Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft;
op gezag van de Rector Magnificus Prof. ir. K.C.A.M. Luyben; voorzitter van het College voor Promoties
in het openbaar te verdedigen op dinsdag 18 september 2012 om 10 : 00 uur
door
Derong KONG
Bachelor of Science in Mathematics Yangzhou University, China
Samenstelling promotiecommissie: Rector Magnificus, voorzitter
Prof. dr. F.M. Dekking, Technische Universiteit Delft, promotor Prof. dr. W. Li, East China Normal University, Shanghai Prof. dr. V. Komornik, Universit´e de Strasbourg, Frankrijk Prof. dr. R.W. Meester, Vrije Universiteit, Amsterdam Prof. dr. J. Bruining, Technische Universiteit Delft Prof. dr. A.W. Heemink, Technische Universiteit Delft Dr. C. Kraaikamp, Technische Universiteit Delft
Copyright c 2012 by Derong Kong
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without the prior permission of the author.
ISBN: 978-94-6203-051-0
1 Introduction 1
1.1 Fractals . . . 2
1.2 Moving plumes . . . 9
1.3 List of publications . . . 15
2 Intersections of homogeneous Cantor sets and beta expansions 17 2.1 Introduction . . . 17
2.2 Geometrical description of Γβ,N ∩ (Γβ,N + t) . . . 20
2.3 The Self-similar structure of Γβ,N ∩ (Γβ,N + t) . . . 26
2.4 The critical point forUβ,±N . . . 29
2.5 The critical point forSβ,±N . . . 39
2.6 Final remarks . . . 43
3 Self similarity of generalized Cantor sets 45 3.1 Introduction . . . 45
3.2 Preliminaries and the main results . . . 48
3.3 Proof of Theorem 3.2.3 . . . 51
3.4 Proof of Theorem 3.2.4 . . . 53
3.5 Proof of Theorem 3.2.5 . . . 59
3.6 Intersections of generalized Cantor sets . . . 62
3.7 Final remarks and open questions . . . 64
4 Multimodality of the Markov binomial distribution 67 4.1 Introduction . . . 67
4.2 The variance of the Markov binomial distribution . . . 69
4.4 The probability mass function of the Markov binomial distribution . . . 73
4.5 The conditional probability mass functions . . . 81
5 A simple stochastic kinetic transport model 85 5.1 Introduction . . . 85
5.2 The PDE reactive transport model . . . 86
5.3 A simple stochastic reactive transport model . . . 87
5.4 Probability generating functions of Kn, KnF and KnA . . . 88
5.5 Towards continuous time . . . 90
5.6 Densities and partial differential equations . . . 91
5.7 Moments of S(t), SF(t) and SA(t) . . . . 94
5.8 Double-peak behavior in reactive transport models . . . 96
5.9 Final remarks . . . 99
6 Plumes in kinetic transport: how the simple random walk can be too simple 101 6.1 Introduction . . . 101
6.2 A discrete time stochastic model for kinetic transport . . . 102
6.3 Conditional variance . . . 104
6.4 A symmetry property of asymmetric random walk . . . 106
6.5 Forty five degree dispersion . . . 108
6.6 Gaussian dispersion . . . 111
References 115
Index 121
Acknowledgements 123
Summary 125
Samenvatting (Dutch summary) 127
Introduction
Intersections of Cantor sets play an important role in planar homoclinic bifurca-tions from nonlinear dynamical systems (cf. [PT93], see also [DH95]). It is exciting that such an intersection can be a self-similar set, which is the most basic fractal set, defined in Section 1.1. Recently, Deng et al [DHW08] gave a necessary and sufficient condition for which the intersection of Cantor set with its translation is a self-similar set. Later Li et al [LYZ11] generalized these results to the N -part homogeneous Can-tor sets Γβ,N for β ≤ 1/(2N − 1). Here β corresponds to the scale parameter in the
construction of N -part homogeneous Cantor set. However, when β > 1/(2N − 1) the intersection presents a more complicated structure and the analysis is intricate. Using some techniques from beta expansions we give in Section 1.1.2 two critical points which are important to study the self-similar structure of intersections of homogeneous Cantor set with its translations. In Section 1.1.3 we consider the intersections of generalized Cantor sets and give a sufficient and necessary condition for which the intersection is a homogeneously generated self-similar set.
Reaction diffusion equations are widely used in civil engineering literature (cf. [MK00]). Simulations are always performed by using a large number of particles. The solution can be explained as the contaminant plume spreads in the ground water. The movement of the plume, terminologically called the kinetic transport, is influenced by dispersion, advection and reaction between the free and the adsorbed parts of the plume. From the microscopic way the moving plume contains tens of thousands of molecules which behave independently of each other. By using the laws of large numbers, we introduce in Section 1.2.2 a discrete time microscopic single particle model for the kinetic
trans-port, and a continuous time model will be introduced in Section 1.2.3. To model the kinetics, we consider a two-state {F, A} Markov chain {Zk, k ≥ 1}. It is important
to study how long the process {Zk, k ≥ 1} stays in state F up to time n ≥ 1, i.e.,
Kn=Pnk=11{Zk=F }, which has a Markov binomial distribution.
Motivated by the double peaking behavior of the concentration of the free part of the solute in the reaction diffusion equations model, we study in Section 1.2.1 the shape of the Markov binomial distribution. Surprisingly, its shape can be unimodal, bimodal or trimodal.
We also consider in Section 1.2.4 the two dimensional stochastic kinetic transport models with respectively two dimensional ninety degree dispersion, forty five degree dispersion and Gaussian dispersion. In particular, we study the lateral spread of the plume. Some strange phenomenon appears when we consider ninety degree dispersion.
1.1
Fractals
Fractals come into great interest until Benoit Mandelbrot wrote his foundational essay in 1975 (cf. [Man82]). Since then, fractal geometry has attracted widespread attention. From a mathematical point of view fractals and fractal measures have many applications to dynamical systems as strange attractors, to number theory via Dio-phantine approximation, to probability theory via Brownian motion and so on (cf. [Fal90]).
Fractal dimensions, such as Hausdorff dimension, packing dimension, box-counting dimension etc, are the central topics in fractal geometry. Hausdorff dimension, as the most important fractal dimension, is defined in terms of Hausdorff measures. Let E be a subset of Rn and s ≥ 0. For all δ > 0 we define
Hs δ(E) := inf nX∞ i=1 |Ui|s : |Ui| ≤ δ for all i, E ⊆ ∞ [ i=1 Ui o , (1.1)
where | · | stands for the Euclidean metric in Rn. The s-dimensional Hausdorff measure of the set E is defined as
Hs(E) := lim δ→0H
s
δ(E). (1.2)
It can be shown that Hs(·) is a Borel regular measure on Rn (cf. [Mat95]). Hausdorff measure generalizes Lebesgue measure, so that H1(E) gives the ‘length’ of a set or curve E, H2(E) gives the (normalized) ‘area’ of a region or surface, etc.
It is easy to show from Equation (1.1) and (1.2) that for all sets E ⊆ Rn there
is a number dimHE, called the Hausdorff dimension of E, such that Hs(E) = ∞ if
s < dimHE and Hs(E) = 0 if s > dimHE. Thus
dimHE = inf{s :Hs(E) = 0} = sup{s :Hs(E) = ∞}.
When s = dimHE the Hausdorff measure Hs(E) can be zero or infinite, but in some
nicest situation 0 <Hs(E) < ∞.
Self-similar sets, as basic sets having a fractal structure, were first theoretically studied by Hutchinson [Hut81] in 1981. Let K be the class of all nonempty compact subsets of Rn. ThenK is a complete metric space with the Hausdorff metric ρ defined by
ρ(E, F ) := inf{δ : E ⊆ Fδ, F ⊆ Eδ},
where Aδ := {x ∈ Rn : |x − a| ≤ δ for some a ∈ A} denotes a δ-neighborhood of a set
A ∈K.
A map S : Rn → Rn is called a similitude if there is r, 0 < r < 1, such that
|S(x) − S(y)| = r|x − y| for x, y ∈ Rn. Here r is called the contraction ratio of the
similitude. Let {S1, · · · , SN} be a finite sequence of similitudes with contraction ratios
r1, · · · , rN, and let ¯S be the map onK defined by
¯ S(E) := N [ i=1 Si(E) for E ∈K.
It is easy to check that ¯S is a contraction map on the complete metric space (K, ρ). Hence, by using Banach fixed-point theorem (cf. [Rud87]) ¯S has a unique fixed point K ∈K, i.e., K = SNi=1Si(K). Moreover, for any set E ∈K,
¯ Sk(E) = N [ i1=1 · · · N [ ik=1 Si1 ◦ · · · ◦ Sik(E) 7−→ K
in the Hausdorff metric ρ as k → ∞. The attractor K is called the self-similar set gen-erated by the itgen-erated function system (IFS) {S1, · · · , SN}. The Hausdorff dimension
dimHK is given as the unique positive solution of (cf. [Hut81]) N
X
i=1
Figure 1.1: The classical middle-third Cantor set for the first few levels.
and its corresponding Hausdorff measure 0 <HdimHK(K) < ∞.
The classical middle-third Cantor set C is a self-similar set; one can easily write down an IFS generating C as S1(x) = x/3, S2(x) = x/3 + 2/3 (see Figure 1.1). It
then follows from Equation (1.3) that its Hausdorff dimension dimHC = log 2/ log 3.
Moreover, by a intricate computation one can find that Hlog32(C) = 1. Obviously, the unit interval [0, 1] is a self-similar set having Hausdorff dimension 1 and Hausdorff measure 1.
Generally, given a fractal set K it is not so easy to determine whether K is a self-similar set or not. This is one of the main problems in fractal geometry (cf. [Fal97]). However, we have positive answers when we study the intersections of Cantor set with its translations.
