Self-similar sets of Hausdorff measure zero and
positive packing measure
Master’s Thesis
Tuomas Pöyhtäri
Department of Mathematical Sciences
University of Oulu
2013
Contents
Introduction
ii
1 Preliminaries
1
1.1
Basic notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
The space of continuously differentiable functions . . . . . . . . . . . . . . . . . .
2
1.3
General measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.4
Measures and dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.5
The space of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.6
Self-similar sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.7
Orthogonal projections of fractal sets . . . . . . . . . . . . . . . . . . . . . . . . .
13
2 Self-similar sets on a line
16
2.1
One-parameter families of iterated function systems . . . . . . . . . . . . . . . .
16
2.2
Families of projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3 Zero Hausdorff measure
3.1
21
The Lebesgue measure of the set of intersection parameters . . . . . . . . . . . .
26
4 Positive packing measure
31
5 Examples
36
Bibliography
39
i
Introduction
Fractals are a rather new concept in mathematics. The word fractal was first used by Benoit
Mandelbrot in 1975. However, ideas relating to fractals have appeared as early as in the 17th
century. In the 19th century, they became mathematically more rigorous and in the 20th century
they grew to be a major field of study. The word fractal originates from the Latin word fractus,
meaning broken or fractured. This word illustrates well the difference between fractals and classical
geometric objects. If we examine the boundary of an object of classical geometry by zooming in
on the picture, at some point we see a solid interior and a clear exterior separated by a simple
curve. If we zoom in more, the curve will also begin to resemble a line. With fractals, this is not
the case. Fractals contain fine details at every level of this kind of zoom – no matter how far we
go, the picture will not simplify. Fractals are shredded figures with arbitrarily small shreds. There
is no formal general definition for what a fractal is.
If we would like to study the geometric properties of a fractal, we have to assume some
properties for it. The most classic examples are self-similar sets, which include the Cantor set
and Sierpinski gasket. Self-similar sets are a kind of fractal which can be divided into a finite
amount of pieces, each of which is similar to the whole set. Being similar here means that they
are the same sets up to translation, rotation, scaling and reflection. Self-similar sets are usually
defined by a family of functions executing these operations.
Some of the tools that are used for studying classical geometry are inadequate when studying
fractals. These common ideas have to be refined so that we can describe the ’length’, ’area’,
’volume’ or ’dimension’ of a fractal set. One useful tool is the Hausdorff measure and Hausdorff
dimension introduced by Felix Hausdorff in 1918. The Hausdorff dimension is a generalization of
the common notion of dimension and it does not have to be an integer. One can then measure
the set in its own dimension with the corresponding Hausdorff measure, which is a generalization
of a length, area or volume. This definition is convenient, since it works for every set. However, it
is only one definition. Claude Tricot introduced the packing measure and packing dimension in
1982. These two dimensions are different from each other but work in the same way.
When self-similar sets are studied, they are usually assumed to satisfy some separation
conditions, as the open set condition or strong separation condition. These conditions limit
how much the individual similar parts may overlap each other. Without any kind of separation
ii
condition, the self-similar set can be very complex. However, it is known that the Hausdorff and
packing dimensions agree on self-similar sets even without separation conditions. The behavior of
the corresponding Hausdorff and packing measures remain quite mysterious. Another question
concerning self-similar sets, is what happens to the measure when we take a projection of
a self-similar set? Marstrand’s theorem [5, Theorem 6.1] guarantees that, in a projection the
Hausdorff dimension is maintained for almost every direction, but how does the Hausdorff measure
change in the process? The purpose of this thesis is to address these problems. It is based on an
article by Peres, Simon and Solomyak [11]. We restrict our focus to self-similar sets which are
defined by functions which only translate and scale, thus rotations and reflections are not allowed.
We will see that there are self-similar sets that have zero Hausdorff measure but positive and
finite packing measure in their dimension. This behavior is common for such self-similar sets on
the line, which are projections of a certain type of planar self-similar sets. This kind of self-similar
set on a line does not necessarily satisfy any separation conditions and, of course, the interesting
cases are exactly those projections where overlapping occurs. By projecting certain types of
planar self-similar sets, we achieve whole families of self-similar sets on a line with the above
property. We also see that if there is overlapping in this kind of projection, the Hausdorff measure
drops to zero in almost every case. However, the corresponding packing measure is positive and
finite. This suggests that the packing measure is a more suitable measure for sets of this kind.
The result concerning packing measure is shown for a general Borel set. We need to make the
assumption about the dimension of that Borel set with respect to the Hausdorff measure, since
packing measure may drop for almost all projections, see [9] and [6]. Thus Hausdorff measure is
still needed and cannot be completely omitted.
Chapter 1 contains preliminary information needed in the thesis. In Chapter 2 we introduce
one-parameter families of iterated function systems and present some properties about them.
These families are the abstract counterparts of the projections of a planar self-similar set. The
results concerning the Hausdorff measure and packing measure are in the Chapters 3 and 4,
respectively. Finally in Chapter 5 we present two examples.
iii
Chapter 1
Preliminaries
In this chapter we introduce definitions and general concepts which are used throughout the
paper. Many of the proofs are omitted but they can be found in most books concerning the
subject at hand. For theory concerning fractals, dimensions and other geometric measure theory
see [5] or [10]. For general measure theory see [12].
1.1
Basic notations
The following notations are in the form for metric spaces (X, d), even though we mostly work in
R or R2 with the Euclidean metric.
The closed ball with centre x ∈ X and radius r > 0 is denoted by
B(x, r) = {y ∈ X : d(x, y) ≤ r}.
The diameter of a set A ⊂ X is denoted by
d(A) = sup{d(x, y) : x, y ∈ A}.
In case of an interval A ⊂ R we may write d(A) = |A|. If A is a non-empty subset of X, then for
the distance from x ∈ X to A we set
d(x, A) = inf{d(x, y) : y ∈ A}.
For ε > 0 the ε-neighbourhood of A ⊂ X is the set
A(ε) = {x ∈ X : d(x, A) ≤ ε}.
For a set A ⊂ X, A is the closure, Conv(A) is the convex hull and #A is the cardinality of A.
1
CHAPTER 1. PRELIMINARIES
1.2
2
The space of continuously differentiable functions
The functions of this class will be used frequently.
Definition 1.1. We say that a function f : I → R, I ⊂ R is compact interval, belongs to the set
C 1 (I) if and only if it has a continuous derivative on I.
Sets C k are defined similarly for all k = 0, 1, . . . , but we only deal with the case k = 1. One can
define different kinds of metrics and norms for these sets to get function spaces.
Theorem 1.2. Let I ⊂ R be a compact interval. Then
kf kC 1 (I) = sup|f (x)| + sup|f 0 (x)|
x∈I
x∈I
is a norm on C 1 (I). Moreover, this norm makes C 1 (I) a Banach space. This norm is called the
C 1 norm.
Note that supx∈A |·| is usually called the uniform norm, supremum norm or infinity norm and
denoted by k·k∞ . One can easily show that this is a norm for bounded functions. The proof
follows from the fact that the uniform norm combined with continuous functions on a compact
metric space form a Banach space. This norm also seems meaningful intuitively, since it considers
both the function and its derivative.
1.3
General measure theory
We assume that the reader has some knowledge of basic measure theory and is familiar with the
Lebesuge measure L.
Definition 1.3. Let X be a set. A function µ : {A : A ⊂ X } → [0, ∞] is called a measure if
1. µ(∅) = 0
2. µ(A) ≤ µ(B), when A ⊂ B ⊂ X
P∞
S∞
3. µ i=1 Ai ≤ i=1 µ(Ai ), when A1 , A2 , . . . ⊂ X.
So it is a non-negative, monotonic and (countably) subadditive set function, which vanishes on
the empty set. In measure theory the above kind of function is usually defined to be only an outer
measure, while a measure is also countably additive (see condition 3. in the next theorem) and
defined on some σ-algebra of X. Our definition seemingly differs from this, and the justification
is presented below.
Definition 1.4. A set A ⊂ X is µ measurable if
µ(E) = µ(E ∩ A) + µ(E \ A)
for every E ⊂ X.
CHAPTER 1. PRELIMINARIES
3
Here we collect the basic properties of measurable sets.
Theorem 1.5. Let µ be an outer measure on X and let M be the set of all µ measurable sets.
1. M is a σ-algebra, that is,
(a) ∅ ∈ M,
(b) if A ∈ M, then X \ A ∈ M,
S∞
(c) if A1 , A2 , . . . ∈ M, then i=1 Ai ∈ M.
2. If µ(A) = 0, then A ∈ M.
3. If A1 , A2 , . . . ∈ M are disjoint, then µ
S∞
i=1
P∞
Ai = i=1 µ(Ai ).
4. If A1 , A2 , . . . ∈ M, then
S∞
(i) µ i=1 Ai = limi→∞ µ(Ai ) provided A1 ⊂ A2 ⊂ · · · ,
T∞
(ii) µ i=1 Ai = limi→∞ µ(Ai ) provided A1 ⊃ A2 ⊃ · · · and µ(A1 ) < ∞.
It is a convenience rather than a restriction to define measures as in Definition 1.3. If ν is a
countably additive non-negative set function on a σ-algebra Σ of X, then it can be extended to a
measure, as in Definition 1.3, by
ν ∗ (A) = inf{ν(B) : A ⊂ B ∈ Σ}.
On the other hand, a measure µ gives a countably additive set function when we restrict it to the
σ-algebra of µ measurable sets.
Definition 1.6. Let X be a topological space. The smallest σ-algebra containing all open sets of
X is called the Borel family. A member of this family is a Borel set.
This kind of smallest σ-algebra always exists [12, Theorem 1.10]. Note that if we speak about
some σ-algebra or Borel sets on X, it implies that we have some topology on X.
Definition 1.7. Let µ be a measure on X.
1. µ is locally finite if for every x ∈ X there is r > 0 such that µ B(x, r) < ∞.
2. µ is a Borel measure if all Borel sets are µ measurable.
3. µ is Borel regular if it is a Borel measure and if for every A ⊂ X there is a Borel set B ⊂ X
such that A ⊂ B and µ(A) = µ(B).
4. µ is a Radon measure if it is a Borel measure and
(a) µ(K) < ∞ for compact sets K ⊂ X,
(b) µ(V ) = sup{µ(K) : K ⊂ V is compact} for open sets V ⊂ X,
(c) µ(A) = inf{µ(V ) : A ⊂ V, V is open} for A ⊂ X.
Definition 1.8. The support of a Borel measure µ, spt µ, on a separable metric space X is the
smallest closed set S ⊂ X such that µ(X \ S) = 0.
CHAPTER 1. PRELIMINARIES
4
Definition 1.9. Let f : X → Y and µ be a measure on X. The image measure of µ under f is
defined by
f∗ µ(A) = µ f −1 (A)
for A ⊂ Y .
Clearly f∗ µ is a measure on Y . Also, it can be shown that if µ is a Borel measure and f a Borel
function, then f∗ µ is a Borel measure. Moreover, spt f∗ µ = f (spt µ).
Definition 1.10. Let µ and ν be measures on metric spaces X and Y , with collection of
measurable sets Mµ and Mν respectively. For every A ⊂ X × Y we set
(µ × ν)(C) = inf{
∞
X
µ(Ai )ν(Bi ) : C ⊂
∞
[
Ai × Bi , Ai ∈ Mµ and Bi ∈ Mν }.
i=1
i=1
This is the product measure µ × ν on X × Y .
In the above definition we treat 0 · ∞ = ∞ · 0 = 0. It can be shown that if both µ and ν are Borel,
Borel regular or Radon measures, then µ × ν has the same property.
Next, we give two useful theorems about integration, Fubini’s theorem and integration with
respect to an image measure.
Theorem 1.11. Let X and Y be separable metric spaces and µ and ν locally finite Borel measures
on X and Y , respectively. If f is a non-negative Borel function on X × Y , then
Z
ZZ
ZZ
f d(µ × ν) =
f (x, y) dµ(x) dν(y) =
f (x, y) dν(y) dµ(x).
One can also use this in cases where there are seemingly no iterated integrals or product measures.
