Fractals: from the fractional dimension to the heat equation

Abstract
In this thesis we will introduce the world of fractals and explore some of their
properties and capabilities. These structures can be characterized in different
ways, but the main feature for which they are known is that many of them have a
“fractional dimension”, something that defies the usual perception that a line has
dimension one and the plane has dimension two.
In Chapter 1, we will understand different notions of dimension and we will define
a fractal as a set that behaves in a specific way with respect to some of these
definitions; in particular, we will state some results characterizing these sets in
terms of their dimension. After that, we will conclude the chapter by showing a
graph theoretical approach to fractals that will enable us to give a closed formula
for the dimension of the fractal.
In Chapter 2, we will deepen the analysis of fractals with the scope of studying the
heat equation on a fractal domain. The first obstacle to this program is to define
correctly the differential operators on fractal sets: we will overcome it by taking
the limit of a suitable difference operator on an “increasing” sequence of resistance
networks. Under suitable assumptions, the sequence of resistance networks will
stem naturally from the so-called “harmonic structure” associated with the fractal
and it will converge to it.
Similarly to what happens in the modern theory of partial differential equations, it
will be appropriate to define the Sobolev spacesH
1
(Ω) andH
1
0
(Ω) on a fractal set Ω in order to encode the boundary conditions within the operator itself. The definition
of these spaces is not immediate and is related to the need of a more general notion
of the Laplacian operator than the usual one on regular domains. The definition
of Laplacian operator that we will provide enjoys many interesting properties,
among which some unique spectral properties, such as possessing eigenfunctions
that satisfy both Neumann and Dirichlet boundary conditions.
Although it is possible to study partial differential equations on fractals with
all type of boundary condition, for the scope of this thesis, we will only cover
Dirichlet- and Neumann-type boundary conditions. The chapter will culminate
with the study of the heat equation ∂
t
u− ∆ u = 0 with assigned starting condition
u
0
∈L
2
(K,µ ) , where we will give the solution in terms of convolution of the initial
datum u
0
with a specific function that will be referred to as the heat kernel (recall
that in the classical framework it is the fundamental solution to the heat equation
∂
t
u− ∆ u =δ
0
).
Two appendices complement this work. The first one may be interesting for the
curious reader that wants to see some exotic examples of fractals with astonishing
properties such as curves that fill the two-dimensional space or the famous Mandel-
bröt set, that was inspirational to many mathematicians to progress in the study
of fractal sets.
The second appendix offers a different, more abstract perspective to the material
presented in Chapter 2. Indeed, it contains some results that the author deems
interestingandinspiringtobeincludedinthisthesis, withoutdivertingtheattention
from the main discussion in Chapter 2.
ii
Introduction
In this thesis, we will explore deeply the properties of fractal sets. These spaces
show features that do not appear in regular subsets ofR
n
. Behavior of fractal
type appear everywhere in nature, and this is the reason why fractals are often the
perfect objects to model chaotic events such as the erosion of beach coasts, the lips
of a fracture in a material, and so on.
The first distinctive property they manifest is that their dimension is not uniquely
defined and we have to distinguish between their covering dimension and the
Hausdorff dimension since, by definition, fractal sets have been categorized as the
objects having different values associated to these two dimensions.
In particular we will see how the dimension of euclidean spaces such asR orR
2
is
always well defined, whereas the dimension of fractal sets like the Sierpiński gasket
can assume different values (for this set the Lebesgue covering dimension is 1 and
the Hausdorff dimension is about 1.585).
Besides the many definitions of dimension that we will give, we will also provide
a closed formula for the determination of the Hausdorff dimension of the space.
We will in fact define the sim-value of the fractal set, which is the value s for
which the function Φ(s) :=
 N
i=1
r
s
i
is equal to 1. The equation Φ(s) = 1 has a
unique solution and it can be determined by using the Perron–Frobenius theory of
matrices. Assuming Moran’s open set condition, the sim-value and the Hausdorff
dimension will be equal.
We will use the theory of directed weighted multigraphs to simplify the construction
of fractal sets. This new description will be suitable for the calculation of the
sim-value, but we will use another approach to begin the study of the analytical
properties of fractals: thanks to the Shift space we will construct a measure space
on top of the fractal set.
The key to understand how measures and similarities work on fractals is to de-
velop an independent study on spaces of strings. A string space is a triplet
(E
(ω)
,E, (f
e
)
e∈E
), where E is an alphabet of N letters, E
(ω)
is the set of all words
1
Introduction
of infinite length formed by letters in the alphabet andf
e
(w) =ew
1
... for every
w =w
1
...∈E
(ω)
and eache∈E. The functions (f
e
)
e∈E
will constitute the system
of similarities defining the fractal. We will associate to every wordw =w
1
...w
N
of length N the subspace of the string space E
w
:=f
w
1
◦...◦f
w
N
(E).