1.1.1 Intersections of homogeneous Cantor sets
In the past two decades, intersections of Cantor sets have been studied by sev-eral authors (cf. [DH95, KP91, Kra92, Kra94, Kra00, LX98b]). Recently, Deng et al [DHW08] have given a necessary and sufficient condition for which C ∩ (C + t) is a self-similar set, where t is a real number, C is the classical middle third Cantor set plotted in Figure 1.1 and C + t := {c + t : c ∈ C}. These results were extended to the
case Γβ,N ∩ (Γβ,N + t) with N ≥ 2 and β ∈ (0, 1/(2N − 1)] by Li et al [LYZ11]. Here
Γβ,N, called a N -part homogeneous Cantor set, is a self-similar set generated by the
IFS {φd(·)}N −1d=0, where
φd(x) = βx + d(1 − β)/(N − 1), x ∈ R. (1.4)
Obviously, Γ1/3,2 is the classical middle-third Cantor set. Figure 1.2 is the 3-part
homogenous Cantor set Γ1/4,3.
Figure 1.2: The 3-part homogeneous Cantor set Γ1/4,3for the first few levels.
Let ΩN = {0, 1, · · · , N − 1}. An important tool in the study of self-similar sets is
that one can establish the surjective map πN : Ω∞N → Γβ,N by letting
πN({dj}∞j=1) = lim k→∞φd1 ◦ φd2◦ · · · ◦ φdk(0) = ∞ X j=1 djβj−1(1 − β) N − 1 . (1.5)
Thus one can rewrite the N -part homogeneous Cantor set Γβ,N in an algebraical way
as Γβ,N = πN(Ω∞N) = nX∞ j=1 djβj−1(1 − β) N − 1 : dj ∈ ΩN, j ≥ 1 o . Note that Γβ,N ∩ (Γβ,N + t) 6= ∅ if and only if t ∈ Γβ,N − Γβ,N,
where we denote for a real number a, and sets A, B ⊆ R, aA := {ax : x ∈ A}, A + B := {x + y : x ∈ A, y ∈ B} and A + a := A + {a}. It then follows from Equation (1.5) that the difference set Γβ,N − Γβ,N can be rewritten as
Γβ,N − Γβ,N = nX∞ j=1 djβj−1(1 − β) N − 1 : dj ∈ Ω±N, j ≥ 1 o ,
where Ω±N = ΩN − ΩN = {0, ±1, · · · , ±(N − 1)}. We call the infinite sequence
{dj}∞
β ∈ (1/(2N − 1), 1), some t ∈ Γβ,N − Γβ,N may have multiple Ω±N-codes. This gives
the intersection Γβ,N ∩ (Γβ,N + t) a more complicated structure, since it follows from
[LX98b] that for t ∈ Γβ,N − Γβ,N, Γβ,N ∩ (Γβ,N + t) = [ ˜ t πN O∞ j=1 ΩN ∩ (ΩN + tj) ,
where the union is taken over all Ω±N-codes ˜t = {tj}∞j=1 of t. Since Γβ,N = [0, 1] when
β ∈ [1/N, 1), it is interesting to ask to what extent will the intersection Γβ,N∩(Γβ,N+t)
with β ∈ (1/(2N − 1), 1/N ) still be a self-similar set.
An infinite sequence {dj}∞j=1∈ Ω∞N is called strongly eventually periodic (SEP) with
period p ∈ N (cf. [DHW08]) if there exist two finite sequences {a`}p`=1, {b`}p`=1 ∈ ΩpN
such that
{dj}∞j=1= {a`}`=1p {a`+ b`}p`=1,
where {c`}∞`=1 strands for an infinite repetition of a finite string {c`}p`=1 ∈ ΩpN. Clearly,
a periodic sequence {a`}p`=1 with period p is SEP with period p since {a`}p`=1 = {a`}p`=1{a`+ 0}p`=1. Obviously, a SEP sequence is eventually periodic. But it is not true the other way around. For example, the sequence 1 0∞ is eventually periodic but not SEP.
We will show in Chapter 2 that if t ∈ Γβ,N− Γβ,N has a unique Ω±N-code {tj}∞j=1,
then Γβ,N∩ (Γβ,N+ t) is a self-similar set if and only if {N − 1 − |tj|}∞j=1is SEP. Then it
is natural to ask whether there exists a t ∈ Γβ,N− Γβ,N which has a unique Ω±N-code.
If so, how many of them? These questions will be positively answered by using some techniques from beta expansions.
1.1.2 Beta expansions and critical points
Given m ≥ 2 and β ∈ (1/m, 1), let Ωm := {0, 1, . . . , m − 1}. The sequence {s`}∞`=1 ∈
Ω∞m is called a β-expansion of x with digit set Ωm if we can write
x =
∞
X
`=1
s`β` with s` ∈ Ωm.
The largest number we can obtain in this way is xmax := (m − 1)β/(1 − β). Clearly
1 ∈ (0, xmax] since β > 1/m. Now for any x ∈ (0, xmax], let us define a sequence
{s`}∞
if s` is already defined for all ` < n, then let snbe the largest element in Ωm satisfying
Pn
`=1s`β` < x. Obviously, P∞`=1s`β` = x, and we call {s`}∞`=1 the quasi-greedy
β-expansion of x with digit set Ωm. It is easy to see that {s`}∞`=1 is an infinite expansion
(i.e., infinitely many s` are non-zeros).
Let {λ`}∞`=1 ∈ Ω ∞
2N −1 be the sequence defined by (cf. [KL02])
λ1 = N, λ2n+1 = λ2n+ 1 = 2N − 1 − λ2n for n = 0, 1, . . . , λ2n+` = λ`= 2N − 2 − λ` for 1 ≤ ` < 2n, n = 1, 2, . . . . Then the sequence {λ`}∞`=1 starts with
N (N − 1)(N − 2)N (N − 2)(N − 1)N (N − 1) (N − 2)(N − 1)N (N − 2) . . . .
Let βc= βc(N ) be the unique positive solution of the following equation: ∞
X
`=1
λ`β`= 1.
We remark here that βc is a transcendental number. Let Uβ,±N be the set of
t ∈ Γβ,N− Γβ,N which has a unique Ω±N-code. Using techniques from beta expansions
we will show in Chapter 2 that βc is the critical point for the set Uβ,±N, i.e., for
βc< β < 1/N the setUβ,±N contains countably infinite many elements, for β = βc the
set Uβ,±N contains uncountably infinite many elements but has Hausdorff dimension
zero, and for 1/(2N − 1) < β < βc the set Uβ,±N has positive Hausdorff dimension.
Similarly, let Sβ,±N be the set of t ∈Uβ,±N such that Γβ,N ∩ (Γβ,N + t) is a
self-similar set. We also show in Chapter 2 that αc = [N + 1 −p(N − 1)(N + 3)]/2 is
the critical point for the set Sβ,±N, i.e., for αc ≤ β < 1/N the set Sβ,±N contains
countably infinite many elements, and for 1/(2N − 1) < β < αc the set Sβ,±N has
positive Hausdorff dimension.
1.1.3 Intersections of generalized Cantor sets
Let 0 < β < 1, and let Dn, n ≥ 1, be nonempty subsets of Z. The generalized
Cantor set Γβ,D of typeD = N∞n=1Dn is defined as
Γβ,D:= nX∞ n=1 dnβn: dn∈ Dn, n ≥ 1 o . (1.6)
We assume that the digit sets Dn, n ≥ 1, have bounded cardinality and the sums in
The generalized Cantor set Γβ,D can also be looked at in a geometrical way. To
illustrate this construction, we assume all the digit sets Dn, n ≥ 1, are included in
{γ, γ + 1, · · · , λ} ⊆ Z for some γ < λ. Let F0 = [γβ/(1 − β), λβ/(1 − β)] and then, for
n ≥ 1, inductively define
Fk=
[
d∈Dn
ψd(Fn−1),
where ψd(x) := β(x + d) for x ∈ R. Then we obtain a monotonic decreasing sequence
of compact sets {Fn}∞n=0, and the generalized Cantor set Γβ,D can be written as
Γβ,D= ∞
\
n=0
Fn.
For example, let β = 1/4, D1 = {0, 2}, D2 = {0, 1} and Dn = {0, 1, 2} for all n ≥ 3.
Then Dn⊆ {0, 1, 2} for all n ≥ 1, and the first few generations F0, · · · , F3 of Γ1/4,D are
plotted in Figure 1.3.
Figure 1.3: The first few generations F0, F1, F2, F3 of the generalized Cantor set Γβ,D
of type D = N∞n=1Dn with β = 1/4, D1 = {0, 2}, D2 = {0, 1} and Dn = {0, 1, 2} for all
n ≥ 3.
Let 0 < β < 1/ND, where ND := supn≥1maxd,d0∈D
n(d − d
0 + 1) is the span of
D = N∞
n=1Dn. When all the digit sets Dn, n ∈ N are consecutive, i.e., there exists
some τn≥ 0 such that Dn= {an, an+ 1, · · · , an+ τn} ⊆ Z, the authors in [KLD10] and
[LYZ11] gave a necessary and sufficient condition for which Γβ,Dis a self-similar set—in
fact we will show in Chapter 3 that it is also a necessary and sufficient condition for Γβ,D
to be a self-similar set generated by an IFS {fi(x) = rx + ai : i ∈ I}. However, when
the digit set Dnis not consecutive for some n, we know nothing about the self-similar
By analogy of the definition of strong eventual periodicity of a sequence defined in Section 1.1.1, we call a sequence of sets {Dn}∞n=1 with Dn ⊆ Z strongly eventually
periodic (SEP) with period p if there exist two finite sequence of sets {A`}p`=1, {B`}p`=1
such that
{Dn}∞n=1= {A`}`=1p {A`+ B`}p`=1,
where {C`}p`=1 denotes the infinite repetition of a finite sequence of sets {C`}p`=1. It is
easy to check that the sequence {|Dn|−1}∞n=1is SEP with period p if the sequence of sets
{Dn}∞n=1is SEP with period p, where |A| stands for the cardinality of the set A. But it is not always correct the other way around. For example, take {Dn}∞n=1= {0, 3}{0, 1, 3}.