Corollary 1.12. Let µ be a Borel measure and f a non-negative Borel function on a separable
metric space X. Then
Z
Z
∞
f dµ =
µ {x ∈ X : f (x) ≥ t} dt,
0
where the latter integral is taken with respect to the Lebesgue measure.
Theorem 1.13. Suppose that f : X → Y is a Borel mapping, µ is a Borel measure on X, and g
is a non-negative Borel function on Y . Then
Z
Z
g df∗ µ = (g ◦ f ) dµ.
Finally we present a usful theorem concerning the Lebesgue measure.
Theorem 1.14. Let L be the Lebesgue measure on R. If A ⊂ R is measurable, then the limit
L(A ∩ B(x, r))
r→0
L(B(x, r))
lim
equals 1 for Lebesgue almost every x ∈ A and equals 0 for Lebesgue almost every x ∈ R \ A.
The above theorem is known as the Lebesgue density point theorem. A point x ∈ A for which this
limit is 1 is called a Lebesgue density point.
CHAPTER 1. PRELIMINARIES
1.4
5
Measures and dimensions
In this section we introduce Hausdorff and packing measures, which are general tools in fractal
geometry. They both give a dimension for the set examined, which coincides with the common
notion of dimension in cases of simple objects of classical geometry. However, these definitions of
dimensions are very general. They assign a dimension for every subset of the space, moreover, this
dimension can be non-integral. We begin with a bit simpler generalization of dimension, called
the box-counting dimension.
Box-counting dimension
Definition 1.15. Let A ⊂ Rn be bounded. The lower and upper box-counting dimensions of A
are
dimB A = lim inf
log Nδ (A)
− log δ
dimB A = lim sup
log Nδ (A)
.
− log δ
δ↓0
δ↓0
If these agree, then the box-counting dimension of A is
dimB A = lim
δ↓0
log Nδ (A)
.
− log δ
Here Nδ (A) is any of the following:
(a) the smallest number of sets of diameter at most δ that cover A,
(b) the smallest number of closed balls of radius δ that cover A,
(c) the smallest number of cubes of side length δ that cover A,
(d) the number of δ-mesh cubes that intersect A,
(e) the largest number of disjoint balls of radius δ with centres in A.
The name relates closest to the definition (d). Suppose that A lies in a evenly spaced mesh of
cubes. Then one has to count the amount of n-cubes required to cover A, while the mesh is made
finer and finer. This dimension is easier to calculate than more refined definitions of dimension
which we introduce later. However, it has some unwanted properties. For example, a set has the
same lower and upper box-counting dimension as its closure has. Then, by taking A to be the
rationals on the unit interval we see that dimB A = 1. Thus a countable set can have non-zero
box-counting dimension. Also, since the box-counting dimension of a singleton is zero, we see
S∞
that dimB i=1 Ai = supi dimB Ai does not hold generally. This dimension can still be useful. It
can be used to approximate the more complicated dimensions we meet later on.
CHAPTER 1. PRELIMINARIES
6
Hausdorff measure and dimension
Definition 1.16. Let X be a separable metric space and 0 ≤ s < ∞. Define
Hδs (A) = inf{
∞
X
d(Ei )s : A ⊂
∞
[
Ei , d(Ei ) ≤ δ}
i=1
i=1
and
Hs (A) = lim Hδs (A) = sup Hδs (A)
δ↓0
δ>0
for every A ⊂ X. We call Hs the s-dimensional Hausdorff measure.
Remark 1.17. Here we interpret 00 = 1 and d(∅)s = 0 for every s. The limit exists and agrees
with the supremum since Hδs (A) is non-increasing with respect to δ. Note that this measure is
gained through Carathéodory’s construction, so it is a Borel measure. Also, Hausdorff measure
does not change if we consider only coverings with open sets Ei . By this, it is also Borel regular.
See [10, Theorem 4.4.].
Next we present some properties of the Hausdorff measure.
Theorem 1.18. Let A ⊂ X and 0 ≤ s < t < ∞.
(1) If Hs (A) < ∞, then Ht (A) = 0.
(2) If Ht (A) > 0, then Hs (A) = ∞.
Proof. To prove (1), let A ⊂
Hδt (A)
≤
S∞
1=1
∞
X
Ei with d(Ei ) ≤ δ and
t
d(Ei ) ≤ δ
t−s
∞
X
P∞
i=1
d(Ei )s ≤ Hδs (A) + 1. Since
d(Ei )s ≤ δ t−s Hδs (A) + 1 ,
i=1
i=1
we get (1) as δ → 0. Statement (2) is clearly equivalent to (1).
The previous theorem enables us to make the following definition.
Definition 1.19. The Hausdorff dimension of a set A ⊂ X is
dimH A = sup{s : Hs (A) > 0} = sup{s : Hs (A) = ∞}
= inf{t : Ht (A) < ∞} = inf{t : Ht (A) = 0}.
This dimension has the properties one would expect from a dimension. It is monotonic and
countably stable, that is, it satisfies
dimH A ≤ dimH B
and
dimH
∞
[
i=1
Ai = sup dimH Ai
i
for A ⊂ B ⊂ X
for Ai ⊂ X, i = 1, 2, . . .
CHAPTER 1. PRELIMINARIES
7
It is also convenient that dimH A = s is defined for every set A, though we do not know anything
about the measure Hs (A) in general. It can be zero, finite or infinite. It is worth noting that integral
dimensional Hausdorff measures play a special role. They can be thought of as a generalization of
the Lebesgue measure of the same dimension. H0 is readily seen to be a counting measure. For a
rectifiable curve Γ ⊂ Rn , H1 (Γ) is the length of Γ. More generally, it can be shown that if M is a
sufficiently regular m-dimensional surface in Rn , 1 ≤ m ≤ n, then the restriction Hm |M gives a
constant multiple of the surface measure on M . See [10, Subsection 4.3].
Packing measure and dimension
Definition 1.20. Let 0 ≤ s < ∞. For A ⊂ Rn put
X
Pδs (A) = sup
d(Bi )s ,
i∈I
where the supremum is taken over all countable collections of disjoint balls of diameter smaller
than δ with centres in A. Then we define the packing pre-measure of A by
P s (A) = lim Pδs (A) = inf Pδs (A).
δ↓0
δ>0
Remark 1.21. Again the limit exists and agrees with the infimum as P s (A) is non-decreasing
with respect to δ. Clearly P s is monotonic and P s (∅) = 0, but it is not subadditive, so it really is
not a measure. But we get a measure out of it with the following definiton.
Definition 1.22. Let A ⊂ Rn and 0 ≤ s < ∞. The s-dimensional packing measure of A is
P s (A) = inf{
∞
X
P s (Ai ) : A =
i=1
∞
[
Ai }
i=1
Theorem 1.23. P s is a Borel regular measure on Rn .
Proof. See [10, Subsection 5.10]
Theorem 1.24. Let A ⊂ Rn and 0 ≤ s < t < ∞.
(1) If P s (A) < ∞, then P t (A) = 0.
(2) If P t (A) > 0, then P s (A) = ∞.
Proof. See [10, Theorem 5.11].
As this is the case, we can define a dimension the same way we did with the Hausdorff measures.
Definition 1.25. The packing dimension of a set A ⊂ Rn is
dimP A = sup{s : P s (A) > 0} = sup{s : P s (A) = ∞}
= inf{t : P t (A) < ∞} = inf{t : P t (A) = 0}.
CHAPTER 1. PRELIMINARIES
8
As we now see, the packing measure and dimension behaves much like Hausdorff measure and
dimension. This dimension is monotonic and countably stable. Also, the measure P s behaves as
Hs with integral dimensions. But how do these measures and dimensions relate to each other?
Theorem 1.26. For every A ⊂ Rn , Hs (A) ≤ P s (A).
Proof. See [10, Theorem 5.12.].
There is an alternative definition for the packing dimension, which does not use measures. This
is called the modified upper box-counting dimension, and is presented on the following theorem.
Theorem 1.27. Let A ⊂ Rn . Then dimP A = inf{supi dimB Ei : A ⊂
S∞
i=1
Ei }.
Proof. See [10, Theorem 5.11.].
The modified lower box-counting dimension is defined in similar way, but this does not concern
us.
After these facts the content of the following theorem is not surprising.
Theorem 1.28. Let A ⊂ Rn be bounded. Then
dimH A ≤ dimP A ≤ dimB A.
Proof. The claim follows immediately from Theorem 1.26 and Theorem 1.27.
It is harder to see that all of the above inequalities can be strict. For such examples see [14].
α
Theorem 1.29. Let c > 0, 0 ≤ α ≤ 1 and f : Rn → Rm such that |f (x) − f (y)| ≤ c|x − y| for
all x, y ∈ Rn . Then
1
dim A,
α
where dim can be any of the above definitions for dimension.
dim f (A) ≤
Proof. First we show this for the box-counting dimensions. Let (Ei )∞
i=1 be a δ-cover of A. Then
α
α
(f (Ei ))∞
i=1 is a cδ -cover of f (A), since d(f (Ei )) = supx,y∈Ei |f (x) − f (y)| ≤ c supx,y∈Ei |x − y| =
cδ α . Therefore Nδ (A) ≥ Ncδα f (A). Now
dimB f (A) = lim sup
δ→0
log Ncδα (f (A))
log Nδ (A)
1 log Nδ (A)
≤ lim sup
= lim sup
α
− log cδ α
−
log
cδ
α
− log c − log δ
δ→0
δ→0
1
= dimB (A).
α
The same is true also for the lower box-counting dimension.
α
∞
For Hausdorff dimension, let s ≥ 0, (Ei )∞
i=1 be a δ-cover of A. Again, (f (Ei ))i=1 is a cδ -cover
P∞
P∞
s/α
s/α s
s/α
s/α
s
of f (A). Thus i=1 d(f (Ei ))
Hδ (A). Letting δ → 0
≤c
i=1 d(Ei ) so Hcδ α (f (A)) ≤ c
gives Hs/α (f (A)) ≤ cs/α Hs (A). If s > dimH A, then Hs/α (f (A)) ≤ cs/α Hs (A) = 0. This implies
that dimH f (A) ≤
s
α
for all s > dimH A so dimH f (A) ≤
1
α dimH A.
CHAPTER 1. PRELIMINARIES
9
For the packing dimension, let A ⊂
S∞
i=1
Ei , where Ei is bounded for every i. Using mono-
tonicity and countable stability of packing measure, Theorem 1.28 and the first part of this proof
we get
dimP f (A) ≤ dimP
∞
[
f (Ei ) = sup dimP f (Ei ) ≤ sup dimB f (Ei ) ≤
i=1
i
i
1
sup dimB Ei .
α i
Taking infimum over all coverings with bounded sets we get
dimP f (A) ≤
∞
[
1
Ei },
inf{sup dimB Ei : A ⊂
α
i
i=1
which yields the claim with Theorem 1.27.
A function satisfying the above condition is called an α-Hölder map. Finally we give a powerful
tool called the Frostman’s lemma.
Theorem 1.30. Let A ⊂ Rn be a Borel set. Then Hs (A) > 0 if and only if there exists a Radon
measure µ such that, 0 < µ(Rn ) < ∞, spt µ is compact and contained in A and µ B(x, r) ≤ rs
for x ∈ Rn and r > 0.
Proof. See [10, Theorem 8.8.].
Such a measure µ is sometimes called a Frostman measure.
1.5
The space of sequences
Now we make ourselves familiar with spaces of infinite words. Suppose that A = {1, . . . , m}, a
finite set which contains some elements. Then, for any interger n ≥ 1, An is the set of every
sequence of length n with members from A, that is
An = {i1 i2 . . . in : ij ∈ A for every 1 ≤ j ≤ n}.
Further, the set AN is the set of every sequence of infinite length with members from A, that is
AN = {i1 i2 . . . : ij ∈ A for every j ∈ N}.
We now define some notations concerning sequences. Denote the length of the sequence ω by |ω|.
