Multifractals and Irregular
Measures on the Middle Third
Cantor Set
Jonathan MacDonald Fraser
I certify that this project has been written by me, is a record of work carried out by
me, and is essentially different from work undertaken for any other purpose or
assessment.
In submitting this project report to the University of St Andrews, I give permission
for it to be made available for use in accordance with the regulations of the
University Library. I also give permission for the title and abstract to be published
and for copies of the report to be made and supplied at cost to any bona fide
library or research worker. I retain the copyright of the work.
Jonathan MacDonald Fraser
050001145
2
Abstract
We will first review some basic multifractal theory and then use this theory to
estimate the multifractal spectra for some special classes of irregular measures
by calculating the Legendre spectrum. We will then examine the symmetry of the
Legendre spectrum in some generality.
3
Contents
Chapter
Page
Preface
5
1.
Introduction and Preliminaries
6
2.
Multifractal Theory and Methodology
11
3.
The Middle Third Cantor Set
22
4.
Evaluating β(q) for General Distribution Functions
26
5.
Convergent Distribution Functions
28
6.
Periodic Non-Convergent Distribution Functions
33
7.
Non-Periodic Non-Convergent Distribution Functions
34
8.
Symmetry of Spectra
40
9.
Some Geometry of the Legendre Spectrum
43
10.
A Note on Lop-sided Supporting Sets
48
11.
Conclusion and Summary of Results
51
Bibliography
53
4
Preface
Before beginning work on this project I had taken a basic course in fractal
geometry but had not encountered multifractals in any context. Due to this the
first few chapters will give a description of the basic theory which will be largely
based on Falconer (2003), Mandelbrot (1999), and Olsen (2000). In chapters 5,
6 and 7 I will introduce some classes of measures that I see as a natural
extension to the examples found in Falconer (2003) and will then proceed to
evaluate the Legendre spectrum as an estimate for the fine and coarse
multifractal spectra. In order to do this I will draw upon some classic results from
analysis as well as some results from the theory of uniformly distributed
sequences. I initially expected to find that the irregular measures studied in
these chapters would give rise to non-symmetric Legendre spectra. When this
did not happen I became increasingly interested in the conditions which give rise
to symmetric and asymmetric Legendre spectra and in chapters 7, 8 and 9 I will
prove some of my own conjectures on the matter.
My intended readership is final year undergraduate students who have taken
courses in fractal geometry and measure theory.
I would also like to take this opportunity to extend particular thanks to Professor
Kenneth Falconer for his advice and for many interesting and inspiring
discussions throughout the writing of this report.
5
1 - Introduction and Preliminaries
As with “fractal”, the term “multifractal” has no rigorous mathematical definition.
One tends to use the word “fractal” to describe sets that exhibit irregularity or
intricate detail at all scales. That is, they have a complex structure, which
continues to appear complicated under arbitrarily large magnifications.
Analogously, the word “multifractal” is used to describe measures which are
distributed in a similarly irregular and intricate way across their support.
Multifractals first appeared in the study of physical phenomena which exhibited
wild variation. A physical situation where one is concerned with distributing
numeric values across a field of points may give rise to multifractal measures if
the local intensity of the field varies according to some power law. Multifractal
theory has been used successfully to model many such phenomena. For a
treatment of several examples including the spatial distribution of earthquakes
and a view of multifractal theory from a statistical view point see Harte (2001).
Due to the success of these models, much current research in physics is focused
on further applications of multifractal theory in the description and analysis of wild
variations.
Despite its background in physics and applied maths, multifractal theory has
attracted a great deal of interest from pure mathematicians over the last 20
years. Although many of the ideas date back to two papers: ‘Possible refinement
of the log-normal hypothesis concerning the distribution of energy dissipation in
intermittent turbulence’ and ‘Intermittent turbulence in self similar cascades:
Divergence of high moments and dimension of the carrier’, by Benoit Mandelbrot,
published in 1972 and 1974 respectively, even Mandelbrot himself acknowledges
that multifractals only became widely known via a paper by Halsey et al
published in 1986 entitled: “Fractal measures and their singularities: The
characterisation of strange sets” (Mandelbrot, 2004: 233). In any case, the
theory of multifractals is very much a topic of current research and recently a
large variety of measures have been fully analysed.
Before looking at multifractal theory in detail we will first review some basic
definitions and theorems from both fractal geometry and measure theory.
Measure theory is fundamental to many aspects of modern analysis. It is the
study of “measures”, which are effectively just functions which quantify the “size”
of sets in a general and abstract way.
We first define the notion of a measure. Although this strict definition will not be
used directly in this report it is worthwhile to put the term ‘measure’ on a firm
mathematical footing.
6
Definition 1.1: Let X be a set. A family of subsets E of X is called a σ-algebra if
and only if:
1)   E
2) E  E: X \ E  E
3) E1, E 2 , E 3 ,...  E:
E
n
E
n
The pair (X, E) is then called a measurable space.
Definition 1.2: For a measurable space (X, E), a function μ: E → [0,  ] is called a
measure if and only if:
1)  ( )  0
2) If E1, E2, E3,… are a countable collection of pair wise disjoint sets from E then:


   E n     (E n )

n

n
The triple (X, E, μ) is then called a measure space.
If μ(X) = 1 then (X, E, μ) is called a probability space and μ is called a probability
measure.
The support of a measure is denoted by spt μ and is the smallest closed set A
such that μ(X\A) = 0.
For the purposes of this report we will only be concerned with probability
measures on bounded subsets of R. With this in mind it is important to note that
such measures can be constructed by distributing a unit of “mass” across a
supporting set F by some iterative process. (See chapter 3).
Fractal geometry itself is a relatively new subject. Although objects which we
would now call fractals have been studied for hundreds of years, the subject did
not gain a proper mathematical foothold until the end of the 20 th century. In his
seminal 1975 paper ‘Objets Fractals’, Benoit Mandelbrot coined the word ‘fractal’
and, to a certain extent, gave birth to the modern interest in the subject.
Mandelbrot himself states that the lack of a formal definition had ‘-- the
advantage of not trapping the notion in a narrow context’ and he claims that this
‘led to a very rich scientific harvest.’ (Cherbit, 1991: 1)
The subject provides a general method for describing the geometric properties of
fractal objects in a way that classical Euclidean geometry and calculus fail. The
concept of dimension is one of the most important aspects involved in the study
of fractals and will subsequently play an important role in this report. One of the
7
reasons that fractals are interesting objects to study is that they often have the
remarkable property that their “fractal dimension” is greater than (or at least
different from) their topological dimension (See chapter 3). Since the previous
notions of dimension were insufficient to describe this property, a number of
more rigorous and appropriate definitions of dimension have been introduced.
We shall now examine some of these that will be relevant to this report.
Hausdorff dimension is arguably the most important form of dimension. It was
introduced by Felix Hausdorff (1868-1942) in his seminal paper ‘Dimension and
Outer Measure’ published in 1918 (Edgar, 1993: 75).
Hausdorff first constructed the Hausdorff measure using a deep result from
measure theory called Carathéodory’s Extension Theorem. This theorem allows
the construction of useful measures by extending measures defined on algebras
to cleverly defined σ-algebras. For a full description of Carathéodory’s Extension
Theorem see Bartle (1995:100-101). Hausdorff praised Carathéodory lavishly for
his insights and gave us a good indication of his own modesty by referring to his
work as a ‘small contribution’ (Edgar, 1993: 75) to Carathéodory’s theory.
We will now define the Hausdorff measure as follows:
Definition 1.3: Let F be a subset of Rn and s  R+. Then for any δ > 0 we define
the δ-approximating s-dimensional Hausdorff measure by:

H s (F )  inf  U i
 i 1
s

: U i  is a  - cover of F

And then the s-dimensional Hausdorff measure is defined by:
H s (F )  lim H s (F )
 0
Note that, as we decrease δ, the possibilities {Ui} for δ-covers of F decrease and
so H s (F ) increases. Hence, H s (F ) must converge up to a limit H s (F ) and we
have that the above limit always exists (although it may be infinity).
Hausdorff measure generalises Lebesgue measure to allow for sets with noninteger dimensions. It is possible, although quite difficult, to show that H s is a
measure, in accordance with definitions 1.1 and 1.2. Evans and Gariepy (1992:
61-63) give a proof of the fact that H s is a regular Borel measure in their book
‘Measure Theory and Fine Properties of Functions’. There are some technical
differences between their definition of a Regular Borel Measure (See pages 4-5
of Evans and Gariepy) and the definition of a measure given in definition 1.1 and
1.2 but we do not pursue these here.
8
Note that increasing s will necessarily decrease H s (F ) for δ < 1 and so H s (F ) is
a decreasing function of s. Now, let F  Rn and let δ < 1 and, suppose that r, s
are positive real numbers with r > s and that {Ui} a δ-cover of F with H s (F ) <  ,
then we have:
U
i
r
i
  Ui
r s
Ui
i
s
  r s  U i
s
i
which, upon taking infimums, gives H r (F )   r s H s (F ) and now by letting
  0 , we have that H r (F ) = 0.
Hence, there is a specific value for which H s (F ) jumps from  to 0. This value
is defined to be the Hausdorff dimension of F, and is written dimH F .
The second definition of dimension which we will require for this report is BoxCounting Dimension. This is calculated by counting the minimum number of sets
in a δ-cover of F and then analysing how this number changes as δ tends to 0.
We define the Box-Counting Dimension of a set F  rn by:
 log N (F ) 
Definition 1.4: dimB F  lim 