Obviously, {|Dn|}∞n=1 = 2 3∞ = 2(2 + 1)∞ is SEP with period 1, but {Dn}∞n=1 is not
SEP.
In Chapter 3 we will show that for ND ≥ 3 and 0 < β ≤ 1/[(3ND− 1)/2] the generalized Cantor set Γβ,D of type D = N∞n=1Dn is a self-similar set generated by
an IFS {fi(x) = rx + ai : i ∈ I} if and only if the sequence of sets {Dn− γn}∞n=1 is
SEP, where [x] denotes the integer part of a real number x, and γn= min{d : d ∈ Dn}.
Moreover, we prove for 0 < β ≤ 1/(2ND− 1) that the generalized Cantor set Γβ,D is a self-similar set generated by an IFS {hk(x) = rkx + ck : k ∈ K} with rk= βqk for some
qk ∈ N if, and only if, the sequence of sets {Dn− γn}∞n=1 is SEP. These results can be
applied to study the self-similarity of intersections of generalized Cantor sets.
1.2
Moving plumes
Moving plumes as solutions of reaction diffusion equations are always numerically studied in the civil engineering literature (cf. [MK00]). Given is a solute that has an adsorbed part that does not move, and a free part that moves in the x-direction by advection and dispersion. Let CF(t, x) and CA(t, x) denote the concentration functions
of the free and the adsorbed parts of the solute at time t and at position x. Then the reaction diffusion equations are expressed as
∂CF(t, x) ∂t + ∂CA(t, x) ∂t = D ∂2C F(t, x) ∂x2 − v ∂CF(t, x) ∂x , ∂CA(t, x) ∂t = −µCA(t, x) + λCF(t, x). (1.7)
Here D is called the dispersion coefficient and v the advection velocity. The parame-ters λ and µ denote respectively the rates of changes from free to adsorbed and from adsorbed to free.
Figure 1.4: The free concentration function CF(t, ·) with R = 20, v = L = DaI =
1.0, D = 0.01 and t∗= 4.2. The left picture is a copy from [MK00] using simulation with 102400 particles; and the right picture is the density of SF
400(t) in our stochastic model.
Double peaks in the free concentration function CF are discussed by Michalak and
Kitanidis in [MK00] using simulations. They point out that the double peaking behavior of CF is controlled by the so called Damk¨ohler number of the first kind DaI = µLR/v
and the dimensionless time t∗= λt, where R = 1 + λ/µ is the dimensionless retardation coefficient and L is the length of the initial solute. This will be explained by our stochastic kinetic transport model in Section 1.2.2 (for example, see Figure 1.4). Later we generalize our stochastic model to the two dimensional case in Section 1.2.4 (for example, see Figure 1.5).
Figure 1.5: The density fS(50) and a contour plot of the density with ν = π, α = β =
1.2.1 Multimodality of Markov binomial distribution
Motivated by the double peaking behavior of the concentration of the free part of the solute in the partial differential equations model (see Figure 1.4), we study the shape of the Markov binomial distribution.
Let {Zk, k ≥ 1} be a Markov chain on the two states {F, A} with initial distribution
ν = (νF, νA) and transition matrix
PF,F PF,A PA,F PA,A = 1 − a a b 1 − b , (1.8)
where we assume 0 < a, b < 1. The Markov binomial distribution (MBD) is defined for n ≥ 1 as the distribution of the random variable Kn, which counts the occupation time
of the process {Zk, k ≥ 1} staying in state F up to time n:
Kn= n
X
k=1
1{Zk=F }. (1.9)
Clearly the MBD generalizes the binomial distribution, where a + b = 1 and (νF, νA) =
(b, a).
A finite sequence of real numbers {xi}ni=0 is said to be unimodal if there exists an
index 0 ≤ n∗ ≤ n, called a mode of the sequence, such that x0 ≤ x1 ≤ · · · ≤ xn∗ and xn∗ ≥ xn∗+1 ≥ · · · ≥ xn. Clearly, a monotonic sequence is unimodal. A nonnegative sequence {xi}ni=0 is called log-concave (or strictly log-concave) if xi−1xi+1 ≤ x2i (or
xi−1xi+1 < x2i) for all 1 ≤ i ≤ n − 1. It is well known that log-concavity implies
unimodality (cf. [Pit97]).
Figure 1.6: The probability mass function f50of K50with ν = (0.8, 0.2). In the left graph
a = b = 0.15; in the middle graph a = 0.05, b = 0.15; and in the right graph a = b = 0.05.
We will study in Chapter 4 the shape of the probability mass function fn of Kn.
The authors in [VBB94] mentioned that the shape of fn can be unimodal, bimodal or
mass function fnis unimodal, and that the probability mass function fnof Knrestricted
to the interval [1, n − 1] is always unimodal (see Figure 1.6 for an example) by proving the log-concavity of the sequence {fn(k)}n−1k=1.
1.2.2 Stochastic kinetic transport model
The moving plume, as the solution of reaction diffusion equations (1.7), contains tens of thousands of molecules (particles) behaving independently of each other. Using the laws of large numbers we will introduce in Chapter 5 a simple stochastic kinetic transport model to describe the behavior of a single particle in the plume. Time t is discretized by choosing some n, and dividing [0, t] into n intervals of the same length
∆t = t/n.
We suppose in such an interval of length ∆t that the particle can only be in one of the two states: ‘free’ or ‘adsorbed’, which we code by the letters F and A. The particle can only move when it is ‘free’, and in this case its displacement has two components: dispersion and advection. Let Xk, k ≥ 1 be the displacement of the particle due to the
dispersion the kth time that it is ‘free’. We model the Xk as independent identically
distributed random variables satisfying
Eν[Xk] = 0, EνXk2 = 2D∆t and EνXk3 = o(∆t) as ∆t ↓ 0,
where D > 0, and ν = (νF, νA) is the initial distribution describing the state of the
particle at time 0. When the particle is free during the interval [(k − 1)∆t, k∆t] for some k, the displacement due to advection is given by v∆t with v the (deterministic) advection velocity. The kinetics, i.e., the movement from ‘free’ to ‘adsorbed’ and vice versa, is modeled by a two state Markov chain {Zk, k ≥ 1} defined in Section 1.2.1 by
substituting
a = λ∆t and b = µ∆t
in the transition matrix (1.8), since the probability that a particle changes its state is proportional to the length of the time step ∆t. Then the Markov binomial distributed random variable Kn in (1.9) counts the number of intervals [(k − 1)∆t, k∆t] when the
Now let Sn(t) be the position of the particle at time t = n∆t, where Sn(0) = 0.
Then by the above we can write Sn(t) as
Sn(t) = Kn X
k=1
(Xk+ v∆t). (1.10)
Here we assume that Kn is independent of the dispersion Xk, k = 1, . . . , Kn.
In a similar way, let Snτ(t), τ ∈ {F, A} be the position of the particle at time t = n∆t conditioned on the particle is in state τ at time t. These two conditional random variables SnF(t), SnA(t) are used to model the behavior of the free and adsorbed parts of the solutes in the moving plume.
We will show in Chapter 5 that the random variables Sn(t), SnF(t) and SAn(t)
con-verges in distribution to some random variables S(t), SF(t) and SA(t) respectively by letting the time step ∆t go to zero. Moreover, we show for instantaneous injection of the solute, i.e., the initial distribution ν = (1, 0), that the partial densities of SF(t) and SA(t) do exist, and satisfy the partial differential equations (1.7). Our model explains, via the multimodality of the Markov binomial distribution studied in Section 1.2.1, the double peaking behavior of the concentration of the free part of the solutes in the partial differential equations model.
1.2.3 Continuous time model
The random variables S(t), SF(t) and SA(t), as the limits of discrete random vari-ables involving discrete time Markov chains, are important to study the reaction diffu-sion equations as described in Section 1.2.2. It is worthwhile to compare these results with a model that involves a continuous time Markov chain.
Let Y (t), t ≥ 0 denote the state of the particle at time t. Recall from Section 1.2 that λ and µ are the rates of changes from ‘free’ to ‘adsorbed’ and from ‘adsorbed’ to ‘free’ respectively. It is natural to model the kinetics by a two-state continuous time Markov chain {Y (t), t ≥ 0} with initial distribution P (Y (0) = τ ) = ντ, τ ∈ {F, A} and
generator matrix Q = QF,F QF,A QA,F QA,A = −λ λ µ −µ .
Let U (t) be the occupation time of the chain {Y (t), t ≥ 0} in state F up to time t, and let fU (t) be its probability density function. We model the displacement of the solute in the free phase as a Brownian motion with drift v. Then the position H(t) of
the particle at time t can be written as a normal distribution with mean v U (t) and variance 2D U (t), i.e.,
H(t) =N(vU(t), 2DU(t)).
One can show that H(t) and S(t) have the same characteristic function. Hence by Lerch’s theorem (cf. [Sch99, page 24]) H(t) and S(t) have the same distribution.
For τ ∈ {F, A} let Hτ(t) = H(t) | {Y (t) = τ } be the conditional random variable denoting the position of the particle at time t conditioned on it is in state τ at time t. Similarly one can show that Hτ(t) and Sτ(t) have the same distribution.
1.2.4 Plume shapes
We consider in Chapter 6 a discrete time two dimensional stochastic kinetic trans-port model, which is an extension of the one dimensional stochastic model described in Section 1.2.2. We point out that going from 1D to 2D introduce some new prob-lems, but that the 3D case is very similar to the 2D case. Some strange phenomenon appears when we describe it by a two dimensional simple random walk with a drift in the x-direction.
Let (Xk, Yk) be the displacement of the particle due to the dispersion the kth time
that it is ‘free’. We model the (Xk, Yk), k ≥ 1 as independent identically distributed
random vectors. When the particle is free, the displacement during a unit time interval due to advection is given by one unit in the x direction.
Now let S(n) be the position of the particle at time n, where S(0) = (0, 0). Then by an obvious extension of the one dimensional case in (1.10), we can write S(n) for n ≥ 1 as S(n) = (SX(n), SY(n)) = Kn X k=1 (Xk+ 1, Yk).
where Kn is the Markov binomial distribution defined in Equation (1.9).
