’ complex
dimensions.
Assumption 7.1. There exist real constants κ ≥ 0 and C(ω) > 0 and a sequence{Tn }n∈Z of real numbers tending to ±∞ as n → ±∞ with T−n < 0 < Tn for n ≥ 1
and limn→∞ Tn /|T−n | = 1, such that for P-almost every ω ∈ Ω,
(H1) For all n ∈ Z and all σ ≥ r(Tn ),|ζη(ω) (σ + iTn )| ≤ C(ω)|Tn |κ .
(H2) For all t ∈ R, |t| ≥ 1,|ζη(ω) (r(t) + it)| ≤ C(ω)|t|κ .
We will show in Lemma 7.5 that there are examples of fractal strings which
satisfy this assumption.
We now give a distributional formula for the counting function of the string, aversion of [14], Theorem 4.12. We regard the measure η generated by the string asa Schwartz distribution. It acts on test functions ϕ byη, ϕ = ∞ϕ(x)η(dx).
0

We write P [k] η for the k-th primitive of the distribution, deﬁned by its action on
ϕ, through ∞ ∞(x − y)k−1[k]ϕ(x)dxη(dy)P η, ϕ =(k − 1)!
0yand P [0] η = η.
Theorem 7.2. Let η be a random fractal string satisfying Assumption 7.1. Thenfor every k ∈ Z the explicit formula for the distribution P [k] η on (0, ∞) is given for310

B. M. HAMBLY AND MICHEL L. LAPIDUS

P-almost all ω ∈ Ω by

P [k] η(x) =Res(z∈Dη(ω) (W)+xs+k−1 ζη(ω) (s); s = z)Γ(s + k)/Γ(s)


k−1
[k]
(−1)j x−j ζη (−j) + Rη(ω) (x),
j

k−1


1
(k − 1)!

j=0−j∈W\Dη (ω)

where
1
[k]
Rη(ω) , ϕ =
2πi

  ∞
ζη(ω) (s)ϕ(x)S0

xs+k−1dxds.
Γ(s + k)/Γ(s)Remark 7.3. Extending to the random case, [14], Theorem 4.20, we conclude that[k]under the same assumptions as for Theorem 7.2, the error term R(x) = Rη(ω) (x)is given for P-a.e. ω by R(x) = O(xσu +k−1 ) as x → ∞, where σu = supt∈R r(t).
Further, if r(t) < σu for all t ∈ R (that is, the screen lies entirely to the left ofthe line Re(s) = σu ), then R(x) = o(xσu +k−1 ) as x → ∞. Note that here, asin [14], Deﬁnition 4.22, we write that R(x) = O(xβ ) (resp. o(xβ )) as x → ∞ ifR, φa  = O(aβ ) (resp. o(aβ )) as a → ∞, where φa (x) = a−1 φ(a−1 x).
We note that the case we will be most concerned with is when k = 1 and thescreen is a vertical line to the right of 0.
Corollary 7.4. Let η be a random fractal string satisfying Assumption 7.1 withscreen given by r(t) = γ. Then P-a.s., for x → ∞, the counting function satisﬁesxs ζη (s); s = z) + o(xγ+δ ),η(x) =
Res(sz∈Dη (W)for every δ > 0.
Lemma 7.5. Assumption 7.1 holds for the random recursive fractal string under
Assumption 3.3 and it also holds for the stable string.
Proof. This is proved using the control we obtained on the zeta function from theEuler–Maclaurin approach to meromorphic continuation. We recall from (4.4) thatthe random zeta function can be written as ∞ζη (σ + it) = W Eζη (σ + it) +x−σ−it V (dx).
0

Now to control the vertical and horizontal growth of this function we use that ∞|ζη (σ + it)| ≤ W E|ζη (σ + it)| +|x−σ−it |V (dx),0≤ W E|ζη (σ + it)| + C(σ, ω)for all σ > α − τ and t ∈ R. The constant C(σ, ω) → 0 as σ → ∞.
Thus the problem is reduced to that of controlling the growth of the mean zetafunction.
(1) The stable string: For the stable string we have an explicit expression for themean zeta function, and the control we require just follows from the properties of
the Gamma function. To prove (H1) we consider Γ(s−α)/Γ(s). Note that the ratioof two Gamma functions, with the parameter of the numerator of smaller real value
than that of the denominator, is bounded as Re(s) → ∞. Hence we can choose

RANDOM FRACTAL STRINGS

311

κ = 0 and a suitable random constant C from the sum of the deterministic boundand the random control on the ﬂuctuation term to obtain (H1). For (H2) we notethat there are no poles except at α and hence we have the vertical control.
(2) The random recursive string: As we are assuming that the probability space
is ﬁnite, we can follow the ideas of [14]. For a lattice string, the poles of thezeta function occur on a ﬁnite number of straight lines, leading to (H2), while
the periodicity ensures that we can satisfy (H1). In [14] it is observed that adeterministic non-lattice self-similar string can be approximated by a sequence oflattice strings. As the structure of our mean zeta function is the same as thedeterministic case, we can also approximate the non-lattice case by the lattice case.
Thus a suitable screen can be constructed which gives a bound on the growth of themean zeta function (see [14], Theorem 2.33) and hence the result we require.