For example, if u ∈ An , then |u| = n and if ω ∈ AN , then |ω| = ∞. The length of the empty
sequence is zero. Denote ω ∧ τ the unique longest common prefix of sequences ω and τ . For
example if u = 112233 and ω = 111 . . . , then u ∧ ω = 11. The longest common prefix always
exists since the empty sequence is always a prefix of any sequence. If u = u1 . . . un and ω is any
CHAPTER 1. PRELIMINARIES
10
sequence, then uω = u1 . . . un ω1 ω2 . . . . Note that this notion has meaning only when the first
sequence is finite. For any sequence ω with |ω| ≥ n, we denote ω|n = ω1 . . . ωn . Suppose we have
some collection of numbers {r1 , . . . , rm } and some sequence ω with members from A, then
rω =
|ω |
Y
rωi .
i=1
We agree that for the empty sequence this is zero. Now we restrict our attention to the set AN
and denote it by Ω. We equip it with the product topology, while {1, . . . , m} carries the discrete
topology. So, by definition, we obtain the topology by taking as a base the sets of the form
∞
Y
Ai ,
i=1
where Ai is open in A for every i and Ai 6= A only for finitely many i. Note that as A has discrete
topology, every subset is open. However, Ω is not discrete as the product is infinite.
Theorem 1.31. The set Ω is compact.
Proof. This follows from Tychonoff’s theorem.
Now we will introduce a metric to the set Ω.
Theorem 1.32. Suppose that A = {1, . . . , m} and 0 < ri < 1 for each i = 1, . . . , m. Then
d(ω, τ ) = rω∧τ is a metric on Ω.
Proof. Let ω, τ , η ∈ Ω. Clearly d(ω, τ ) = 0 if and only if ω = τ and d(ω, τ ) = d(τ, ω), so we
only need to check that the triangle inequality holds. Now |ω ∧ τ | ≥ min{|ω ∧ η|, |τ ∧ η|}, so
rω∧τ ≤ max{rω∧η , rτ ∧η } ≤ rω∧η + rτ ∧η .
Theorem 1.33. The topology induced by the metric d(ω, τ ) = rω∧τ is the same as the product
topology on Ω.
Proof. We prove this by showing that the topologies are equally strong. Let U =
Q∞
i=1
A i , U 6= ∅
such that Ai 6= A only for finitely many i. Let n be the smallest integer such that Ai = A for
every i ≥ n. Choose some τ ∈ U and r > 0 small enough to r < rτ |n . Now, if η ∈ B(τ, r), then
d(τ, η) < rτ |n . This implies that |τ ∧ η| ≥ n so η ∈ U and B(τ, r) ⊂ U .
On the other hand let τ ∈ Ω and r > 0. Let n be smallest integer such that rτ |n < r. Choose
U = {ω ∈ Ω : ω|n = τ |n}. Clearly U ⊂ B(τ, r) and U is of the form required.
∞
Remark 1.34. A sequence (ω n )n=1 in Ω converges to ω if and only if |ω n ∧ ω| → ∞ as n → ∞.
S∞
Also note that Ω is uncountable but separable. Let W = i=1 Ai , the countable set of all
finite sequences with members from A. Then {u111 . . . : u ∈ W } is countable dense set in Ω.
CHAPTER 1. PRELIMINARIES
11
Lemma 1.35. Suppose that A = {1, . . . , m} and 0 < ri < 1 for each i = 1, . . . , m. Then, with
Pm
the metric d(ω, τ ) = rω∧τ , dimH Ω = s and Hs (Ω) > 0, where i=1 ris = 1.
Proof. As Ω is compact and we want to calculate the infimum of
P∞
i=1
d(Ei )s over all open
coverings (Ei )∞
i=1 ⊃ Ω, it is enough to consider finite coverings by open sets. By definition, an
Q∞
open set U ⊂ Ω is of the form i=1 Ai , where Ai = A when i ≥ k, k ∈ N. Therefore, the set
{d(ω, τ ) : ω, τ ∈ U } has only finite amount of elements. Thus, if U ⊂ Ω is some open set, then
d(U ) = maxω,τ ∈U = rω0 ∧τ0 for some ω0 , τ0 ∈ U . Denote Su := {ω ∈ Ω : |ω ∧ u| ≥ |u|}, where
S∞
u ∈ i=1 Ai . We have d(Sω0 ∧τ0 ) = rω0 ∧τ0 and U ⊂ Sω0 ∧τ0 , so we may also assume that the cover
consists of sets of the form Su . Finally, if S and S 0 are covering sets of this kind and S ∩ S 0 6= ∅,
then S ⊂ S 0 or S 0 ⊂ S, so we may also assume that these sets are disjoint. This kind of cover is
of the form {Su }u∈I , where I is a finite set such that for every ω ∈ Ω there exists a unique u ∈ I
which is a prefix of ω. Then we claim that
X
d(Su )s =
u∈I
X
rus = 1,
(1.1)
u∈I
whence Hs (Ω) = 1 and dimH Ω = s. To see why (1.1) holds, start with the set Q1 = {1, . . . , m}.
If Q1 = I then we are done. If it is not, then construct Q2 as follows. For every u ∈ Q1 and u ∈
/ I,
add the sequences u1, . . . , um to the set Q2 . For every u ∈ Q1 and u ∈ I, add u ∈ Q2 . At every
step k ∈ N we have
X
rus = 1
u∈Qk
by the definition of s. After a finite amount of repetitions, namely when k = maxu∈I |u|, we have
Qk = I.
1.6
Self-similar sets
It is hard to study fractals in general. To get a grip of them we have to restrict ourselves to certain
type of fractals. Self-similar sets are roughly those kind of fractals which consists of arbitrarily
small geometrically similar copies of itself. We will now present some theory concerning self-similar
sets. For a more detailed introduction see [5] or [8].
Definition 1.36. A function S : Rn → Rn is called a similitude if there exists 0 < r < 1 such
that
|S(x) − S(y)| = r|x − y|
for x, y ∈ Rn .
So a similitude can rotate, reflect and zoom out a set with respect to the similarity ratio r. To
state the description of a self-similar set more rigorously, we want it to be a compact set K ⊂ Rn ,
which satisfies
CHAPTER 1. PRELIMINARIES
12
K=
m
[
Si (K)
(1.2)
i=1
for some similitudes S = {S1 , . . . Sm }. We may write Si (K) = Ki for simplicity. Some definitions
also require this union to be ’nearly disjoint’, that is,
Hs Ki ∩ Kj = 0
(1.3)
for i =
6 j and dimH K = s, but we omit this condition. We have a nice definition, but how can we
find these sets? To this end some work has to be done.
Definition 1.37. A collection {f1 , . . . , fm } of contractions on Rn , that is, functions fi : Rn → Rn
which satisfy |fi (x) − fi (y)| < ci |x − y|, 0 < ci < 1 for every x, y ∈ Rn is called an iterated
function system (i.f.s.).
Clearly similitudes are contractions.
Definition 1.38. Let C be the set of all non-empty compact subsets of Rn . For all A, B ∈ C set
h(A, B) = max{sup d(a, B), sup d(b, A)}
a∈A
b∈B
which is called the Hausdorff metric.
Theorem 1.39. The set C equipped with the Hausdorff metric forms a complete metric space.
It is easy to show that this is a metric space. It requires a bit more work to show that the space
is complete. For detailed proof see [2]. Next, observe that for any i.f.s. {f1 , . . . , fm }, especially
Sm
with similitudes, the function F : C → C, F(A) = i=1 fi (A) is a contraction on C. Then we use
the Banach fixed point theorem to conclude that there exists a unique fixed point, the attractor
of the i.f.s, a compact set on Rn for which (1.2) holds. The set can be very complex if arbitrary
overlaps are permitted in the union. Therefore we usually have some separation conditions for
self-similar sets. In the case of the union (1.2) being disjoint, we say that S satisfies the strong
separation condition. This condition really is strong, it doesn’t allow any overlap. To allow some,
but not too much overlap, we have the following condition.
Definition 1.40. Let S = {S1 , . . . , Sm } be an i.f.s. of similitudes. S then satisfies the open set
condition if there is a non-empty open set O such that
m
[
Si (O) ⊂ O
and Si (O) ∩ Sj (O) = ∅
i=1
for i 6= j.
Strong separation condition clearly implies open set condition, but the inverse is not true. If a
self-similar set K is generated by i.f.s. which satisfies, say, open set condition, then for simplicity
we may just say that K satisfies the open set condition.
CHAPTER 1. PRELIMINARIES
13
Theorem 1.41. If S = {S1 , . . . , Sm } satisfies the open set condition, then for the self-similar
set K we have 0 < Hs (K) ≤ P s (K) < ∞, where s = dimH K = dimP K = dimB K. Moreover, s is
the unique number for which
m
X
ris = 1,
(1.4)
i=1
where ri is the similarity ratio of Si .
The number s in the above theorem is called the similarity dimension of S. The value s is easily
determined if the i.f.s. is homogenous, that is, ri = r for each i. In this case s = log m/ log r−1 .
Observe that a self-similar set on R generated by similitudes Si (x) = rx + ai , i = 1, . . . m is the
set
K=
∞
nX
o
rn cn : cn ∈ {a1 , . . . am } .
n=0
This is the case since
Si
∞
n X
rn cn : cn ∈ {a1 , . . . am }
o
=
n
ai +
n=0
∞
X
rn cn : cn ∈ {a1 , . . . am }
o
n=1
and thus it is the unique set satisfying condition (1.2). Now the set
K−K =
∞
nX
o
rn cn : cn ∈ Γ ,
n=0
where Γ = {ai − aj : i, j ≤ m}, is also self-similar. The set K − K is generated by a homogenous
i.f.s. which contains #{ai − aj : i, j ≤ m} similitudes.
Self-similar sets satisying the open set condition are quite well understood, but with arbitrary
overlaps permitted they are still quite mysterious. It is known that even for a general self-similar
set the Hausdorff, packing and box-counting dimensions agree [4], but much less is known about
the Hausdorff and packing measures of these sets. This is a problem which motivates this thesis.
1.7
Orthogonal projections of fractal sets
This section consists of some known results concerning projections of fractals and some related
questions. Theorem 1.43 is known as Marstrand’s theorem and can be generalized to higher
dimensions naturally [5, Theorem 6.1]. The projection of a planar self-similar set results in a new
self-similar set on a line, which may have significant overlaps. This connection, however, allows us
to make deductions about the resulting set based on the properties of the original set.
Definition 1.42. Denote by Lθ the line y cos θ = x sin θ. Let projθ : R2 → Lθ , θ ∈ [0, π) be the
orthogonal projection on Lθ , that is, projθ (x, y) = (x, y) · (cos θ, sin θ) (cos θ, sin θ), where · is
the inner product on R2 .
CHAPTER 1. PRELIMINARIES
14
Theorem 1.43. Let A ⊂ R2 be a Borel set. Then
dimH (projθ A) = min{dimH A, 1}
for almost all θ ∈ [0, π).
So the dimension is maintained in almost every case, unless the set projected has a dimension
greater than that of the line. This raises a question what happens to the measure of the set?
In case of a self-similar set K generated by i.f.s satisfying the open set condition we know that
0 < Hs (K) < ∞ and dimH (projθ K) = s with the similarity dimension s and a typical θ, but how
does Hs (projθ K) behave?
From now on we make a restriction to consider only iterated function systems that consists
of similitudes of the form S(x) = rx + a, thus only a translation is allowed in addition to the
compulsory contraction. So the self-similar sets we work with hereafter are usually generated by
m
an i.f.s. {Si }i=1 , where
Si (x) = ri x + ai
(1.5)
for some ri ∈ (0, 1) and ai ∈ Rn , where n is one or two.
e ⊂ R2 be a self-similar set generated by i.f.s. { Sei }m , where Sei : R2 → R2 , Sei (x) = ri x+ci ,
Let K
i=1
0 < ri < 1 and ci = (ai , bi ) ∈ R2 for i = 1, . . . , m. Then projθ ◦ Sei (x, y) = ri · projθ (x, y) +
m
projθ (ai , bi ) is a similitude when restricted to the set Lθ , so the set {projθ ◦ Sei }i=1 is an i.f.s.
on Lθ . For A ⊂ R2 , let Pθ (A) = {Lθ+ π2 + (x, y) : (x, y) ∈ A}. Then projθ A = Lθ ∩ Pθ (A). Sets
Sei (Pθ (x, y)) and Pθ (Sei (x, y)) are both lines perpendicular to Lθ and they share the point Sei (x, y).