We are interested in the spread of the plume. A natural quantitative way to measure the lateral spread of the plume at position x at time n is to consider the conditional variance σx2 = Var (SY(n) | SX(n) = x). Intuitively, σx2 should be increasing in x. This
is the case (for example, see the second and last row of Figure 1.7) when we consider forty five degree dispersion:
Figure 1.7: The probability mass function of S(50), a list contour plot of its mass function and the conditional variance Var (SY(50) | SX(50) = x) with ν = (0.5, 0.5) and α = β =
1/4. In the first row we have ninety degree dispersion; in the second row we have forty five degree dispersion; and in the last row we have Gaussian dispersion.
and the natural Gaussian dispersion:
(Xk, Yk) d =N(0, 0), 2α 0 0 2β , where α, β ≥ 0 and α + β = 1/2.
However, this is not the case (for example, see the first row of Figure 1.7) when we consider ninety degree dispersion
P ((Xk, Yk) = (j, 0)) = α, P ((Xk, Yk) = (0, j)) = β for j = ±1.
Here too, α, β ≥ 0, α + β = 1/2. When Knfollows a binomial distribution, S(n) is then
a two dimensional simple random walk with a drift in the x-direction. Surprisingly, there is symmetry in σx2, i.e., σx2 = σ2n−x2 .
1.3
List of publications
Ch. 2: DeRong Kong, Wenxia Li, and F. Michel Dekking. Intersections of homogeneous Cantor sets and beta-expansions. Nonlinearity, 23 (11): 2815-2834, 2010.
Ch. 3: DeRong Kong. Self similarity of generalized Cantor sets. Preprint arXiv: 1207.3652v1, 2012.
Ch. 4: Michel Dekking and DeRong Kong. Multimodality of the Markov binomial dis-tribution. Journal of Applied Probability, 48: 938-953, 2011.
Ch. 5: Michel Dekking and DeRong Kong. A simple stochastic kinetic transport model. To appear in Advances in Applied Probability, 2012.
Ch. 6: Michel Dekking and DeRong Kong. Plumes in kinetic transport: how the simple random walk can be too simple. To appear in Stochastic Models, 2012.
Intersections of homogeneous
Cantor sets and beta expansions
Let Γβ,N be the N -part homogeneous Cantor set with β ∈ (1/(2N − 1), 1/N ). Anystring (j`)∞`=1with j` ∈ {0, ±1, . . . , ±(N − 1)} such that t =P∞`=1j`β`−1(1 − β)/(N − 1)
is called a code of t. Let Uβ,±N be the set of t ∈ [−1, 1] having a unique code, and
let Sβ,±N be the set of t ∈ Uβ,±N which make the intersection Γβ,N ∩ (Γβ,N + t)
a self-similar set. We characterize the set Uβ,±N in a geometrical and algebraical
way, and give a sufficient and necessary condition for t ∈ Sβ,±N. Using techniques
from beta-expansions, we show that there is a critical point βc ∈ (1/(2N − 1), 1/N ),
which is a transcendental number, such that Uβ,±N has positive Hausdorff dimension
if β ∈ (1/(2N − 1), βc), and contains countably infinite many elements if β ∈ (βc, 1/N ).
Moreover, there exists a second critical point αc = N + 1 −p(N − 1)(N + 3) /2 ∈
(1/(2N − 1), βc) such that Sβ,±N has positive Hausdorff dimension if β ∈ (1/(2N −
1), αc), and contains countably infinite many elements if β ∈ [αc, 1/N ).
2.1
Introduction
Let {fi(x) = rix + bi}pi=1 be a family of functions on R with 0 < |ri| < 1. It is well
known (cf. [Fal90]) that there exists a unique nonempty compact set Γ ⊆ R such that Γ =
p
[
i=1
In this case, Γ is called the self-similar set generated by the iterated function system (IFS) {fi(·)}pi=1.
We will be interested in the self-similar set Γβ,Ω generated by an IFS {φd(·) : d ∈ Ω},
where Ω is a finite set of integers, and
φd(x) = βx + d(1 − β)/(N − 1), x ∈ R
for some N ≥ 2 and β ∈ (0, 1/N ). It is well known that one can establish a surjective map πΩ: Ω∞→ Γβ,Ω by letting πΩ(J ) = ∞ X `=1 j`β`−1(1 − β) N − 1 (2.1) for J = (j`)∞`=1∈ Ω
∞. The infinite string J is called an Ω-code of π
Ω(J ). Note that an
element x ∈ Γβ,Ω may have multiple Ω-codes. These Ω-codes are closely related to the
classical beta-expansions (cf. [EJK90, GS01, KL02, Par60, R´en57, Sid07, dVK09]). A sequence (s`)∞`=1 ∈ Ω∞ is called a β-expansion of x with digit set Ω if we can write
x =
∞
X
`=1
s`β`, s` ∈ Ω.
Let ΩN := {0, 1, . . . , N − 1}. We simplify the notation Γβ,ΩN to Γβ,N, so this set satisfies
Γβ,N =
[
d∈ΩN
φd(Γβ,N).
The set Γβ,N is called the N -part homogeneous Cantor set. Thus Γ1/3,2 is the classical
middle-third Cantor set and Γβ,2 is the middle-α Cantor set with α = 1 − 2β.
In terms of (2.1), let πN := πΩN. Thus we can rewrite Γβ,N as
Γβ,N = πN Ω∞N = (∞ X `=1 j`β`−1(1 − β) N − 1 : j` ∈ ΩN, ` ≥ 1 ) . (2.2)
We consider the intersection of Γβ,N with its translation by t. It is easy to check that
Γβ,N ∩ (Γβ,N + t) 6= ∅ if and only if t ∈ Γβ,N − Γβ,N.
Here we denote for a real number a, and sets A, B ⊆ R, aA := {ax : x ∈ A}, A + B := {x + y : x ∈ A, y ∈ B}, and A + a := A + {a}.
It follows from Equation (2.2) that the difference set Γβ,N − Γβ,N can be written as Γβ,N − Γβ,N = (∞ X k=1 t`β`−1(1 − β) N − 1 : t`∈ Ω±N ) = π±N Ω∞±N = Γβ,Ω±N,
where Ω±N := ΩN − ΩN = {0, ±1, . . . , ±(N − 1)} and π±N := πΩ±N. Since Ω2N −1 = {0, 1, . . . , 2N − 2} = Ω±N + N − 1, it is easy to see that (t`)∞`=1 is a Ω±N-code of
t ∈ Γβ,N−Γβ,N if and only if (t`+ N − 1)∞`=1is an β-expansion of (t + 1)β(N − 1)/(1 − β)
with digit set Ω2N −1. Thus some results and techniques from beta-expansions can be
used to deal with the difference set Γβ,N − Γβ,N.
In the past two decades, intersections of Cantor sets have been studied by several au-thors (cf. [DH95, KP91, Kra92, Kra94, Kra00, LX98b]). Recently, Deng et al. [DHW08] gave a necessary and sufficient condition for t ∈ [−1, 1] such that Γ1/3,2∩ (Γ1/3,2+ t) is a self-similar set. Their results were extended to the case Γβ,N ∩ (Γβ,N + t) with
β ∈ (0, 1/(2N − 1)] by Li et al. [LYZ11], and to the case Γβ,2 ∩ (Γβ,2 + t) with
β ∈ (1/3, 1/2) and t having a unique Ω±2-code by Zou et al. [ZLL08].
In this chapter we consider arbitrary N ≥ 2, and β ∈ (1/(2N − 1), 1/N ). Then Lebesgue a.a. t ∈ Γβ,N− Γβ,N = [−1, 1] have a continuum of distinct Ω±N-codes. This
gives the set Γβ,N∩ (Γβ,N+ t) a more complicated structure. We summarize the results
in the following. In Section 2.2, an algebraical and geometrical description of the set
Uβ,±N :=t ∈ [−1, 1] : |π−1±N(t)| = 1
(i.e., the set of t ∈ [−1, 1] having a unique Ω±N-code) is given in Theorem 2.2.2, where
throughout the paper |A| denotes the number of members in the set A. Section 2.3 is mainly devoted to investigating the self-similar structure of Γβ,N ∩ (Γβ,N + t). Let
Sβ,±N :=t ∈Uβ,±N : Γβ,N ∩ (Γβ,N + t) is a self-similar set .
Theorem 2.3.2 gives a sufficient and necessary condition for t ∈ Sβ,±N. In Section 4,
we study the set Uβ,±N for different β ∈ (1/(2N − 1), 1/N ) culminating in Theorem
2.4.6. Using techniques from beta-expansions, we obtain a critical point βc∈ (1/(2N −
1), 1/N ) such that Uβ,±N has positive Hausdorff dimension if β ∈ (1/(2N − 1), βc),
and contains countably infinite many elements if β ∈ (βc, 1/N ). We point out that
the critical point βc is a transcendental number which is related to the famous
[N + 1 −p(N − 1)(N + 3) ]/2 ∈ (1/(2N − 1), βc) (see Theorem 2.5.1) such thatSβ,±N
has positive Hausdorff dimension if β ∈ (1/(2N −1), αc), and contains countably infinite
many elements if β ∈ [αc, 1/N ). In the following table, we give the critical points βc=
βc(N ) and αc= αc(N ) calculated for different integers N by means of Mathematica.
N 2 3 4 5 6 7 8 9
βc≈ 0.39433 0.27130 0.21004 0.17221 0.14625 0.12722 0.11265 0.10111
αc≈ 0.38197 0.26795 0.20871 0.17157 0.14590 0.12702 0.11252 0.10102
Thus for β ∈ [αc, βc), the set Uβ,±N (the set of t ∈ [−1, 1] having a unique Ω±N
-code) has positive Hausdorff dimension, but only countably many t ∈Uβ,±N make the
intersection Γβ,N ∩ (Γβ,N + t) a self-similar set.