The simplest way to define a self-similar measure to the string space is to associate
to each finite length word w = w
1
...w
N
a weight computed as the product of
the weights associated to its letters with their multiplicity and then defining the
measure of E
w
equal to that weight. It can be proven that this set function is
actually a Borel regular probability measure over the whole string space and its
image through the natural projection between the string space and the fractal set
will be a Borel regular probability measure on the fractal.
We will be able to study analytically only a subset of all the existing fractals, the
ones having the so-called post-critically finite (p.c.f., for short) structure. In fact,
we will get to the fractal set by a limit of a sequence of resistance networks. The
interesting part of the study is that every metric, Laplacian, and Green’s operator
associated to each fractal will be different from fractal to fractal, but the strategy
to derive them will be the same for each set. In fact, what changes among the
fractals that we will going to analyze is their harmonic structure, which is a pair
(D,r), where D is a Laplacian operator on the boundary of the set and r is the
list of similarity ratios that define the fractal set. From only these two pieces of
information we will be able to construct naturally a sequence of networks that will
eventually converge to the fractal itself.
The sequence{(V
m
,H
m
)}
m
will be monotonically increasing in the sense that
V
n
⊂V
m
for every n<m and the resistance relations defined on the networkV
m
by the Dirichlet formE
m
associated with the Laplacian H
m
will also include the
resistance relations defined on all the smaller networks: i.e.E
m
(p,q) =E
n
(p,q) for
every n<m and every p,q∈V
n
.
Since every network is a finite set, the limit V
∗
:=∪
m∈N
V
m
will be at most a
countable set, therefore we will state some convergence results to ensure that
V
∗
will be at least a dense subset of the fractal. The discussion about how to
achieve from V
∗
the fractal under consideration is contained in the second part of
appendix B. In particular, the closure of V
∗
will coincide with the fractal if and
only if a specific condition on the harmonic structure is matched, showing that our
derivation is valid and the density problems will not arise when the fractal presents
some sort of symmetry.
With the limiting procedure we will recover a metric on the fractal set K and a
scalar product on the space L
2
(K,µ ), thus we will be able to start the study of
partial differential equations on fractal sets.
2
Introduction
Once the fractal is given a metric and a measure, it also needs a suitable definition
of the Laplacian operator, which we will denote as ∆ µ (this definition is presented
in Section 2.4). In the same section, it will be given also the definition of the
Neumann derivative of a function: this object is a derivative in the sense that it
will satisfy the Gauss-Green’s formula using ∆ µ as the Laplacian in the classical
formula. Moreover it will appear clear how the Neumann Laplacian defined as the
limit of the Laplacians H
m
will be related with the Laplacian ∆ µ : the Neumann
Laplacian corresponds to the restriction of ∆ µ to the functions having Neumann
derivative equal to 0 on the boundary of the fractal.
The discussion is then focused on the spectral properties of the Laplacian and the
study of the heat equation. From the classical theory it is known that the spectrum
of the Laplacian on a regular bounded subset ofR
n
does not admit eigenfunctions
satisfying both Neumann and Dirichlet boundary conditions, but in this settings
not only we prove their existence, but we also prove that for each open set O⊂K,
there exists at least one of such eigenfunctions whose support is contained in O.
Although we can obtain an uncountable number of eigenfunctions with support
as small as we want, the set of pre-localized eigenfuncions does not form a dense
subset of L
2
(K,µ ); in fact L
2
(K,µ ) can be decomposed as the orthogonal sum of
the space of pre-localized eigenfunctions and the space of polynomials, where we
identify polynomials with functionsf∈L
2
(K,µ ) satisfying ∆ m
µ (f) = 0. This result
proves that the Stone-Weierstrass theorem does not hold for fractal sets.
In the last section of Chapter 2, we will discuss about the properties of the solutions
to the heat equation. There will be given regularity results: the solution u(x,t)
will be a continuous function in K× [0,∞) and u(x,·)∈C
∞
(0,∞) (regularizing
effect).
Can curves in R
n
become so dense to cover R
n
itself? The answer is positive
and such objects are called space-filling curves. This and many other topics like
the Julia sets and the Mandelbröt set will be treated in the first appendix. In
particular, Julia sets are fractals that appear frequently in physics, ergodic theory
and geometry, but they are mostly famous because of their relationship with the
Mandelbröt set (expressed via the Decoration Theorem). These objects are some
of the earliest fractal sets studied; in fact, it was because of the properties that
emerged from the Mandelbröt set that fractal theory was developed in the last
century.
In the second appendix, we will give mostly some formal justifications to the results
contained in Chapter 2, and we invite the reader to read them carefully in order to
gain some more insights on the behavior of fractal sets. Noteworthy is the paragraph
about the existence of a harmonic structure for a fractal set. That discussion will
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