Hence Sei (Pθ (x, y)) = Pθ (Sei (x, y)), and Sei (Pθ (A)) = Pθ (Sei (A)). Therefore
e = Lθ ∩ Pθ (K)
e = Lθ ∩
projθ K
m
[
e =
Pθ (Sei (K))
=
e =
Lθ ∩ Sei (Pθ (projθ K))
i=1
e =
Lθ ∩ Pθ (Sei (K))
m
[
e =
Lθ ∩ Pθ (Sei (projθ K))
m
[
e
Lθ ∩ Sei (Pθ (K))
i=1
i=1
i=1
m
[
m
[
m
[
e
projθ (Sei (projθ K)).
i=1
i=1
e is the self-similar set generated by the i.f.s. {projθ ◦ Sei }m . To examine projθ K
e
Thus projθ K
i=1
we identify Lθ with R in natural way by rotating the set to zero angle. Now we have a family
m
of iterated function systems {projθ ◦ Sei }
which generates a self-similar set on R for every
i=1
θ ∈ [0, π). There is also a different way to represent this family, one which we use more often.
For the parameter we do not take the angle θ, but λ = tan θ, with the similitudes Siλ : R → R,
m
Siλ (x) = ri x + (ai + bi λ). The attractor of {Siλ }i=1 is not technically the same as projθ K but
e = √ 1 Kλ . Now we reach
affine equivalent with it. It can be normalized by scaling as projθ K
1+λ2
every case by λ ∈ R as we would with θ, with the exception of a projection on the y-axis, which
would be the case ’λ = ∞’. However, one point is not a problem, especially since the results are
of the type ’almost every’.
CHAPTER 1. PRELIMINARIES
15
Example 1.44. The s-dimensional Sierpinski gasket G r of ’unit size’ is generated by similitudes
Se1 (x, y) = r(x, y), Se2 (x, y) = r(x, y)+(1 − r, 0) and Se3 (x, y) = r(x, y)+(0, 1 − r), where 0 < r ≤ 1
2
and s = log 3/ log r−1 . By above, the orthogonal projections of G r on Lθ is up to scaling the
set generated by similitudes S1λ (x) = rx, S2λ (x) = rx + 1 − r and S3λ (x) = rx + (1 − r)λ, where
λ = tan θ. Moreover, this set is
∞
nX
o
rn cn : cn ∈ C ,
(1.6)
n=0
where C = {0, 1 − r, (1 − r)λ}. As affine transformations won’t matter much when studying the
properties of a self-similar set we can adopt a simpler representation for this set. Dividing by
1 − r we can assume that C = {0, 1, λ}. Further, observe that it is enough to consider λ ≥ 2, as
we can scale C so that the distance between the two points closest to each other is one and then
translate and reflect accordingly, unless λ is 0 or 1. Thus essentially every projection of G r is
achieved with the set (1.6) by assuming for example that C = {0, 1, λ} and λ ≥ 2.
Chapter 2
Self-similar sets on a line
2.1
One-parameter families of iterated function systems
The goal is to study orthogonal projections of a planar self-similar sets onto lines of different
angles. We restrict our attention only to self-similar sets generated by similitudes of the form
(1.5), that is, similitudes which do not rotate or reflect. As discussed above, the projection of this
kind of self-similar set is also self-similar set on the line, which depends on the original set by one
parameter. The parameter can be taken as the direction of the line the set is being projected
onto. The problem is that the resulting set need not satisfy any separation conditions. We now
λ
}λ∈J where
consider general one-parameter families of iterated function systems {S1λ , . . . , Sm
Siλ (x) = ri (λ)x + ai (λ)
(2.1)
for x ∈ R and J ⊂ R is a compact interval. For our purposes we can assume that ai (λ) and ri (λ)
belong to C 1 (J), the space of continuously differentiable functions. Because the similitudes are
contractive, we also have
0 < β ≤ ri (λ) ≤ ρ < 1
(2.2)
for all i ≤ m and λ ∈ J. Let Kλ be the self-similar set corresponding to λ and s(λ) the similarity
dimension of the i.f.s. Denote A = {1, . . . , m} and Ω = AN , the space of all infinite words from
A. For u ∈ An we write Suλ = Suλ1 ◦ · · · ◦ Suλn .
Definition 2.1. The map Π(λ, ·) : Ω → Kλ defined by
Π(λ, ω) = lim Sωλ1 ...ωn (0) =
n→∞
∞
X
rω1 ...ωn−1 (λ)aωn (λ)
n=1
will be called the natural projection map.
Next we present some properties of the natural projection map.
16
(2.3)
CHAPTER 2. SELF-SIMILAR SETS ON A LINE
17
Lemma 2.2. With the above definitions
(i) Π(·, ω) ∈ C 1 (J) for all ω ∈ Ω.
(ii) With Ω equipped with the product topology, the map Ω → C 1 (J) : ω 7→ Π(·, ω) is continuous.
Proof.
(i) Let ω ∈ Ω. From equation (2.2) we have
sup{ri (λ) : λ ∈ J, i = 1, . . . , m} ≤ ρ
(2.4)
and since J is compact and ai is continuous for all i ∈ N, we can choose
a := maxkai kC(J) .
i
Then
∞
N
X
X
rω1 ...ωn−1 (λ)aωn (λ) = sup
rω1 ...ωn−1 (λ)aωn (λ) sup Π(λ, ω) −
λ∈J
λ∈J n=N +1
∞
X
n=1
≤ sup
∞
X
|rω1 ...ωn−1 (λ)aωn (λ)| ≤
λ∈J n=N +1
∞
X
n
N
ρ =
= aρ
n=0
This shows that
Sωλ1 ...ωN (0),
ρn−1 a
n=N +1
N
aρ
N →∞
−−−−→ 0.
1−ρ
as function of λ, converges to Π(λ, ω) (even uniformly) as
d
λ
dλ Sω1 ...ωN (0),
N → ∞. For Π(·, ω) to be of class C 1 (J), we show that
as a function of λ,
converges uniformly. First choose
M1 := maxkri0 kC(J) and M2 = maxka0i kC(J) .
i
i
Note that these exist since the derivatives are continuous and J is compact. By calculating
d
rω1 ...ωn−1 (λ)aωn (λ) | = |rω0 1 (λ) · rω2 (λ) · · · rωn−1 (λ) · aωn (λ)
|
dλ
+rω1 (λ) · rω0 2 (λ) · · · rωn−1 (λ) · aωn (λ)
..
.
+rω1 (λ) · rω2 (λ) · · · rωn−1 (λ) · a0ωn (λ)|
and using above approximations we get
d
|
rω1 ...ωn−1 (λ)aωn (λ) | ≤ (n − 1)M1 aρn−2 + ρn−1 M2 ≤ (n − 1)M1 aρn−2 + ρn−2 M2
dλ
= ((n − 1)M1 a + M2 )ρn−2 ≤ c(n − 1)ρn−2 ,
where c > 0 and n > 1. Now
∞
∞
X
X
d
d
sup|
rω1 ...ωn−1 (λ)aωn (λ) | ≤
rω1 ...ωn−1 (λ)aωn (λ) |
sup|
dλ
λ∈J
λ∈J dλ
n=N
n=N
∞
X
≤c
n=N
1
since ρ < 1. Hence Π(·, ω) ∈ C (J).
N →∞
(n − 1)ρn−2 −−−−→ 0
CHAPTER 2. SELF-SIMILAR SETS ON A LINE
18
(ii) Let ω ∈ Ω and a, M1 and M2 be as in (i). Let ω N → ω as N → ∞ and let kN = |ω ∧ ω N |.
Note that by Remark 1.34 kN → ∞ as N → ∞. Now
∞
X
sup|Π(λ, ω) − Π(λ, ω N )| = sup|
λ∈J
rω1 ...ωn−1 (λ)aωn (λ) −
λ∈J
n=kN
∞
X
n−1
≤2
ρ
∞
X
N
(λ)aωnN (λ)|
rω1N ...ωn−1
n=kN
a = 2aρkN
n=kN
1
N →∞
−−−−→ 0.
1−ρ
Also, as in (i),
sup|Π0 (λ, ω) − Π0 (λ, ω N )| = sup|
λ∈J
λ∈J
≤2
∞
∞
X
X
d
d
rω1 ...ωn−1 (λ)aωn (λ) −
r N N (λ)aωnN (λ) |
dλ
dλ ω1 ...ωn−1
n=kN
∞
X
n=kN
(n − 1)M1 ρn−2 a + ρn−1 M2
n=kN
= 2M1 a
∞
X
(n − 1)ρn−2 + 2M2
n=kN
∞
X
N →∞
ρn−1 −−−−→ 0.
n=kN
Denote fω,τ (λ) = Π(λ, ω) − Π(λ, τ ).
Remark 2.3. Lemma 2.2 implies the following facts.
(i) The set {fω,τ : ω, τ ∈ Y } is compact in C 1 (J) as long as Y ⊂ Ω is compact.
(ii) The function (ω, τ, λ) 7→ fω,τ (λ) : Ω × Ω × J → R is continuous with compact domain, so it
also is uniformly continuous function.
Next we give two useful definitions.
Definition 2.4. The transversality condition holds on J if for any ω, τ ∈ Ω
if there is
λ∈J
such that
0
fω,τ (λ) = fω,τ
(λ) = 0 then fω,τ ≡ 0.
Definition 2.5. The set of intersection parameters is
IP = {λ ∈ J : fω,τ (λ) = 0 for some ω, τ ∈ Ω but fω,τ 6≡ 0}.
We are to study projections of a planar self-similar set K. We want to assume that K satisfies
strong separation condition and that it is not on a line. The analogues of these assumptions in
this general setting are
for any
λ ∈ J,
Kλ
is not a singleton
(2.5)
and
fω,τ ≡ 0
implies ω = τ.
(2.6)
CHAPTER 2. SELF-SIMILAR SETS ON A LINE
19
If K is on a line, then the projection of that set onto a line of a certain direction is a singleton,
hence the condition (2.5). If x, y ∈ K and |projθ (x) − projθ (y)| = 0 for every θ, then x = y.
If K satisfies strong separation condition, then the point x has a unique representation with a
sequence, hence the condition 2.6.
Next we give some consequences of the latter condition (2.6).
Lemma 2.6. Suppose that (2.6) holds. Then
(i) IP = {λ ∈ J : Kiλ ∩ Kjλ 6= ∅ for some i 6= j },
(ii) IP is compact,
(iii) If λ ∈
/ IP , then Hs(λ) (Kλ ) > 0,
(iv) Suppose that transversality holds on J. There exists δ > 0 such that for all ω, τ ∈ Ω with
ω1 6= τ1 ,
if
|fω,τ (λ)| < δ,
then
0
|fω,τ
(λ)| > δ.
(2.7)
Proof.
(i) Let λ ∈ IP . Then there are ω 6= τ such that fω,τ (λ) = 0. Choose α ∈ An such that αω 0 = ω,
ατ 0 = τ and ω10 6= τ10 . (If ω1 6= τ1 , then α is not needed.) Now
0 = fω,τ (λ) = Π(λ, αω 0 ) − Π(λ, ατ 0 )
which is equivalent to Sα Π(λ, ω 0 ) = Sα Π(λ, τ 0 ) . Because Sα is a bijection, we have
6 ∅, hence IP ⊂ {λ ∈ J : Kiλ ∩Kjλ =
Π(λ, ω 0 ) = Π(λ, τ 0 ). This clearly states that Kωλ 0 ∩Kτλ0 =
6
1
1
∅ for some i 6= j }.
Suppose now that Kiλ ∩ Kjλ =
6 j. Then there exists ω, τ ∈ Ω such that ω1 = j 6=
6 ∅ and i =
i = τ1 and Π(λ, ω) = Π(λ, τ ) so fω,τ (λ) = 0. By (2.6) and ω =
6 τ we have fω,τ 6≡ 0. So
{λ ∈ J : Kiλ ∩ Kjλ 6= ∅ for some i 6= j } ⊂ IP .
n→∞
(ii) Let λ0 ∈ IP . This, with (i), means that there is a sequence λn −−−−→ λ0 such that for every
n ∈ N there are ω n , τ n ∈ Ω with fωn ,τ n (λn ) = 0 and ω1n 6= τ1n . Using compactness, we can
form two subsequences ω ni and τ ni which converges to ω 0 and τ 0 respectively. Note also
that ω10 6= τ10 . By Lemma 2.2 we have
fω0 ,τ 0 (λ0 ) = lim fωni ,τ ni (λni ) = 0,
i→∞
but as in (i), fω0 ,τ 0 6≡ 0 since ω10 6= τ10 . This means that λ0 ∈ IP , so as a closed subset of a
bounded set, the set is compact.
m
(iii) If λ ∈
/ IP , then by (i) we have Kiλ ∩ Kjλ = ∅ for every i 6= j. So {Siλ }i=1 meets the strong
separation condition and the claim follows by Theorem 1.41.