2.2
Geometrical description of Γ
β,N∩ (Γ
β,N+ t)
We say that the IFS {fi(·)}pi=1satisfies the open set condition (OSC) if there exists
a nonempty bounded open set O ⊆ R such that O ⊇Sp
i=1fi(O), with a disjoint union
on the right side. An IFS {fi(·)}pi=1 is said to satisfy the strong separation condition
(SSC) if the union Γ =Sp
i=1fi(Γ) is disjoint.
When β ∈ (0, 1/(2N − 1)) the IFS {φd(·) : d ∈ Ω±N} satisfies the SSC, so each
point in Γβ,Ω±N has a unique Ω±N-code. In case β = 1/(2N − 1), the IFS {φd(·) : d ∈ Ω±N} fails to satisfy the SSC but satisfies the OSC, so each point has a unique
Ω±N-code except for countably many points having two Ω±N-codes. However, for the
case β ∈ (1/(2N − 1), 1/N ) the IFS {φd(·) : d ∈ Ω±N} fails to satisfy the OSC and
Γβ,Ω±N = [−1, 1]. In this case, Lebesgue a.a. t ∈ [−1, 1] have a continuum of distinct Ω±N-codes (cf. [Sid07]). This gives Γβ,N ∩ (Γβ,N + t) a more complicated structure,
since it follows ([LX98b]) that for t ∈ Γβ,Ω±N
Γβ,N ∩ (Γβ,N + t) = [ ˜ t πN ∞ O `=1 D`,˜t ! (2.3)
where the union is taken over all Ω±N-codes of t, and for each code ˜t = (t`)∞`=1∈ Ω∞±N
D`,˜t= ΩN ∩ (ΩN + t`) = {0, 1, . . . , N − 1} ∩ ({0, 1, . . . , N − 1} + t`).
(P1) the union on the right side of (2.3) consists of pairwise disjoint sets; (P2) for each Ω±N-code ˜t = (t`)∞`=1 of t, we have
1 + t − πN ∞ O `=1 D`,˜t ! = πN ∞ O `=1 D`,˜t ! ,
i.e., πN(N∞`=1D`,˜t) is centrally symmetric. Furthermore, 1 + t − Γβ,N ∩ (Γβ,N + t) =
Γβ,N ∩ (Γβ,N + t).
These properties can be obtained as follows. Let (t`)∞`=1 be a Ω±N-code of t and let
J = (j`)∞`=1∈ Ω∞N. If πN(J ) = ∞ X `=1 j`β`−1(1 − β) N − 1 ∈ πN ∞ O `=1 ΩN ∩ (ΩN + t`) ! ,
then (j`− t`)∞`=1 ∈ Ω∞N. Note that the IFS {φd(·) : d ∈ ΩN} satisfies the SSC (since
β < 1/N ). This implies that each point x ∈ Γβ,N has a unique ΩN-code. Thus
(j`− t`)∞`=1 is the unique ΩN-code of πN(J ) − t, implying (P1). In addition, one can
check that for each ` ≥ 1,
N − 1 + t`− ΩN∩ (ΩN + t`) = ΩN ∩ (ΩN + t`),
implying (P2).
Let Ω be a nonempty finite subset of Z. Denote by ε the empty word and put Ω0 = {ε}. For I ∈ S∞ `=0Ω` and J ∈ Ω∞ ∪ S∞ `=0Ω`, let IJ ∈ Ω∞ ∪ S∞ `=0Ω` be
the concatenation of I and J . So in particular εJ = J . For a nonnegative integer
k and a finite string I ∈ S∞
`=1Ω`, let Ik := k
z }| {
I . . . I be the k times repeating of I and I∞ := III · · · ∈ Ω∞ be the infinite repeating of I. In particular, I0 = ε. For
J = (j`)∞`=1∈ Ω∞and k ∈ N, let J|k= (j`)k`=1∈ Ωk. We define the algebraic difference
between two infinite strings I = (i`)∞`=1, J = (j`)∞`=1 ∈ Ω∞ by I − J = (i`− j`)∞`=1, and
for a positive integer k let I|k− J |k= (I − J )|k= (i`− j`)k`=1.
Given β ∈ (1/(2N − 1), 1/N ) and t ∈ [−1, 1], for an integer d ∈ Z, let ψd(x) = βx + d(1 − β)/(N − 1) + t(1 − β), x ∈ R. Then Γβ,N + t = [ d∈ΩN ψd(Γβ,N + t).
For J = (j`)k`=1 ∈ ΩkN with k ∈ N, let ψJ := ψj1◦ · · · ◦ ψjk (the same for φJ). For a real number x, it is easy to see that ψd(t + x) = φd(x) + t for all d ∈ ΩN. Thus by induction
we obtain ψJ(t + x) = φJ(x) + t for all J ∈ ∞ [ `=1 Ω`N, x ∈ R. (2.4)
The sets Γβ,N and Γβ,N + t can be represented in a geometrical way as (cf. [Fal90])
Γβ,N = ∞ \ k=1 [ J ∈ΩkN φJ([0, 1]) and Γβ,N + t = ∞ \ k=1 [ J ∈ΩkN ψJ([t, 1 + t]).
We call φJ([0, 1]), ψJ([t, 1+t]) with J ∈ ΩkN thek-level components of Γβ,N and Γβ,N+t,
respectively. The 1-level components of Γβ,N are φ0([0, 1]), φ1([0, 1]), . . . , φN −1([0, 1]) of
length β. All gaps between them have the same length (1−β)/(N −1)−β. The left end-point of φ0([0, 1]) is 0 and the right endpoint of φN −1([0, 1]) is 1. For a `-level component
φJ([0, 1]), J ∈ Ω`N, the (`+1)-level components φJ 0([0, 1]), φJ 1([0, 1]), . . . , φJ (N −1)([0, 1])
have the same length β`+1 and all gaps (called (` + 1)-level gaps) between them have the same length β`(1 − β)/(N − 1) − β`+1. The left endpoint of φJ 0([0, 1]) coincides
with the left endpoint of φJ([0, 1]) and the right endpoint of φJ (N −1)([0, 1]) coincides
with the right endpoint of φJ([0, 1]). The requirement β ∈ (1/(2N − 1), 1/N ) implies
the following simple properties:
(P3) the length of a k-level gap is less than the length of a k-level component, i.e.,
βk−1(1 − β)/(N − 1) − βk< βk;
(P4) if φI([0, 1]) ∩ ψJ([t, t + 1]) 6= ∅ for I, J ∈ ΩkN with k ∈ N, then
φI([0, 1]) ∩ ψJ([t, 1 + t]) ∩ Γβ,N ∩ (Γβ,N + t) 6= ∅.
For J ∈ ΩkN with k ∈ N, the neighborhood of φJ([0, 1]) with respect to the k−level
components of Γβ,N + t is defined as (see Figure 2.1)
Nt(J ) :=
n
ψI([t, 1 + t]) : I ∈ ΩkN, φJ([0, 1]) ∩ ψI([t, 1 + t]) 6= ∅
o .
The setNt(J ) may be empty and |Nt(J )| ∈ {0, 1, 2}. For k ≥ 1 let
Λk:= n J ∈ ΩkN : |Nt(J )| ≥ 1 o and Λ := n J ∈ Ω∞N : J |k∈ Λk for all k ∈ N o .
Figure 2.1: N = 3, β = 0.28, t = 0.19. The 1-level components of Γβ,Nare φ0([0, 1]), φ1([0, 1])
and φ2([0, 1]). The 1-level components of Γβ,N+ t are ψ0([t, 1 + t]), ψ1([t, 1 + t]) and ψ2([t, 1 + t]).
Here Nt(0) = {ψ0([t, 1 + t])}, Nt(1) = {ψ0([t, 1 + t]), ψ1([t, 1 + t])} and Nt(2) = {ψ1([t, 1 +
t]), ψ2([t, 1 + t])}.
Then Γβ,N ∩ (Γβ,N + t) can be rewritten in a geometrical way as
Γβ,N ∩ (Γβ,N + t) = πN(Λ) = ∞ \ k=1 [ J ∈Λk φJ([0, 1]).
A set D ⊆ ΩN is said to be consecutive if D = ΩN ∩ (ΩN + d) for some d ∈ Ω±N.
Proposition 2.2.1. Given N ≥ 2 and β ∈ (1/(2N − 1), 1/N ), let t ∈ [−1, 1]. If |Nt(J )| ≤ 1 for all J ∈S∞`=1Ω`N, then
Λ =
∞
O
`=1
D`
with each D` consecutive.
Proof. The condition β ∈ (1/(2N − 1), 1/N ) implies (P3), i.e., all gaps between the intervals φd([0, 1]), d ∈ ΩN have the same length strictly less than β, the length of
φd([0, 1]) (see Figure 2.1). Thus since t ∈ [−1, 1], either |Nt(0)| = 1 or |Nt(N − 1)| = 1,
which implies that
D1:=
n
d ∈ ΩN : |Nt(d)| = 1
o 6= ∅.
It follows from |Nt(d)| ≤ 1 for all d ∈ ΩN that D1 is consecutive and Λ1 = D1.
Now for k ∈ N let the consecutive sets D1, . . . , Dk be chosen such that Λk =
Nk
`=1D`. Fix a J ∈ Λk and take
Dk+1:=
n
d ∈ ΩN : |Nt(J d)| = 1
o .
Then Dk+1 is nonempty by (P3), and is consecutive by the same argument as above.
Note that Dk+1 is independent of the choice of J ∈ Λk. Thus Λk+1=Nk+1`=1 D` which
implies Λ =N∞
The following theorem characterizes the set of t ∈ [−1, 1] having a unique Ω±N-code
from a geometrical and an algebraical aspect.
Theorem 2.2.2. Given N ≥ 2 and β ∈ (1/(2N − 1), 1/N ), let Uβ,±N be the set of
t ∈ [−1, 1] which have a unique Ω±N-code. Then the following conditions are equivalent.