We now conclude by stating some results that follow from our earlier theorems.
Let V () denote the volume of the inner -neighbourhood of the string L. Thatis, we recall that the string is deﬁned as a bounded open subset U of the real line
and set V () = |U |, the one-dimensional Lebesgue measure of the (one-sided) neighbourhood of the boundary of the string, U = {x ∈ U : dist(x, ∂U ) < }. Thiscan be expressed as2 +lj = v , η,V () =
j:lj ≥2

j:lj <2

where v (x) = 2 for x ≤ 1/(2) and v (x) = 1/x for x > 1/(2).
The following ‘random tube formula’ is obtained by a simple adjustment of[14], Theorem 6.1, to the random string setting.
Theorem 7.6. Under the hypotheses of Theorem 7.2, the volume of the tubularneighbourhood of the boundary of the random string η is given as  → 0+ , forP-almost all ω ∈ Ω, by
V () =
(7.1)



res(

z∈Dη(ω) (W)ζη(ω) (s)(2)1−s; s = z)s(1 − s)+ 2ζη(ω) (0)I{0∈W\Dη(ω) (W)} + Rη(ω) (),where
1Rη(ω) () =2πi


(2)1−s ζη(ω) (s)S1−σu

The remainder satisﬁes Rη () = O(ds.
s(1 − s)) with σu as given in Remark 7.3.

Remark 7.7. If z is a simple pole of ζη (s), thenres(ζη(ω) (s)(2)1−s(2)1−z; s = z) = res(ζη(ω) (s); s = z).
s(1 − s)z(1 − z)We next apply Theorem 7.6 and adapt the techniques of [14], Section 6.3.1, tothe random strings we considered in Sections 4 and 5. We recall that W is the limit
random variable arising in the general branching process and L is the limit randomvariable arising from the random sequence with Poisson-Dirichlet distribution.

312

B. M. HAMBLY AND MICHEL L. LAPIDUS

Theorem 7.8. (1) The volume of the tubular neighbourhood of the random recursive string L is given P-almost surely by (7.1), where W = {s : Re(s) > α − τ },with α = df and τ as in Theorem 4.5, and R() = o(1−α+τ −δ ), for every ﬁxed
δ > 0 as  → 0+ . Moreover, for a non-lattice random recursive string, we have
V () = W Res(E(ζ(s)); s = α)(2)1−α α(1 − α) + o(1−α ), as  → 0+ , P-a.s.
(2) For the α-stable string, the volume V () of a tubular neighbourhood, for every
ﬁxed δ > 0, is given by
V () =

L(2)1−α+ o(1−α/2−δ ), as  → 0+ , Pα -a.s.
(1 − α)A consequence of this result is that, in the case of the stable string and the(strongly) non-lattice random recursive string, the boundary of the string, that is,
the associated random Cantor set, is almost surely Minkowski measurable. Some
results in this direction appear in [7].
We now turn to the eigenvalue counting function. In the case of a random fractal,
the result of [9] shows that the coeﬃcient of the leading term in the asymptotic
expansion of the counting function can be a random constant. As we are considering
a fractal boundary value problem, we observe a similar result for the second orderterm. As the string consists of a sequence of one-dimensional intervals, it is easy
to write down the eigenvalues (up to a normalization) for the Dirichlet Laplacianas klj−1 for k, j = 1, 2, . . . . Thus there is a simple relationship between the spectral
zeta function, ζsp (s), for the string and the geometric zeta function as
ζsp (z) =

∞

j=1

λsj =

∞


ks ljs = ζr (z)ζη (z),j,k=1where ζr (z) is the usual Riemann zeta function. Hence we can transfer any results
about the geometry of the string through to its spectrum.
Let Nsp (x) denote the number of eigenvalues of the Laplacian of the fractal stringwith value less than x.
Theorem 7.9. The eigenvalue counting function for the non-lattice random recursive fractal string is given by
Nsp (x) = x + W ζr (α)Res(Eζη (s); s = α)xα
+ o(xα ), as x → ∞, P-a.s.
αFor the α-stable string
Nsp (x) = x + Lζr (α)xα + o(xα/2+ ), as x → ∞, Pα -a.s.
In closing we mention an open problem that naturally arises from the resultspresented here. In [14] it was proposed that the deﬁnition of fractality should be
the existence of at least one non-real complex dimension (with positive real part).
As we have seen in Section 5 there are natural random strings for which this is not
the case. However we have not been able to establish meromorphic continuation of
the geometric zeta functions of these strings to all of C. This raises the question ofwhether there is a natural boundary for such strings beyond which the zeta function
cannot be extended.

RANDOM FRACTAL STRINGS

313

Acknowledgement
The authors would like to thank the organisers of the Isaac Newton Instituteprogramme ‘The mathematics and applications of fractals’, where this work was
initiated.
Added after posting
It may be helpful to the reader to specify that in either the introduction orin Theorem 7.9 (and the text preceding it), we have used the following implicit
convention. As in reference [14], N = N (x) really denotes the normalized frequency (rather than eigenvalue) counting function for the Dirichlet
Laplacian on√
the bounded open set U ; that is, we have the relation x = λ/π, where λ is the
eigenvalue parameter.
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Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB,
United Kingdom
E-mail address: hambly@maths.ox.ac.uk
Department of Mathematics, University of California, Riverside, California 925210135
E-mail address: lapidus@math.ucr.edu