CHAPTER 2. SELF-SIMILAR SETS ON A LINE
20
(iv) If (2.7) does not hold, we have ω n , τ n ∈ Ω, ω1n 6= τ1n and λn ∈ J such that
|fωn ,τ n (λn )| <
1
n
and |fω0 n ,τ n (λn )| ≤
1
n
for every n ∈ N. As in (ii), by choosing suitable subsequences we can get an index sequence
ni
6 τ10 and also λni → λ0 as i → ∞, since J is
→ ω 0 , τ ni → τ 0 , ω10 =
(ni )∞
i=1 such that ω
compact too. When i → ∞ we get
fω0 ,τ 0 (λ0 ) = 0 and fω0 0 ,τ 0 (λ0 ) = 0
by Lemma 2.2. This is a contradiction with Definition 2.4, since again by (2.6) we have
fω,τ 6≡ 0.
2.2
Families of projections
Let K =
Sm
i=1 (ri K
+ bi ) ⊂ R2 be a self-similar set. Recall Definition 1.42. As mentioned in
θ
},
Section 1.7, projθ K is also a self-similar set on Lθ . This set is the attractor of {S1θ , . . . , Sm
where Siθ (x) = ri x + projθ bi . We now have a one-parameter family of i.f.s.
θ
Lemma 2.7. The one-parameter family of i.f.s. {S1θ , . . . , Sm
}θ∈[0,π) satisfies the transversality
condition Definition 2.4.
Proof. Note first that if (x, y) ∈ R2 , then, with · being the inner product, (x, y) · (cos θ, sin θ) =
x cos θ + y sin θ is the projection of (x, y) on the line Lθ . Now
d
π
π
(x cos θ + y sin θ) = −x sin θ + y cos θ = x cos(θ + ) + y sin(θ + )
dθ
2
2
π
π
= (x, y) · (cos(θ + ), sin(θ + )
2
2
e : Ω → K be the
so the derivative is the projection of (x, y) on the line perpendicular to Lθ . Let Π
natural projection map. Then
e
e
Π(θ, ω) = projθ ◦ Π(ω)
= Π(ω)
· (cos θ, sin θ),
so simply by Pythagoras’ theorem we have
2
|Π(θ, ω) − Π(θ, τ )| + |
2
d
e
e )|2 .
− Π(τ
Π(θ, ω) − Π(θ, τ ) | = | Π(ω)
dθ
0
e
e ), hence the claim.
Now we see that fω,τ (λ) = 0 and fω,τ
(λ) = 0 if and only if Π(ω)
= Π(τ
Remark 2.8. For a family of projections θ ∈ IP if and only if projθ : K → R is not an injection.
Chapter 3
Zero Hausdorff measure
This chapter concentrates on the Hausdorff measures of the projections of planar self-similar sets.
From this perspective the main result is Theorem 3.6. This follows easily from more general result
Theorem 3.5, for which we need the following preliminary theorems.
Let A = {1, . . . , m} as in Chapter 2 and recall the notations from Section 1.5 regarding
S∞
sequences in A. Denote A∗ = n=1 An . Also, for S : Rd → Rd let kSk = sup{|Sx| : |x| ≤ 1}.
Theorem 3.1 (Bandt and Graf). Let K be the self-similar set generated by an i.f.s. {Si }i∈A of
the form (2.1), and let s be the similarity dimension. Then Hs (K) = 0 if and only if for any
ε > 0 there exists u, v ∈ A∗ , u 6= v such that
kSu−1 ◦ Sv − Idk < ε.
Proof. See [1].
The similitudes we consider are of the form Siλ (x) = ri (λ)x + ai (λ), so
r (λ)
S λ (0) − Suλ (0)
v
(Suλ )−1 ◦ Svλ − Id (x) =
−1 x+ v
ru (λ)
ru (λ)
r (λ)
Svλ x0 (λ) − Suλ x0 (λ)
v
− 1 x − x0 (λ) +
.
=
ru (λ)
ru (λ)
Here x0 (λ) is arbitrary and is added for conveniency. We choose x0 (λ) = Π(λ, 1), where
1 = 111 . . . ∈ Ω, so we have Svλ x0 (λ) = Π(λ, v1). With the upcoming proof in mind, we make
the following definition in the spirit of the previous theorem.
Definition 3.2. Let Vε be the set of λ ∈ J such that there exists u, v ∈ A∗ , u 6= v such that
e−ε <
ru (λ)
< eε
rv (λ)
(3.1)
and
|fv1,u1 (λ)| = |Π(λ, v1) − Π(λ, u1)| < εru (λ).
21
(3.2)
CHAPTER 3. ZERO HAUSDORFF MEASURE
22
Lemma 3.3. For any γ > 0, β ∈ (0, 1), and λ0 ∈ J there exists N ∈ N with the following
property: For any s, t ∈ A∗ , with rs (λ0 )/rt (λ0 ) ∈ [β, β −1 ], there exists u, v ∈ A∗ such that s, t
are their respective prefixes, |u| − |s| ≤ N , |v| − |t| ≤ N , and
e−γ <
ru (λ0 )
< eγ
rv (λ0 )
(3.3)
Proof. Let γ > 0, β ∈ (0, 1) and λ0 ∈ J be fixed. For any finite sequence s = s1 , s2 , . . . sn ∈ A∗
P
Psn
we denote i∈s f (i) = i=s
f (i). Suppose that s, t ∈ A∗ such that rs (λ0 )/rt (λ0 ) ∈ [β, β −1 ].
1
We want to find p, q ∈ A∗ such that
rs (λ0 ) rp (λ0 )
·
∈ (e−γ , eγ )
rt (λ0 ) rq (λ0 )
(3.4)
and |p|, |q| ≤ N , where N ∈ N is a uniform bound, that is, it depends only on γ, β and λ0 . In
other words
X
log ri (λ0 ) −
i∈s
X
log ri (λ0 ) ∈ [log β, − log β],
(3.5)
i∈t
and we want to find p, q ∈ A∗ as above such that
X
X
X
X
log ri (λ0 ) −
log ri (λ0 ) +
log ri (λ0 ) −
log ri (λ0 ) ∈ (−γ, γ).
i∈s
i∈t
i∈p
Let ai = log ri (λ0 ). Suppose first that
ai
aj
i∈q
∈ Q for all i, j ∈ A. Therefore for any i ∈ A we can
write ai = qi a1 , where qi ∈ Q. The set of all possible values in (3.5) is
nX
o
X
L :=
ai −
ai : s, t ∈ A∗ ∩ [log β, − log β].
i∈s
(3.6)
i∈t
m
Scale this set with the lowest common denominator of the rationals {qi }i=1 and with the inverse
of a1 to achieve the set
nX
i∈s
ki −
X
o
ki : s, t ∈ A∗ ∩ [−c, c],
i∈t
where ki ∈ N for every i ∈ A and c > 0. This set contains only integers from a bounded
interval. Therefore L is finite. For every point x ∈ L choose some sx , tx ∈ A∗ such that x =
P
P
i∈tx ai . Since L is finite, there is N ∈ N such that max{|sx |, |tx |} ≤ N for every
i∈sx ai −
x ∈ L. Thus for any rs (λ0 )/rt (λ0 ) ∈ [β, β −1 ] we can choose p = tx and q = sx . Then
X
X
X
X
ai = 0,
ai −
ai −
ai +
i∈s
which means that
log
i∈tx
i∈t
i∈sx
r (λ ) r (λ ) rstx (λ0 )
s 0
0
t
= log
· x
=0
rtsx (λ0 )
rt (λ0 ) rsx (λ0 )
and (3.4) is satisfied. Next we deal with the opposite case. Suppose that there are some i, j ∈ A
such that
ai
aj
∈
/ Q. Then the set
{x : x = nai (modaj ) for some n ∈ N}
CHAPTER 3. ZERO HAUSDORFF MEASURE
23
is dense in [0, aj ). For this see for example [3, page 4]. By this we can choose k, l ∈ N such
that kai − laj = δ < 2γ. Let p = iii . . . i and q = jjj . . . j, with |p| = k and |q| = l. Then
P
P
β
we have
δ = i∈p ai − i∈q ai . Now by choosing a suitable b ∈ Z, |b| ≤ 2 log
δ
X
i∈s
ai −
X
ai + bδ ∈ (−γ, γ).
i∈t
Equivalently
rs (λ0 ) rp (λ0 ) b
·
∈ (e−γ , eγ ).
rt (λ0 )
rq (λ0 )
Thus for any rs (λ0 )/rt (λ0 ) ∈ [β, β −1 ] we can extend s and t by not more than
2 log β
δ
max{k, l}
elements to satisfy (3.3).
Lemma 3.4. For any u ∈ A∗ and λ1 , λ2 ∈ J,
L
ru (λ2 )
≤ e β |u||λ2 −λ1 |
ru (λ1 )
where β is from (2.2) and L := maxi∈A kri0 kC(J) .
Proof. By applying the mean value theorem we get
log
ri (λ2 )
= log ri (λ2 ) − log ri (λ1 ) ≤ |log ri (λ2 ) − log ri (λ1 )|
ri (λ1 )
L
1
≤ |ri (λ2 ) − ri (λ1 )| ≤ |λ2 − λ1 |
β
β
for any i = 1, . . . , m and the claim follows.
Denote C1 =
1
1−ρ
maxi∈A kai kC(J) , where ρ is as in (2.2). Observe that
|fω,τ (λ)| ≤ |Π(λ, ω)| + |Π(λ, τ )| ≤ 2C1 ,
that is, two points of Kλ cannot be farther away from each other than this. However, this estimate
can be easily tightened by every common step they share in their characterization. In other words,
|fω,τ (λ)| ≤ 2C1 rω∧τ (λ).
(3.7)
Recall that a set which is a countable intersections of opens sets is called a Gδ set.
λ
Theorem 3.5. Suppose that the one-parameter family of iterated function systems {S1λ , . . . , Sm
}λ∈J
satisfies (2.1), (2.2) and Definition 2.4. Then
(i) Hs(λ) (Kλ ) = 0 for Lebesgue almost every λ ∈ IP .
(ii) Suppose, in addition that (2.5) and (2.6) hold. Then IP is a compact perfect set, and the
set {λ ∈ IP : Hs(λ) (Kλ ) = 0} is a dense Gδ set in IP .
CHAPTER 3. ZERO HAUSDORFF MEASURE
24
Note that (i) has content only when the set IP has positive Lebesgue measure. Checking this
is discussed at the end of this chapter.
Proof.
(i) Theorem 3.1 states that Hs(λ) (Kλ ) = 0 if and only if λ ∈ Vε for every ε > 0. To
show that Hs(λ) (Kλ ) = 0 for Lebesgue almost every λ ∈ IP , we show that L(IP \ Vε ) = 0
for every ε > 0. For this we check that IP \ Vε has no Lebesgue density points for any ε > 0.
By (3.1) and (3.2) the set Vε is open as an inverse image of an open set for a continuous
function. Therefore it is indeed Lebesgue measurable. Let ε > 0, λ0 ∈ IP and ρ, β as in
(2.2). By definition, there are such ω, τ ∈ Ω that fω,τ (λ0 ) = 0 but fω,τ 6≡ 0. Let k ∈ N and
choose minimal n, p ∈ N such that rω|n (λ0 ) ≤ ρk and rτ |p (λ0 ) ≤ ρk . Note that since n and
p are minimal, we have n, p ≤ k and
βρk ≤ rω|n (λ0 ) ≤ ρk
This shows that
β≤
and βρk ≤ rτ |p (λ0 ) ≤ ρk .