(A) t ∈Uβ,±N;
(B) |Nt(J )| ≤ 1 for all J ∈S∞`=1Ω`N;
(C) t has a Ω±N-code (t`)∞`=1 such that for all k ≥ 1
( P∞
`=1tk+`β` < 1−N β1−β , if tk< N − 1
P∞
`=1tk+`β` > −1−N β1−β , if tk> 1 − N.
(2.5)
Proof. (A) ⇒ (B). Suppose that |Nt(J )| = 2 for some J = (j`)k`=1 ∈ ΩkN with k ≥ 1.
Then either |Nt(J |k−10)| = 2 or |Nt(J |k−1(N − 1))| = 2. Without loss of generality,
let |Nt(J |k−10)| = 2. Then there exists d ∈ ΩN such that |Nt(J |k−1d)| = 1 by the
geometric structure of Γβ,N ∩ (Γβ,N + t) (see Figure 2.2).
Figure 2.2: N = 3. Here J0= J |k−11, J00= J |k−12 andNt(J0) ∩Nt(J00) = {ψI([t, 1 + t])}. Let J0 = J |k−1(d − 1) and J00= J |k−1d. Then
Nt(J0) ∩Nt(J00) =ψI([t, 1 + t])
for some I = i1i2· · · ik−1(N − 1) ∈ ΩkN. By (P4) we can pick
x ∈ φJ0([0, 1]) ∩ ψI([t, 1 + t]) ∩ Γβ,N ∩ (Γβ,N + t)
and
y ∈ φJ00([0, 1]) ∩ ψI([t, 1 + t]) ∩ Γβ,N ∩ (Γβ,N + t).
Let (x`)∞`=1and (y`)∞`=1be the unique ΩN-code of x and y, respectively. Then xk = d−1
and yk = d. On the other hand, x − t, y − t ∈ Γβ,N and by (x∗`) ∞ `=1, (y ∗ `) ∞ `=1 we denote
their unique ΩN-code, respectively. It follows from (2.4) that
x ∈ ψI([t, 1 + t]) = φI([0, 1]) + t and y ∈ ψI([t, 1 + t]) = φI([0, 1]) + t,
which imply x − t, y − t ∈ φI([0, 1]). Thus x∗k = y∗k= N − 1. Hence t = x − (x − t) =
(B) ⇒ (A). By Proposition 2.2.1, we have Γβ,N ∩ (Γβ,N + t) = πN(N∞`=1D`) with
D` consecutive. Thus, it follows from (2.3) that t has a unique Ω±N-code (t`)∞`=1 with
each t` determined by D`= ΩN ∩ (ΩN+ t`).
(B) ⇒ (C). It follows from Proposition 2.2.1 that Γβ,N∩ (Γβ,N+ t) = πN(N∞`=1D`)
with each D` consecutive. Take J = (j`)∞`=1 ∈
N∞
`=1D`. Then πN(J ) ∈ Γβ,N∩ (Γβ,N +
t). Let J∗ = (j`∗)∞`=1 be the unique ΩN-code of πN(J ) − t ∈ Γβ,N. Thus it follows by
(2.4) that for each k ≥ 1
πN(J ) ∈ φJ |k([0, 1]) ∩ (φJ∗|k([0, 1]) + t) = φJ |k([0, 1]) ∩ ψJ∗|k([t, 1 + t]), and
J − J∗= (j`− j`∗)∞`=1 = (t`)∞`=1
is the unique Ω±N-code of t (the uniqueness is given by (B) ⇒ (A)). We shall prove
(t`)∞`=1 satisfies (2.5) in the following.
Case I. tk6= ±(N − 1).
In this case, (jk, jk∗) /∈ {(N −1, 0), (0, N −1)}. This together with the requirements in
(B) imply that the distance between the left endpoints of φJ |k([0, 1]) and ψJ∗|
k([t, t + 1]) must be less than the length of the k-th gap (see Figure 2.3), i.e., |ψJ∗|
k(t) − φJ |k(0)| < βk−1(1 − β)/(N − 1) − βk.
Figure 2.3: N = 3. Here φJ |k(0) is the left endpoint of the k-level component φJ |k([0, 1]) of
Γβ,N, and ψJ∗|k(t) is the left endpoint of k-level component φJ∗|k([t, 1 + t]) of Γβ,N+ t.
Thus (2.5) follows by the following computation.
∞ X `=1 tk+`β`−1(1 − β) N − 1 = β−k ∞ X `=k+1 t`β`−1(1 − β) N − 1 = β−k t − k X `=1 t`β`−1(1 − β) N − 1 = β−k t − k X `=1 j`β`−1(1 − β) N − 1 − k X `=1 j`∗β`−1(1 − β) N − 1 ! = β−k|t − (φJ | k(0) − φJ∗|k(0))| = β −k|ψ J∗| k(t) − φJ |k(0)| < 1 − N β β(N − 1). Case II. tk= N − 1.
In this case, (jk, jk∗) = (N − 1, 0). This together with the requirements in (B) imply
that φJ |k(0) − ψJ∗|k(t) < β
I, we have ∞ X `=1 tk+`β`−1(1 − β) N − 1 = β −k(ψ J∗| k(t) − φJ |k(0)) > − 1 − N β β(N − 1), leading to (2.5).
The final case tk= 1 − N can be done in the same way as above.
(C) ⇒ (B). We will prove by induction that for any k ≥ 1 and J ∈ ΩkN
Nt(J ) = ( ψJ −(t `)k`=1([t, 1 + t]) , if J ∈ N k `=1 ΩN ∩ (ΩN + t`) ∅, otherwise. (2.6)
For k = 1, let J ∈ ΩN ∩ (ΩN + t1). In view of the proof of (B) ⇒ (C), (2.5) becomes
(
ψJ −t1(t) − φJ(0) < (1 − β)/(N − 1) − β, if t1 < N − 1 φJ(0) − ψJ −t1(t) < (1 − β)/(N − 1) − β, if t1 > 1 − N.
This implies (2.6) from the geometrical structure of Γβ,N ∩ (Γβ,N + t).
Suppose that (2.6) is true for k = n. Let J = (j`)n+1`=1 ∈ Ωn+1N . Then Nt(J ) = ∅
if J |n∈/ Nn`=1 ΩN ∩ (ΩN + t`). Thus we assume J|n∈Nn`=1 ΩN ∩ (ΩN + t`). For
jn+1∈ ΩN ∩ (ΩN+ tn+1), (2.5) becomes ( ψJ −(t `)n+1`=1(t) − φJ(0) < β n(1 − β)/(N − 1) − βn+1, if t n+1 < N − 1 φJ(0) − ψJ −(t `)n+1`=1(t) < β n(1 − β)/(N − 1) − βn+1, if t n+1 > 1 − N,
which implies (2.6) for k = n + 1.
2.3
The Self-similar structure of Γ
β,N∩ (Γ
β,N+ t)
Let Ω be a nonempty finite subset of Z. An infinite string K ∈ Ω∞ is called strongly eventually periodic (SEP) with period q if there exist two finite strings I = (i`)q`=1, J = (j`)q`=1 ∈ Ωq with q ≥ 1 such that K = I J
∞
and I 4 J, where I 4 J means i` ≤ j`, 1 ≤ ` ≤ q. For two infinite strings I, J ∈ Ω∞, we say I 4 J if I|k4 J|k
for all k ∈ N. The following lemma (cf. [LYZ11, Lemma 3.1]) gives a description of SEP infinite strings.
Lemma 2.3.1. Let (j`)∞`=1∈ Ω∞N. If there exists a positive integer q such that j`+q ≥ j`
When t has a unique Ω±N-code (t`)∞`=1, from the proof of Theorem 2.2.2 it follows
that there exists a sequence of consecutive subsets ΩN ∩ (ΩN+ t`) such that
Γβ,N ∩ (Γβ,N + t) = πN ∞ O `=1 ΩN∩ (ΩN + t`) ! .
Let γ∗ be the smallest member of Γβ,N ∩ (Γβ,N + t). It is easy to check that
Γt:= Γβ,N ∩ (Γβ,N + t) − γ∗ = πN ∞ O `=1 {0, . . . , N − 1 − |t`|} ! . (2.7)
Thus the Hausdorff and packing dimensions of Γβ,N∩(Γβ,N+t) are given by (cf. [LX98a])
dimHΓβ,N ∩ (Γβ,N + t) = dimHΓt= − 1 log βlimk→∞ Pk `=1(N − |t`|) k ; dimPΓβ,N ∩ (Γβ,N + t) = dimP Γt= − 1 log βlimk→∞ Pk `=1(N − |t`|) k .
The following properties make it easier to deal with Γt.
(P5) For I, J ∈ Ω∞N, if I 4 J and πN(J ) ∈ Γt, then πN(I) ∈ Γt;
(P6) Γt= γ∗− Γtwhere γ∗ = πN (N − 1 − |t`|)∞`=1 is the largest member in Γt.
Thus, when Γtis generated by an IFS, say {fi(x) = rix + bi}pi=1, we can require all
ri > 0 : if ri < 0 we can replace fi(x) by fi∗(x) = −rix + bi+ riγ∗. This follows from a
simple computation (cf. [DHW08, LYZ11])
fi∗(Γt) = −riΓt+ bi+ riγ∗ = ri(γ∗− Γt) + bi = riΓt+ bi = fi(Γt).
Furthermore, we can assume 0 = b1 ≤ b2≤ · · · ≤ bp since 0 = πN(0∞) ∈ Γt by (P5).
The following theorem gives a sufficient and necessary condition for t ∈Sβ,±N, i.e.,
the set of t ∈ [−1, 1] which have a unique Ω±N-code and at the same time make the
intersection Γβ,N ∩ (Γβ,N + t) a self-similar set.
Theorem 2.3.2. Given N ≥ 2 and β ∈ (1/(2N − 1), 1/N ), let (t`)∞`=1 be the unique
Ω±N-code of t ∈Uβ,±N. Then t ∈Sβ,±N if and only if (N − 1 − |t`|)∞`=1 is SEP.
Proof. It suffices to prove that Γt, given by (2.7), is a self-similar set if and only if
(N − 1 − |t`|)∞`=1 is SEP. Firstly, we prove the sufficiency. If (N − 1 − |t`|)∞`=1 ∈ Ω∞N
is strongly periodic, it can be written as (N − 1 − |t`|)∞`=1 = σ (σ + τ )∞ ∈ Ω∞N where
σ = (σ`)q`=1, τ = (τ`)q`=1∈ Ω q
N for some q ∈ N and σ + τ = (σ`+ τ`)q`=1 ∈ Ω q N. Let S := ( β−q 2q X `=1 j`β`−1(1 − β) N − 1 : Ω 2q N 3 (j`) 2q `=1 4 στ ) .
One can check that Γt can be generated by the IFS {fs(x) = βq(x + s) : s ∈ S}
(cf. [LYZ11]).
Next, we will prove the necessity. By (P6), we can assume that Γt is generated
by an IFS {fi(x) = rix + bi}pi=1 with ri ∈ (0, 1) and 0 = b1 ≤ b2 ≤ · · · ≤ bp. Note
that the union (0, 1) = S∞
q=0[βq+1, βq) is disjoint, there exist some q ≥ 0 such that
r1 ∈ [βq+1, βq).
Case I. r1 = βq+1. Then for each ` ≥ 1, it follows from (P5) that
(N − 1 − |t`|)β`−1(1 − β) N − 1 = πN 0 `−1(N − 1 − |t `|)0∞ ∈ Γt. Thus f1 (N − 1 − |t`|)β`−1(1 − β) N − 1 = (N − 1 − |t`|)β `+q(1 − β) N − 1 ∈ Γt
which implies that N − 1 − |t`| ≤ N − 1 − |t`+q+1| for each ` ≥ 1. So (N − 1 − |t`|)∞`=1
is SEP with period q + 1 by Lemma 2.3.1.
Case II. βq+1 < r1 < βq. Let r1= βq+γ with 0 < γ < 1.
(IIa) γ is rational. Take k ∈ N such that kγ ∈ N. Note that the IFS {f0(x) =
r1kx, fi(x) = rix + bi, 1 ≤ i ≤ p} generates Γt. Thus the conclusion can be proved in
the same way as that in Case I.
(IIb) γ is irrational. Take k ∈ N such that
β < β1−kγ+[kγ]< 1 − β
N − 1. (2.8)
This is possible since the set {kγ − [kγ] : k ∈ N} is dense in the interval (0, 1). Let f0(x) = r1kx. Then for some β`−1(1 − β)/(N − 1) ∈ Γt we have
f0 β`−1(1 − β) N − 1 = β kq+kγ+`−1(1 − β) N − 1 < ξ := βkq+[kγ]+`−1(1 − β) N − 1 .
Figure 2.4: ξ = (βkq+[kγ]+`−1(1 − β))/(N − 1), η = βkq+[kγ]+`. From the geometrical con-struction of Γt, it is easy to see that (η, ξ) ∩ Γt= ∅.
On the other hand, from (2.8) it follows that
βkq+kγ+`−1(1 − β)
N − 1 > η := β
kq+[kγ]+`.
Thus f0(β
`−1(1−β)
In fact, the above proof gives a general result on the structure of a class of subsets of the N -part homogeneous Cantor set.
Corollary 2.3.3. Given N ≥ 2 and β ∈ (0, 1/N ), let (i`)∞`=1, (j`)∞`=1 ∈ Ω ∞
N satisfying
(i`)∞`=1 4 (j`)∞`=1. Then πN(N∞`=1{i`, i`+ 1, . . . , j`}) is a self-similar set if and only if
(j`− i`)∞`=1 is SEP.
2.4
The critical point for
U
β,±NAccording to a result of Sidorov [Sid07, Proposition 3.8] pertaining to the general digit sets, we have that Lebesgue a.a. t ∈ [−1, 1] have a continuum of distinct Ω±N
-codes if β ∈ (1/(2N − 1), 1/N ). However, we will show in this section, for the same set of β’s, that there are infinitely many t ∈ [−1, 1] having a unique Ω±N-code. Note
that these t form exactly the set Uβ,±N defined earlier. Moreover, there is a critical
point βc ∈ (1/(2N − 1), 1/N ) such that Uβ,±N has positive Hausdorff dimension if
β ∈ (1/(2N − 1), βc), and contains countably infinite many elements if β ∈ (βc, 1/N ).
This can be seen in Theorem 2.4.6 which is proved by using techniques from beta-expansions.
Given m ≥ 2 and β ∈ (1/m, 1), let Ωm := {0, 1, . . . , m−1}. Recall that the sequence
(s`)∞`=1 ∈ Ω ∞
m is called a β-expansion of x with digit set Ωmif we can write x =P∞`=1s`β`
with s` ∈ Ωm. The largest number we can obtain in this way is xmax:= (m−1)β/(1−β).
Now for any x ∈ (0, xmax], let us define a sequence (s`)∞`=1 ∈ Ω∞m recursively by the
quasi-greedy algorithm (cf. [dVK09]): let s0 = 0, and if s` is already defined for all
` < n, then let sn be the largest element in Ωm satisfying Pn`=1s`β` < x. Obviously,
P∞
`=1s`β`= x, and we call (s`)∞`=1 the quasi-greedy β-expansion of x with digit set Ωm.
We always call (s`)∞`=1 a quasi-greedy expansion of x if there is no confusion about
β and the digit set Ωm. It is easy to see that (s`)∞`=1 is an infinite expansion (i.e.,
infinitely many s` are non-zeros).
We use systematically the lexicographical order between sequences: we write (a`)∞`=1 <
(b`)∞`=1 or (b`)∞`=1 > (a`)∞`=1 if there exists an n ∈ N such that a` = b` for ` < n and
an < bn. Furthermore, we write (a`)∞`=1 ≤ (b`)∞`=1 or (b`)∞`=1 ≥ (a`)∞`=1 if we also allow
the equality of the two sequences. Similarly, for two s-blocks c1. . . cs and d1. . . ds, we
and cn < dn. Moreover, we write (c`)s`=1 ≤ (d`)s`=1 if we allow the equality of the two
blocks.
Therefore, the quasi-greedy expansion of x ∈ (0, xmax] is the largest infinite
ex-pansion among all the β-exex-pansions of x in the sense of lexicographical order. Note that 1 ∈ (0, xmax] since β > 1/m. In the remainder of the paper we will reserve the
notation (δ`)∞`=1 = (δ`(β))∞`=1 for the quasi-greedy β-expansion of 1 with digit set Ωm.
The following important properties of the quasi-greedy expansion of 1, will be used in the proof of Theorem 2.4.6.
Proposition 2.4.1 (Parry [Par60]). Given m ≥ 2, the map β → (δ`(β))∞`=1 ∈ Ω ∞ m, with
β ∈ (1/m, 1), is strictly decreasing in the sense of lexicographical order. Moreover, the map is continuous w.r.t. the topology in Ω∞m induced by the metric d (a`)∞`=1, (b`)∞`=1 =
2− min{j:aj6=bj}.
Proposition 2.4.2 (de Vries and Komornik [dVK09]). Given m ≥ 2 and β ∈ (1/m, 1), let (γ`)∞`=1 be an infinite β-expansion of 1 with digit set Ωm. Then (γ`)∞`=1 is the
quasi-greedy expansion of 1 if and only if for all k ≥ 1
γk+1γk+2· · · ≤ γ1γ2. . . (2.9)
in the lexicographical order.
Given m ≥ 2, let d = m − 1 − d be the reflection of the digit d ∈ Ωm. For a sequence
(a`)∞`=1 ∈ Ω∞m, let (a`)`=1∞ = (a`)∞`=1 = (m − 1 − a`)∞`=1 be the reflection of the sequence
(a`)∞`=1 ∈ Ω∞m. A sequence (a`)∞`=1 ∈ Ω∞m is said to be admissible if for all k ≥ 1
ak+1ak+2· · · < a1a2. . . , if ak< m − 1
ak+1ak+2. . . < a1a2. . . , if ak> 0.
Let (τ`)∞`=0∈ Ω∞2 be the classical Thue-Morse sequence, i.e., τ0 = 0, and if τ`is already
defined for some ` ≥ 0, set τ2`= τ` and τ2`+1= τ` = 1 − τ`. Then the sequence (τ`)∞`=0
begins as follows
0 1101 0011 0010 1101 0010 1100 1101 0011 0010 1100 . . . .
We construct a sequence (λ`)∞`=1= (λ`(m))∞`=1 ∈ Ω∞m for the even and odd numbers m
respectively.
(I). λ`= q − 1 + τ` for ` ≥ 1, if m = 2q with q ≥ 1;
(II). λ`= q + τ`− τ`−1 for ` ≥ 1, if m = 2q + 1 with q ≥ 1. (2.10)
Komornik and Loreti [KL02] showed that (λ`)∞`=1 is the smallest admissible sequence in
Proposition 2.4.3 (Komornik and Loreti [dVK09]). Let (λ`)∞`=1 ∈ Ω∞m be defined in
(2.10). Then for all k ≥ 1
λk+1λk+2· · · < λ1λ2. . . , λk+1λk+2. . . < λ1λ2. . . .
For a more general digit set Ω, there also exist some results on the smallest admis-sible sequence which is related to the Thue-Morse sequence (cf. [AF09]).
The following important theorem on the set
Aβ,m:=x ∈ [0, xmax] : x = ∞
X
`=1
ε`β`, ε`∈ Ωm has a unique β-expansion
is due to Parry [Par60], Erd¨os et al. [EJK90], Komornik et al. [KL02] and de Vries et al. [dVK09].
Theorem 2.4.4. Given m ≥ 2 and β ∈ (1/m, 1), let (δ`)∞`=1 be the quasi-greedy
β-expansion of 1 with digit set Ωm. Then P∞`=1ε`β`∈Aβ,m if and only if for all k ≥ 1
(
εk+1εk+2· · · < δ1δ2. . . , if εk< m − 1
εk+1εk+2. . . < δ1δ2. . . , if εk> 0.