(3.8)
rω|n (λ0 )
≤ β −1 .
rτ |p (λ0 )
Now we can use Lemma 3.3 to get u, v ∈ A∗ with |u| − n ≤ N, |v| − p ≤ N , such that
u|n = ω|n, v|p = τ |p and
e−ε/3 ≤
ru (λ0 )
≤ eε/3 .
rv (λ0 )
Lemma 3.4 states that if
|λ − λ0 | ≤
then
εβ
3L(k + N )
εβ
L
ru (λ)
≤ e β |u| 3L(k+N ) ≤ eε/3 .
ru (λ0 )
Here the parameters λ and λ0 are interchangeable, and the same argument holds for v, so
we get an approximation for both values
ru (λ) rv (λ)
,
∈ [e−ε/3 , eε/3 ].
ru (λ0 ) rv (λ0 )
(3.9)
By this we see that also
e−ε ≤
e−ε/3 ru (λ0 )
ru (λ)
eε/3 ru (λ0 )
≤
≤
≤ eε .
·
·
rv (λ)
eε/3 rv (λ0 )
e−ε/3 rv (λ0 )
(3.10)
Next we find a sub interval of J for which (3.2) holds. We have fω,τ (λ0 ) = 0 which in other
words means that Π(λ0 , ω) = Π(λ0 , τ ). Then, by (3.7),
|fv1,u1 (λ0 )| ≤ |Π(λ0 , v1) − Π(λ0 , τ )| + |Π(λ0 , ω) − Π(λ0 , u1)| = |fv1,τ (λ0 )| + |fω,u1 (λ0 )|
≤ 2C1 rv1∧τ (λ0 ) + ru1∧ω (λ0 ) ≤ 2C1 rτ |p (λ0 ) + rω|n (λ0 ) ≤ 4C1 ρk
(3.11)
CHAPTER 3. ZERO HAUSDORFF MEASURE
25
0
Because fω,τ 6≡ 0 and fω,τ (λ0 ) = 0 we have, by transversality Definition 2.4, |fω,τ
(λ0 )| > δ
for some δ > 0. By Remark 2.3(ii) there are γ > 0 and k0 ∈ N such that
0
|fv1,u1
(λ)| > δ
(3.12)
as long as |λ − λ0 | < γ and k ≥ k0 . Denote Fk := [λ0 − 4C1 ρk /δ, λ0 + 4C1 ρk /δ]. If k ≥ k0
and 4C1 ρk /δ < γ, then by (3.11) and (3.12) the function fv1,u1 has to reach 0 at some point
of Fk . Thus, there exists λ1 ∈ Fk such that fv1,u1 (λ1 ) = 0. Suppose that k is so large that
4C1 ρk
εβ
< min{
, γ },
δ
3L(k + N )
then (3.1) holds on Fk by (3.10). We have ω|n = u|n so by (3.8) we get
ru (λ0 ) = ru|n (λ0 )run+1 ...u|u| (λ0 ) ≥ βρk β |u|−n ≥ ρk β N +1 .
Moreover, if we assume that ε is small enough so that eε/3 ≤ 2, then again by (3.9) we have
ru (λ0 ) ≤ 2ru (λ)
0
kC(J) ≤ C2 for
when λ ∈ Fk . Remark 2.3(i) makes it possible to take C2 > 0 such that kfζ,ξ
every ζ, ξ ∈ Ω. Because fv1,u1 (λ1 ) = 0 then |fv1,u1 (λ)| ≤ C2 |λ − λ1 |. Assume that ε is so
small that
β N +1 ρk ε
4C1 ρk
≤
.
2C2
δ
Now, if
|λ − λ1 | <
β N +1 ρk ε
,
2C2
then |fv1,u1 (λ)| < (ε/2)β N +1 ρk ≤ εru (λ), so λ ∈ Vε ∩ Fk . Finally
L(Fk ∩ Vε )
β N +1 ρk ε/(2C2 )
δβ N +1 ε
.
≥
=
L(Fk )
8C1 ρk /δ
16C1 C2
Now Fk is an interval with a middle point λ0 . Because L(Fk ) → 0 as k → ∞, we see with
the help of Theorem 1.14, that λ0 cannot be a Lebesgue density point for the set IP \ Vε .
As λ0 was arbitrary it follows that IP \ Vε has no Lebesgue density points.
(ii) Here condition (2.6) was also assumed. So we can use Lemma 2.6, which states that IP is
compact. This tells us that Baire category theorem holds in IP . Let ε > 0 and let Vε be as
above. As noted in the proof of (i) the set Vε is open. Thus V1/n ∩ IP is open in IP . In (i)
we took arbitrary point from IP and found λ1 ∈ Vε ∩ IP from its neighbourhood. Thus
the set Vε ∩ IP is also dense. Therefore the dense Gδ property follows from the identity
{λ ∈ IP : Hs(λ) (Kλ ) = 0} =
\
(V1/n ∩ IP )
n∈N
CHAPTER 3. ZERO HAUSDORFF MEASURE
26
with Baire category theorem. Finally, to see that IP is perfect, we check that it has no
isolated points. To this end, let λ0 ∈ IP and ω, τ ∈ Ω such that fω,τ (λ0 ) = 0 but fω,τ 6≡ 0.
For every k ∈ N we construct τ k in the following way. For the beginning of the sequence τ k
we choose τ |k. This guarantees that the points ω and τ k will be close enough to each other.
For the second infinite part look at the sequence ω 0 := ωk+1 ωk+2 ωk+3 . . . . Condition (2.5)
ensures that Kλ0 is not a singleton, so we can choose τ 0 ∈ Ω such that fω0 ,τ 0 (λ0 ) 6= 0. Finally
we add these parts together to get τ k = ω|k τ 0 . Then rω|k , rτ k |k ≤ ρk . Now fω,τ k (λ0 ) 6= 0,
but |fω,τ k (λ0 )| ≤ 4Cρk as in (3.11). Again, using transversality we can find δ > 0 such that
0
0
|fω,τ
(λ0 )| > δ and by Remark 2.3(ii) we can find γ > 0 and k0 ∈ N such that |fω,τ
k (λ)| > δ
when λ ∈ [λ0 − γ, λ0 + γ] and k ≥ k0 . Suppose that k ∈ N is big enough that 4Cρk /δ < γ
and k ≥ k0 . Then, as in (i), we see that there exists λ1 ∈ [λ0 − 4Cρk /δ, λ0 + 4Cρk /δ] such
that fω,τ k (λ1 ) = 0. This means that λ1 ∈ IP , so λ0 is not an isolated point.
Theorem 3.6. Let K ⊂ R2 be a self-similar set of dimension s ∈ (0, 1) that is not on a line and
is generated by i.f.s. {Si }i≤m of the form (1.5).
(i) Hs (projθ K) = 0 for Lebesgue almost every θ ∈ IP .
(ii) If the i.f.s. {Si }i≤m satisfies the strong separation condition, then IP is a compact perfect
set, and the set {θ ∈ IP : Hs (projθ K) = 0} is a dense Gδ set in IP .
Proof. Condition (2.6) follows from the strong separation condition, so the claim follows immediately from Theorem 3.5 combined with Lemma 2.7 and Remark 2.8.
3.1
The Lebesgue measure of the set of intersection
parameters
As mentioned before, Theorem 3.5(i) and Theorem 3.6(i) has content whenever IP has positive
Lebesgue measure.
Proposition 3.7. Suppose that the one-parameter family of iterated function systems satisfy
conditions (1.5), (2.2) and Definition 2.4. Then dimH IP ≤ 2smax , where smax = supλ∈J s(λ).
Thus, if smax ∈ (0, 21 ) then L(IP ) = 0.
Proof. When δ > 0, let IPδ be the set of all λ ∈ J for which there exist ω, τ ∈ Ω such that
0
fω,τ (λ) = 0 and |fω,τ
(λ)| > δ.
Transversality Definition 2.4 holds, so IP =
dimH IPδ ≤ 2smax for any δ > 0.
S
n∈N
(3.13)
IP1/n . Thus it is sufficient to show that
CHAPTER 3. ZERO HAUSDORFF MEASURE
27
Let K be the corresponding self-similar set. For u ∈ A∗ let Iuλ := Conv(Kuλ ). As scaling is not
an issue, we can assume that d(K) = 1. Then also |Iuλ | = ru (λ) ≤ d(Su (K)). Let δ > 0, λ0 ∈ IPδ
and ω, τ ∈ Ω satisfy (3.13). By Remark 2.3(ii) we may choose k ∈ N and γ > 0 such that
0
|fζ,ξ
(λ)| ≥
δ
2
(3.14)
for |ζ ∧ ω|, |ξ ∧ τ | ≥ k and |λ0 − λ| ≤ γ. This implies that the intervals Iuλ and Ivλ move relative
to each other with a speed not less than δ/2 as λ varies close to λ0 and u, v are sufficiently
long prefixes of ω, τ . Let J = [λ1 , λ2 ]. Write w ∈ Cε (λ1 ) if |Iwλ1 | ≤ ε and no prefix of w has
this property. For w ∈ Cε (λ1 ), we can limit the size |Iwλ | uniformly for all λ ∈ J by choosing
ε > 0 small enough, since |Iwλ | ≤ d(Sw (K)). Then, if ε > 0 is small enough, (3.14) says that
the intervals Iuλ and Ivλ move past each other as λ ∈ [λ0 − γ, λ0 + γ], where u, v are the unique
prefixes of ω, τ from the set Cε (λ1 ). This means that when λ varies close to λ0 , the sets Iuλ and
Ivλ intersect each other on an interval of length at most 2δ maxλ∈J |Iuλ | + maxλ∈J |Ivλ | , which is a
subinterval of [λ0 − γ, λ0 + γ]. As J is a bounded interval, the number of these intervals where Iuλ
|J |
2γ . Choose
C
+ 2γ
sets of
and Ivλ move past each other, as u, v are fixed and λ is the variable, is at most 1 +
C = max{|J |, 2}. Then {λ ∈ J : Iuλ ∩ Ivλ 6= ∅} can be covered with a union of 1
length Cδ maxλ∈J |Iuλ | + maxλ∈J |Ivλ | .
P
s(λ )
Note that |Iwλ1 | ≥ βε for w ∈ Cε (λ1 ) and w∈Cε (λ1 ) rw 1 = 1, which can be calculated as in
the proof of Theorem 1.35. Therefore
X
1=
s(λ1 )
rw
≥ #Cε (λ1 ) · (βε)s(λ1 )
w∈Cε (λ1 )
giving
β −s(λ1 ) ε−s(λ1 ) ≥ #Cε (λ1 ).
Thus the number of all possible pairs u, v from Cε (λ1 ), u 6= v, such that the sets Iuλ and Ivλ
intersect for some λ ∈ J is then at most β −2s(λ1 ) ε−s2(λ1 ) .
As implied at the beginning of the proof, for a fixed w ∈ Cε (λ1 ) the sets Iwλ are roughly the
same size for all λ ∈ J. More rigorously, if
t := min
o
n log r (λ)
i
: λ ∈ J, i = 1, . . . , m ,
log ri (λ1 )
then
t
|Iwλ1 | = (rw1 (λ1 ) · · · rwn (λ1 ))t = et log(rw1 (λ1 )···rwn (λ1 )) = et log rw1 (λ1 )+···+t log rwn (λ1 )
λ
≥ elog rw1 (λ)+···+log rwn (λ) = rw1 (λ) · · · rwn (λ) = |Iw
|
for all w ∈ A∗ and λ ∈ J, since
t log rj (λ1 ) ≥
log rj (λ)
log rj (λ1 )
log rj (λ1 )
CHAPTER 3. ZERO HAUSDORFF MEASURE
28
for every j = 1, . . . , m. Now maxλ∈J |Iuλ | + maxλ∈J |Ivλ | ≤ 2εt .
In conclusion, if ω, τ ∈ Ω are such sequences that there is λ ∈ J such that (3.13) holds, then
by considering the prefixes u, v ∈ Cε (λ1 ) of ω, τ we see that every such λ belongs to a set which
can be covered by 1 +
C
2γ
sets of length
C
t
δ 2ε .