For m ≥ 2, let βc,m be the unique positive solution of the following equation
1 =
∞
X
`=1
λ`β`, (2.11)
where (λ`)∞`=1 = (λ`(m))∞`=1 ∈ Ω∞m is defined in (2.10). We remark here that βc,m is
a transcendental number for all m ≥ 2 (cf. [KL02]). For m = 2, Glendinning and Sidorov [GS01] have shown that the critical point forAβ,2 is βc,2, i.e.,Aβ,2 has positive
Hausdorff dimension if β < βc,2 and Aβ,2 contains at most countably many elements
if β > βc,2. Their results can be generalized to the even number case, i.e., for an even
number m ≥ 2, the critical point for Aβ,m is βc,m. However, it is more intricate to
find the critical point forAβ,mfor an odd number m. Inspired by [GS01] we show that
for an odd number m ≥ 3, the critical point forAβ,m is still βc,m, the unique positive
solution of Equation (2.11).
Given N ≥ 2 and β ∈ (1/(2N − 1), 1/N ), we will find the critical point for Uβ,±N,
which is the set of t ∈ [−1, 1] having a unique Ω±N-code.
To make the connection with the theory of beta-expansions we shift Ω±N to the set
Thus from [−1, 1] = π±N Ω∞±N it follows that [0, 2] = π2N −1 Ω∞2N −1 = ( ∞ X `=1 ε`β`−1(1 − β) N − 1 : ε`∈ {0, 1, . . . , 2N − 2}, ` ≥ 1 ) , where π2N −1:= πΩ2N −1 is as in (2.1). Let Uβ,2N −1:=t ∈ [0, 2] : |π−12N −1(t)| = 1 ,
i.e., the set of t ∈ [0, 2] having a unique Ω2N −1-code. Thus, it is easy to see that
Uβ,2N −1 =Uβ,±N + 1.
For β ∈ (1/(2N − 1), 1/N ), note that
x ∈Aβ,2N −1 ⇐⇒
1 − β
β(N − 1)x ∈Uβ,2N −1.
Thus Theorem 2.4.4 yields the the following important theorem which could also be shown in a different way by using (2.5).
Theorem 2.4.5. Given N ≥ 2 and β ∈ (1/(2N − 1), 1/N ), let (δ`)∞`=1 be the
quasi-greedy β-expansion of 1 with digit set Ω2N −1. Then (ε`)∞`=1 ∈ π −1
2N −1(Uβ,2N −1) if and
only if for all k ≥ 1
(
εk+1εk+2· · · < δ1δ2. . . , if εk∈ {0, . . . , 2N − 3}
εk+1εk+2. . . < δ1δ2. . . , if εk∈ {1, . . . , 2N − 2},
(2.12)
where εk+1εk+2. . . is the reflection of εk+1εk+2· · · ∈ Ω∞2N −1.
Therefore, dealing with the set Uβ,±N is equivalent to dealing with the set of
se-quences (ε`)∞`=1 ∈ Ω ∞
2N −1 which satisfy (2.12). Substituting m = 2N − 1 in (2.10), we
get the smallest admissible sequence (λ`)∞`=1∈ Ω ∞
2N −1 which starts with
N (N − 1)(N − 2)N (N − 2)(N − 1)N (N − 1) (N − 2)(N − 1)N (N − 2) . . . .
It is helpful to give another equivalent definition of the sequence (λ`)∞`=1 ∈ Ω∞2N −1
(cf. [KL02]), i.e.,
λ1= N, λ2n+1 = λ2n+ 1 = 2N − 1 − λ2n for n = 0, 1, . . . ,
λ2n+`= λ`= 2N − 2 − λ` for 1 ≤ ` < 2n, n = 1, 2, . . . . (2.13) So it is easy to see λ2n = N for n = 0, 2, 4, . . . and λ2n= N − 1 for n = 1, 3, 5, . . . .
Theorem 2.4.6. Given N ≥ 2, β ∈ (1/(2N − 1), 1/N ), let Uβ,±N be the set of t ∈
[−1, 1] having a unique Ω±N-code and βc ∈ (1/(2N − 1), 1/N ) be the unique positive
solution of Equation (2.11) with (λ`)∞`=1 ∈ Ω∞2N −1 defined in (2.13). Then
(1) If β ∈ (1/(2N − 1), βc), then dimHUβ,±N > 0;
(2) If β = βc, then |Uβc,±N| = 2
ℵ0 and dim
HUβc,±N = 0; (3) If β ∈ (βc, 1/N ), then |Uβ,±N| = ℵ0.
Since Uβ,±N = Uβ,2N −1 − 1, the critical point of Uβ,±N is equal to the critical
point ofUβ,2N −1. Thus we only need to show the corresponding conclusions for the set
Uβ,2N −1.
Using Proposition 2.4.2 and Proposition 2.4.3, we obtain (δ`(βc))∞`=1 = (λ`)∞`=1,
i.e., (λ`)∞`=1 is the quasi-greedy βc-expansion of 1 with digit set Ω2N −1. The proof of
Theorem 2.4.6 will be divided into several lemmas.
Lemma 2.4.7. λk. . . λk+2n−2 < λ1. . . λ2n−1for any n ≥ 2 and any k ∈ {2, . . . , 2n−1}; λk. . . λk+2n−2< λ1. . . λ2n−1 for any n ≥ 2 and any k ∈ {1, . . . , 2n− 1}.
Proof. Since for n = 2 the lemma is quickly checked, let n ≥ 3 and k ∈ {2, . . . , 2n−
1}. Then by Proposition 2.4.3 λkλk+1· · · < λ1λ2. . . , which implies λk. . . λk+2n−2 ≤ λ1. . . λ2n−1. It is easy to check that λk. . . λk+2n−2 < λ1. . . λ2n−1 for k < 7. For all other k we can write k = 2s+ 2p+ j with 1 ≤ p < s < n and 1 ≤ j < 2p. It follows from [KL02, Lemma 5.4] that
λk. . . λk+2p+1−j < λj. . . λ2p+1 ≤ λ1. . . λ2p+1−j+1
which implies λk. . . λk+2n−2 < λ1. . . λ2n−1, since n > p + 1.
For the second inequality, ignoring the trivial cases k = 1 and 2, suppose k = 2q+ j with 1 ≤ j < 2q and 1 ≤ q < n. Then it again follows from [KL02, Lemma 5.5] that
λk. . . λk+2q−j < λj. . . λ2q ≤ λ1. . . λ2q−j+1.
which implies that λk. . . λk+2n−2< λ1. . . λ2n−1, since n > q.
Lemma 2.4.8. Let n ≥ 3 be an odd integer. If λk. . . λ2n−1 = λ1. . . λ2n−k for some k ∈ {1, . . . , 2n− 1}, then λ2n−k+1 = N .
Proof. Suppose λk. . . λ2n−1= λ1. . . λ2n−k. It can not happen that k < 2n−1since then we will obtain that λk. . . λk+2n−1−2 = λ1. . . λ2n−1−1 which contradicts Lemma 2.4.7.
It is also impossible that k = 2n−1 since then N − 2 = λ
2n−1 = λ1= N . Thus we must have k > 2n−1. From the definition of (λ`)∞`=0 in (2.13) it follows that
λk−2n−1. . . λ2n−1−1 = λk. . . λ2n−1 = λ1. . . λ2n−k, which implies N ≥ λ2n−k+1≥ λ2n−1 = N by Proposition 2.4.3.
We want to approximate (λ`)∞`=1 by eventually periodic sequences which satisfy
(2.9). This does not work for the obvious choice (λ1. . . λ2n)∞. Thus we define for n ≥ 0
Cn∞= λ1. . . λ2n(λ2n+1. . . λ2n+1)∞. Since for all n ≥ 0 we have λ2n+1 > λ2n, we obtain that
λ1. . . λ2n(λ2n+1. . . λ2n+1)3 > λ1. . . λ2n+1λ2n+1+1. . . λ2n+2, which implies
(P7) C0∞> C1∞> · · · > Cn∞> · · · > (λ`)∞`=1 in the lexicographical order.
Lemma 2.4.9. Let n ≥ 3 be an odd number. Then for any k ≥ 1 we have σk(Cn∞) < Cn∞, where σ is the left-shift map.
Proof. Since Cn∞ is an eventually periodic sequence in Ω∞2N −1, we only have to check the lemma for k ∈ {1, . . . , 2n+1− 1}. For k = 2n− 1 or 2n+1− 1, it is easy to check
that σk(Cn∞) < Cn∞. Then we only need to consider the following two cases. (I) k ∈ {1, . . . , 2n− 2}. It follows from Lemma 2.4.7 that
σk(Cn∞) = λk+1. . . λ2n+k−1· · · < λ1. . . λ2n−1λ2n(λ2n+1. . . λ2n+1)∞= Cn∞. (II) k ∈ {2n, . . . , 2n+1− 2}. Write k = 2n+ `. Then, by the definition of (λ
`)∞`=1,
σk(Cn∞) = λk+1. . . λ2n+1−1λ2n+1(λ2n+1. . . λ2n+1)∞ = λ`+1. . . λ2n−1λ2n+1(λ2n+1. . . λ2n+1)∞.
If λ`+1. . . λ2n−1 < λ1. . . λ2n−`−1, we have shown that σk(Cn∞) < Cn∞. Otherwise, ` ≥ 2 and we have by Proposition 2.4.3 that λ`+1. . . λ2n−1 = λ1. . . λ2n−`−1. Using Lemma 2.4.8 we obtain that also λ2n+1 = N = λ2n−`. Thus it is enough to show
λ2n+1. . . λ2n+1−1 < λ2n−`+1. . . λ2n+1−`−1. Taking reflections on both sides, this is equivalent to showing
λ1. . . λ2n−1 > λ2n−`+1. . . λ2n+1−`−1, which is true by Lemma 2.4.7 since ` ≥ 2.