In fact, this set covers all such λ for any ω 0 , τ 0 ∈ Ω
which have the same prefixes u and v from the set Cε (λ1 ) as ω and τ have. Thus the whole set IPδ
can be covered with (1 +
C
−2s(λ1 ) −2s(λ1 )
ε
2γ )β
sets each of length
C
t
δ 2ε .
This allows us to estimate
the box-counting dimension of IPδ which in turn is an upper limit for Hausdorff dimension.
C
)β −2s(λ1 ) ε−2s(λ1 )
log (1 + 2γ
2s(λ1 )
2smax
dimB IPδ ≤ lim sup
=
≤
.
(3.15)
C
t
t
t
− log δ 2ε
ε→0
To get rid of t, divide J into small intervals and apply the above to each part of the division.
Since Hausdorff dimension is countably stable, it is enough to consider the part with the biggest
dimension. We can make this division as dense as we want, so by continuity of ri , we can make t
in (3.15) go as close to 1 as we want. Therefore dimH IPδ ≤ 2smax .
In homogenous case there is easy way to check whether L(IP ) > 0. This is presented in the
following lemma.
λ
}λ∈J be a one-parameter family of homogenous i.f.s. on R satisfying
Lemma 3.8. Let {S1λ , . . . , Sm
(2.1) and (2.6). Suppose that for some i 6= j there exists a subinterval Je ⊂ J such that
Conv(Kiλ ) ∩ Conv(Kjλ ) 6= ∅
for every
λ ∈ Je
(3.16)
and
Kλ − Kλ
is an interval for every
e
λ ∈ J.
(3.17)
Then Je ⊂ IP .
Proof. For any A ⊂ R we have A − A = {t ∈ R : A ∩ (A + t) 6= ∅}. From condition (3.17) we
have Kλ − Kλ = Conv(Kλ − Kλ ) = Conv(Kλ ) − Conv(Kλ ). The latter equality is true for any set.
Also, Kiλ is a translate of r(λ)K λ for every i = 1, . . . , m and λ ∈ J as the families are homogenous,
e Then Conv r(λ)Kλ + ai ∩ Conv r(λ)Kλ +
in other words Kiλ = r(λ)Kλ + ai . Let λ ∈ J.
aj 6= ∅. This means that aj − ai ∈ t ∈ R : Conv r(λ)Kλ ∩ Conv r(λ)Kλ + t 6= ∅ =
Conv r(λ)Kλ − Conv r(λ)Kλ = r(λ)Kλ − r(λ)Kλ = {t ∈ R : r(λ)Kλ ∩ r(λ)Kλ + t =
6 ∅}. Thus
λ
λ
r(λ)K + ai ∩ r(λ)K + aj 6= ∅ and λ ∈ IP by Lemma 2.6.
Remark 3.9. The conditions (3.16) and (3.17) can be verified by little calculations. Let {S1 , . . . , Sm }i≤m
be a homogenous i.f.s. on R, with Si (x) = rx + ai , and K be the corresponding self-similar set.
Suppose that g is the minimal gap between two consecutive elements of {ai }i≤m and G is
the maximal gap between two consecutive elements of Γ := {ai − aj }i,j≤m . We claim that
Conv(Ki ) ∩ Conv(Kj ) 6= ∅ for some i 6= j if and only if
r
max Γ ≥ g,
1−r
(3.18)
CHAPTER 3. ZERO HAUSDORFF MEASURE
29
and K − K is an interval if and only if
r
1
max Γ ≥ G.
1−r
2
(3.19)
Suppose for simplicity that a1 ≤ · · · ≤ am . Indeed, for Conv(Ki ) and Conv(Kj ) to intersect for
some i 6= j, we need to have
ai + r
∞
X
am rn ≥ ai+1 + r
n=0
∞
X
a1 rn
n=0
r
r
am ≥ ai+1 +
a1
⇔ ai +
1−r
1−r
r
⇔
am − a1 ≥ ai+1 − ai
1−r
for some i = 1, . . . , m − 1, and (3.18) follows. For the equation (3.19), consider the set
K−K =
∞
nX
cn rn : cn ∈ Γ
o
n=0
which is also a self-similar set. Suppose again that Γ = {c1 , . . . , cM }, where c1 ≤ · · · ≤ cM .
Similarly to the first part, if K − K is an interval, then
∞
X
ci + r
n=0
max{cj }rn ≥ ci+1 + r
j≤M
∞
X
n=0
min {cj }rn
j≤M
for every i = 1, . . . , M − 1. Because of the fact that sets of the form A − A are symmetrical with
respect to the origin, we have maxj≤M {cj } = cM and minj≤M {cj } = −cM . Therefore we need
to have
r
r
cM ≥ ci+1 +
(−cM )
1−r
1−r
r
⇔2
cM ≥ ci+1 − ci
1−r
ci +
for every i = 1, . . . , M − 1. Hence (3.19). On the other hand, let K − K be a set with (3.19).
Now K − K is a self-similar set generated by similitudes Si (x) = rx + ci , 1 = 1, . . . M . Evidently
cM
cM
cM
cM
cM
M
SM 1−r
= 1−r
and = S1 − 1−r
= −c
1−r , since c1 = −cM . Therefore Si [− 1−r , 1−r ] ⊂
cM
cM
cM
cM
cM
, 1−r
, 1−r
[− 1−r
] and Si [− 1−r
] is an interval of length r2 1−r
for every i. Since we assumed
cM
cM
cM
r2 1−r
] is the unique
≥ G, every two consecutive images intersect each other. Thus [− 1−r
, 1−r
attractor K − K.
The following two lemmas sharpens Proposition 3.7 for families of projections.
Lemma 3.10. Let K ⊂ R2 be a self-similar set generated by an i.f.s. which satisfies (1.5) and
the strong separation condition. Then dimH IP ≤ dimH (K − K).
Proof. As K is compact, strong separation condition guarantees that there is t > 0 such that
Ki (t) ∩ Kj (t) = ∅ for every i 6= j. If x ∈ Ki and y ∈ Kj , then |x − y| ≥ t, so
[
(Ki − Kj ) ⊂ {x ∈ R2 : |x| ≥ t} =: A.
i6=j
CHAPTER 3. ZERO HAUSDORFF MEASURE
30
Let Υ(x) := x/|x|. If x, y ∈ A we have
x
y x y 1
−Υ
Υ(x) − Υ(y) = Υ
≤ − = |x − y|.
t
t
t
t
t
The inequality holds since x/t and y/t are on the unit circle or outside of it. This means that Υ
is a Lipschitz mapping from the set A to the unit circle. Strong separation condition also gives us
(2.6), so Lemma 2.6(i) states that
IP = {θ ∈ [0, π) : projθ Ki ∩ projθ Kj 6= ∅
for some
i 6= j }.
This means that if θ ∈ IP , then for some i 6= j there are two points x ∈ Ki and y ∈ Kj which are
on the same line perpendicular to Lθ . In other words, x − y is on the line Lθ+π/2 and
[
π
π
(Ki − Kj ) .
cos( + θ), sin( + θ) ∈ Υ
2
2
i6=j
Therefore
[
o
π
π
+ θ), sin( + θ) ∈ Υ
(Ki − Kj )
2
2
i6=j
S
and IP is the same set as Υ i6=j (Ki − Kj ) apart from a rotation by π/2. So for the dimension
IP =
n
θ ∈ [0, π) :
cos(
we get
dimH IP = dimH Υ
[
[
(Ki − Kj ) ≤ dimH (Ki − Kj ) = max dimH (Ki − Kj ) ≤ dimH (K − K),
i6=j
i6=j
i6=j
since Lipschitz functions preserve dimension.
Lemma 3.11. Let K ⊂ R2 be a self-similar set generated by an i.f.s. which satisfies (1.5) and
the strong separation condition. If dimH (K − K) < 1, then Hs (projθ K) > 0 for almost every
θ ∈ [0, π). This happens especially when s = dimH K < 21 .
Proof. The first part follows immediately from Lemma 3.10, Lemma 2.6(iii) and the fact that
the Lebesgue measure of a set of Hausdorff dimension less than one is zero. For the latter part
note that if x ∈ A − A, then x = z − y, y, z ∈ A and (y, z) ∈ A × A. Now proj 3π
(y, z) is rotation
4
equivalent (by
5π
4 )
3π
to y cos 3π
4 + z sin 4 =
√1 (z
2
− y). Therefore A − A and proj 3π
(A × A) are
2
the same sets up to scaling and rotation for every A ⊂ R2 . Because K is self-similar Hausdorff
and packing dimensions agree. Therefore by [10, Corollary 8.11] we have
(K × K) ≤ dimH (K × K) = 2dimH K.
dimH (K − K) = dimH proj 3π
2
Chapter 4
Positive packing measure
Recall the notations from the beginning of Chapter 2. In Chapter 3 we proved that the s-dimensional Hausdorff measure of a projection of s-dimensional self-similar set is typically zero, if there
is some overlap in the projection. In this chapter we restrict our attention to packing measure in
similar conditions. We prove that unlike the Hausdorff measure, packing measure of this kind of
projection is typically positive. Also, this is shown for Borel sets, not just for self-similar sets.
The main result is Theorem 4.1. This yields useful and interesting Theorems 4.7 and 4.8.
Theorem 4.1. Let (Ω, d) be a complete separable metric space, A ⊂ Ω a Borel set and Hγ (A) > 0.
Also, let Πλ : A → R, λ ∈ J be a one-parameter family of continuous maps, where J is a closed
interval. Suppose that α(λ) is some positive function such that for all ω, τ ∈ Ω, ω =
6 τ the
functions
Ψ(λ) = Ψω,τ (λ) =
Πλ (ω) − Πλ (τ )
d(ω, τ )α(λ)
(4.1)
belong to C 1 (J). Suppose also that there are δ > 0 and M > 0 independent of ω and τ such that
kΨkC 1 (J) ≤ M
(4.2)
|Ψ(λ)| + |Ψ0 (λ)| > δ
(4.3)
dimH {λ ∈ J : P σ(λ) (Πλ (A)) = 0} ≤ σmax
(4.4)
and
for all λ ∈ J. Then
where σ(λ) = γ/α(λ) and σmax = sup{σ(λ) : λ ∈ J }.
Remark 4.2.
(a) The theorem has content only when σmax < 1.
(b) Since Ψ is bounded, the functions Πλ are Hölder with exponent α(λ). Thus P γ (A) < ∞
implies P s(λ) Πλ (A) < ∞.
(c) The function α is continuous, since Ψ ∈ C 1 (J).
31
CHAPTER 4. POSITIVE PACKING MEASURE
32
(d) Condition (4.3) is closely related to Definition 2.4 and was mentioned in Lemma 2.6.
Lemma 4.3. Suppose that Ψ ∈ C 1 (J) and conditions (4.2) and (4.3) holds for Ψ. Then for
every ρ ≤ δ/2 the set {λ ∈ J : Ψ(λ) ≤ ρ} is a union of at most 1 + M |J |/δ intervals of length
at most 4ρ/δ.
Proof. If |Ψ(λ)| ≤ ρ ≤ δ/2, then |Ψ0 (λ)| ≥ δ/2. So the set {λ ∈ J : |Ψ(λ)| ≤ ρ} is a union of
disjoint intervals {Ai } on each of which the function Ψ is monotone. The length of such interval
is at most 4ρ/δ. To check how many such intervals there is observe that {λ ∈ J : |Ψ(λ)| ≤ ρ} ⊂
{λ ∈ J : |Ψ(λ)| ≤ δ/2} and the latter set is also a union of disjoint intervals {Bi } on each of
which the function Ψ is monotone. Therefore each interval Bi contains exactly one of the sets Ai .
Since |Bi | ≥
δ
M
for every i we need at most 1 + M |J |/δ such intervals to cover J.
Proposition 4.4. Under the assumptions of Theorem 4.1, let smax < s1 < 1 and let η be a
probability Frostman measure on J such that η B(x, ρ) ≤ cρs1 for any ρ > 0. Suppose also that
µ is a Borel probability measure on Ω such that
(µ × µ){(ω, τ ) ∈ Ω2 : d(ω, τ ) < r} ≤ C1 rγ
for all r> 0. Then
Z Z
J :=
J
(4.5)
νλ B(x, rα(λ) ) dνλ (x) dη(λ) ≤ Crγ
(4.6)
R
for all r> 0, where νλ = Πλ∗ µ is the image measure.
Proof. Using Fubini’s theorem, we get
Z Z
J =
µ {ω ∈ Ω : Πλ (ω) − Πλ (τ ) ≤ rα(λ) } dµ(τ ) dη(λ)
ZJ ZΩ Z
χ
dµ(ω) dµ(τ ) dη(λ)
=
{ω∈Ω : Πλ (ω)−Πλ (τ ) ≤r α(λ) }
J Ω Ω
Z Z Z
=
dη(λ) dµ(ω) dµ(τ )
χ
{λ∈J : Πλ (ω)−Πλ (τ ) ≤r α(λ) }
Ω Ω J
Z Z
η {λ ∈ J : Πλ (ω) − Πλ (τ ) ≤ rα(λ) } dµ(ω) dµ(τ ).
=
Ω
Ω
Decompose this integral as
ZZ
+
d(ω,τ )<r
∞ ZZ
X
k=1
=: J0 +
2k−1 r≤d(ω,τ )<2k r
∞
X
Jk .
k=1
Recall that Ψω,τ (λ) = Πλ (λ) − Πλ (τ ) d(ω, τ )−α(λ) and observe that for d(ω, τ ) ≥ 2k−1 r we have
η {λ : |Πλ (λ) − Πλ (τ )| ≤ rα(λ) } ≤ η {λ : |Ψω,τ (λ)| ≤ rα(λ) (2k−1 r)−α(λ) }
= η {λ : |Ψω,τ (λ)| ≤ 2−(k−1)α(λ) }
≤ η {λ : |Ψω,τ (λ)| ≤ 2−(k−1)αmin } ,
CHAPTER 4. POSITIVE PACKING MEASURE
33
where αmin = inf λ∈J α(λ) > 0, since α is continuous. Choose k0 ∈ N large enough to 2−(k0 −1)αmin ≤
δ/2. Now we can use Lemma 4.3 to see that {λ : |Ψω,τ (λ)| ≤ 2−(k−1)αmin } is contained in a
finite union of intervals of length C 0 2−kαmin , and since η is Frostman measure we get that
η {λ : |Ψω,τ (λ)| ≤ 2−(k−1)αmin } ≤ C2 2−kαmin s1
(4.7)
for every ω, τ and k ≥ k0 . Now we can continue the estimation of J with (4.5) and (4.7) and
recalling that µ and η are probability measures:
J ≤
kX
0 −1
(µ × µ){(ω, τ ) : d(ω, τ ) < 2k r} + C2
k=0
≤ C3
kX
0 −1
∞
X
2−kαmin s1 (µ × µ){(ω, τ ) : d(ω, τ ) < 2k r}
k=k0
(2k r)γ + C4
k=0
∞
X
2−kαmin s1 (2k r)γ .
k=k0
Observe that for the first sum
kX
0 −1
(2k r)γ = rγ
k=0
kX
0 −1
2kγ ≤ rγ 2k0 γ
k=0
and for the second
∞
X
2−kαmin s1 (2k r)γ = rγ
k=k0
∞
X
(2γ−αmin s1 )k = rγ
k=k0
(2γ−αmin s1 )k0
1 − 2γ−αmin s1
because we assumed that s1 > smax = γ/αmin . We conclude that there is C such that J ≤ Crγ
for every r > 0.
Theorem 4.5. Suppose that we are in the setting of Theorem 4.1 and µ is a Borel probability
measure as in Proposition 4.4. Then
Z
dimH λ ∈ J :
Ds(λ) (νλ , x) dνλ (x) = ∞ ≤ smax ,
(4.8)
where
νλ B(x, r)
Ds(λ) (νλ , x) = lim inf
r→0
rs(λ)
is the lower s(λ)-dimensional density of νλ at the point x.
Proof. Denote the set in (4.8) by J0 . Suppose that J0 has Hausdorff dimension greater than smax .
Then there is s1 > smax such that Hs1 (J0 ) > 0, so Theorem 1.30 implies that there is a measure
η as in Proposition 4.4 supported on the set J0 . Since η(J0 ) > 0, we have
Z Z
Ds(λ) (νλ , x) dνλ (x) dη(λ) = ∞.
J
CHAPTER 4. POSITIVE PACKING MEASURE
34
But on the other hand, since s(λ) = γ/α(λ), Proposition 4.4 and Fatou’s lemma yields
Z Z
Z Z
νλ B(x, rα(λ) )
dνλ (x) dη(λ)
lim inf
Ds(λ) (νλ , x) dνλ (x) dη(λ) =
r→0
rα(λ)s(λ)
J
J
Z Z
νλ B(x, rα(λ) )
dνλ (x) dη(λ)
=
lim inf
r→0
rγ
J
Z Z
1
≤ lim inf γ
νλ B(x, rα(λ) ) dνλ (x) dη(λ)
r→0 r
J
1
≤ lim inf γ Crγ < ∞,
r→0 r
which is a contradiction.
We will use the following result to prove Theorem 4.1.
Theorem 4.6 (Taylor and Tricot). For any α > 0 and n ∈ N there exists a constant p(α, n) > 0
with the following property: For any Borel probability measure ν on Rn , Borel set A ⊂ Rn and
C > 0, if Dα (ν, x) ≤ C for every x ∈ A, then
P α (A) ≥ C −1 p(α, n)ν(A).
Proof. See [13].
Proof of Theorem 4.1. Since Hγ (A) > 0 Frostman’s lemma in metric spaces [7] states that there
exists a probability measure µ supported on the set A such that µ B(x, r) ≤ Crγ for any x ∈ a
and r > 0. Then
(µ × µ){(ω, τ ) ∈ Ω2 : d(ω, τ ) < r} =
Z
ZA
=
µ {ω ∈ Ω : d(ω, τ ) < r} dµ(τ )
µ B(τ, r) dµ(τ ) ≤ µ(A)Crγ
A
so condition (4.5) holds for µ. Now νλ is a Borel probability measure supported on Πλ (A) for
every λ ∈ J. If Ds(λ) (νλ , x) < ∞ for almost every x ∈ Πλ (A), then there is A0 ⊂ Πλ (A) of
positive νλ measure such that Ds(λ) (νλ , x) ≤ C for x ∈ A0 . Therefore by Theorem 4.6 we see that
0 < P s(λ) (A0 ) ≤ P s(λ) (Πλ (A)). By (4.8) this happens for all λ ∈ J except on a set of dimension
less than or equal to smax .
Theorem 4.7. Suppose that there is some i.f.s. satisfying (2.1), (2.2), (2.4) and (2.6) hold, and
ϕ(λ)
ri (λ) = ri
for some positive function ϕ(λ) and some reals ri ∈ (0, 1). Then 0 < P s(λ) (Kλ ) < ∞
for all λ ∈ J except a set of Hausdorff dimension smax = supλ∈J s(λ).
Proof. Let Ω = AN with the metric d(ω, τ ) = rω∧τ (λ1 ), where J = [λ1 , λ2 ]. Let also A = Ω,
Πλ = Π(λ, ·) be the natural projection map (2.3). Now, with α(λ) = ϕ(λ)/ϕ(λ1 ),
Ψω,τ (λ) =
Πλ (ω) − Πλ (τ )
Πλ (ω) − Πλ (τ )
= Πλ (ω 0 ) − Πλ (τ 0 ),
=
α(λ)
rω∧τ (λ)
d(ω, τ )
CHAPTER 4. POSITIVE PACKING MEASURE
35
where ω10 6= τ10 . Since (2.6) is assumed (4.3) holds by Lemma 2.6(iv). Also, (4.2) holds by Lemma
2.2. Let γ = dimH A. Then by Lemma 1.35 for γ = s(λ1 ) we have Hγ (A) > 0. Moreover
m
X
ri (λ)
s(λ1 )
α(λ)
=
m
X
ϕ(λ1 )s(λ1 )
ri
= 1,
i=1
i=1
so s(λ1 )/α(λ) = s(λ) and the claim follows by Theorem 4.1.
Theorem 4.8. Let K ⊂ R2 be any Borel set such that Hs (K) > 0 for some s ∈ (0, 1). Then
P s (projθ K) > 0 for almost every θ. Moreover,
dimH {θ ∈ [0, π) : P s (projθ K) = 0} ≤ s.
Proof. Let (Ω, d) be R2 with the Euclidean metric. Further, let A = K and Πθ = projθ . Choose
α(θ) = 1 for all θ whence
Ψx,y (θ) =
projθ (x) − projθ (y)
.
|x − y|
for x, y ∈ R2 . By Lemma 2.2 we see that condition (4.2) holds. Moreover, with calculations
similar to those in the proof of Lemma 2.7 we see that condition (4.3) is satisfied. Also, smax = s,
thus the claim follows by Theorem 4.1.
Chapter 5
Examples
Here we present two examples in which we use the theorems achieved. The examples are very
similar in content. The first one concerns projections of planar Sierpinski gaskets onto lines of
different angles. In the second example the same study is made for planar Cantor dusts.
Example 5.1. Let
Kur =
∞
nX
o
an rn : an ∈ T ,
n=0
with T = {0, 1, u}, where 0 < r < 1 is fixed and u ∈ R is the parameter. As noted in Example
1.44, this set is affine-equivalent to the family of projections of the s-dimensional sierpinski gasket
G r , where s = log 3/ log r−1 . Thus transversality condition holds by Lemma 2.7. Also, as noted in
the same example, we may assume, without loss of generality, that u ≥ 2. Then, by using Lemma
3.8 and Remark 3.9 with the appropriate notations, we have Γ = {0, ±1, ±u, ±(u − 1)}. Now
2(1−r)
max Γ = u, g = 1 and G = max{1, u − 2}. Thus (3.18) and (3.19) implies that [ 1−r
r , 1−3r ] ⊂ IP .
This interval is not empty for r ∈ ( 15 , 31 ), hence L(IP ) > 0. Therefore Theorem 3.5 is meaningful
and states that
Hs (Kur ) = 0
2(1−r)
for every r ∈ ( 15 , 13 ) and for almost every u ∈ [ 1−r
r , 1−3r ]. On the other hand, Theorem 4.7
implies that
0 < P s (Kur ) < ∞
2(1−r)
r
for every r ∈ ( 15 , 13 ) and almost every u ∈ [ 1−r
r , 1−3r ], since smax < 1. Moreover G satisfies the
strong separation condition and, recalling Section 1.6, G r − G r is a self-similar set generated by 7
similitudes. Therefore the similarity dimension is log 7/ log r−1 . So, by Lemma 3.11, Hs (Kur ) > 0
for every r <
1
7
and almost every u.
Example 5.2. Let Cr2 = Cr × Cr , where Cr is the middle-α Cantor set for α = 1 − 2r. The family
of projections {projθ Cr2 }0≤θ<π is affine-equivalent to the self-similar sets generated by iterated
36
CHAPTER 5. EXAMPLES
37
function systems {rx, rx + 1, rx + u, rx + 1 + u}u>0 . We want to have dimH Cr2 < 1, so we assume
that r <
1
4.
As with the previous Example, we may assume, without loss of generality, that
u ≥ 1. Transversality condition holds by Lemma 2.7. With the appropriate notations we have
max Γ = 1 + u, g = min{1, u − 1}, G = max{1, u − 2}, and we can check with conditions (3.18)
1 1
2
and (3.19) that [ 1−2r
r , 1−3r ] ⊂ IP . This interval is non-empty for r ∈ ( 6 , 4 ). As in the previous
2
Example we can conclude that for every r ∈ ( 16 , 14 ) and almost every u ∈ [ 1−2r
r , 1−3r ] we have
Hs (projθ Cr2 ) = 0 and 0 < P s (projθ Cr2 ) < ∞,
where s is the similarity dimension log 4/ log r−1 . Moreover, similar to the previous Example,
Hs (projθ Cr2 ) > 0 for every r <
1
9
and for almost every θ by Lemma 3.11, since Cr2 − Cr2 is a
planar self-similar set with similarity dimension log 9/ log r−r .
The case r ∈ ( 17 , 15 ) in the first example and the case r ∈ ( 19 , 61 ) in the second example remains
open. Anyhow, Peres, Simon and Solomyak [11] suspects that L(IP ) > 0 for almost every such r.
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