 0
  log  
where Nδ(F) is the least number of sets required in a δ-cover of F.
The above limit does not always exist and so sometimes we work with upper and
lower box-counting dimensions defined as follows:
 log N (F ) 
dimB F  lim 

 0
  log  
 log N (F ) 
dimB F  lim 

 0   log  
Although, for a given set F, the box-counting dimension and the Hausdorff
dimension are often the same, this is not true in general. If they are different the
box-counting dimension is always larger and we can show that for all sets F
dim H F  dim B F  dim B F . (Falconer, 2003: 46)
9
The following lemma concerning box dimension will be useful later on:
Lemma 1.4: Assuming dimB F exists we have that:
  0 :  1  0 :    1    dimB F  N (F )    dimB F
Proof:
From the above definition:
  0 :  1  0 :    1  dimB F   
log N (F )
 dimB F  
 log 
And we have that:
dimB F   



log N (F )
 dimB F    log   dim B F  log N (F )  log   dim B F
 log 

   dimB F  N (F )    dimB F
And we have proved the lemma.
This lemma indicates that dimB F is essentially a description of how Nδ(F)
behaves as   0 , i.e. N (F )    dimB F for small δ.
10
2 - Multifractal Theory and Methodology
The key difference between fractals and multifractals is that multifractals cannot
be described by a single quantity in the way fractals can be described by their
dimension. Instead, multifractals appear as an infinite hierarchy of fractal sets,
with each ‘layer’ representing a different fractal with a different dimension.
Hence, multifractals require a ‘spectrum’ of dimensions to fully describe them.
This spectrum can be viewed in two different ways.
Coarse multifractal analysis looks to estimate the number of ‘r-mesh cubes’ C for
which μ(C) varies approximately as rα for some   0 and examines the power law
behaviour of this number as r→ 0 in a similar vein to box-counting dimension.
Fine multifractal analysis, on the other hand, looks at the fractals determined by
considering points in the support of μ which have similar ‘local intensities’.
We will now look at these two views separately and in some detail.
We define the coarse multifractal spectrum as follows:
Definition 2.1: For a finite measure μ on Rn, the coarse multifractal spectrum of
μ is defined by:
log  (N r (   )  N r (   )) 
fC ( )  lim lim

 0 r 0
 log r


if both limits exist.
Note that for   0 , Nr ( ) is given by:
Nr ( ) # {r - mesh cubes C with (C )  r  } .
Also note that we define log  ( x )  max{ 0,log x } to ensure that fC ( )  0 .
In the instance that the limit as r → 0 does not exist, we define the upper and
lower coarse spectrum analogously as:
log  (N r (   )  N r (   )) 
fC ( )  lim lim

 0 r 0
 log r


log  (N r (   )  N r (   )) 
fC ( )  lim lim

 0 r 0
 log r


11
Note that the second limit, i.e. the limit taken as   0 , always exists. We can
see this by noting that, as we decrease  , Nr (   ) necessarily increases as
there are more r-mesh cubes C with  (C )  r   and likewise Nr (   )
necessarily decreases and so log  (N r (   )  N r (   )) increases and we have
log  (N r (   )  N r (   )) 
log  (N r (   )  N r (   )) 
that
and
lim
lim


r 0
r 0
 log r
 log r




converge up to a limits fC ( ) and fC ( ) respectively as   0 .
Lemma 2.2
  0 :  1  0 : r1  0 :    1 & r  r1  r  fC ( )  N r (   )  N r (   )  r  fC ( )
Proof: From the above definition we have that:
  0 :   0 : r  0 : fC ( )   
log  (N r (   )  N r (   ))
 fC ( )  
 log r
And that:
fC ( )   

log  (N r (   )  N r (   ))
 fC ( )  
 log r


 log r  fC ( )  log  (N r (   )  N r (   ))  log r  fC ( )

 r  fC ( )  N r (   )  N r (   )  r  fC ( )
And we have proved the lemma.
This result is somewhat analogous to the result obtained for box-counting
dimensions in Lemma 1.4, and gives an indication of the parallels between
coarse multifractal theory and the analysis of fractals via their box-counting
dimension.
We can see that fC ( ) provides information about the global fluctuations of the
measure but no information about the behaviour of the measure at any specific
point. (Falconer, 2003: 279)
We will now describe fine multifractal analysis and give the necessary definitions.
12
Definition 2.3: For a finite measure μ on rn we define the local dimension of μ at
a point x by:
 log  (B( x, r )) 
dimloc  ( x )  lim

r 0
log r


if this limit exists.
The local dimension focuses on a single point x spt μ and measures how
intense the measure is in the neighbourhood of x. Indeed for small r we
have  (B( x, r ))  r dim loc  ( x ) .
Now we consider sets of points which exhibit similar local intensities and
measure their dimension.
Definition 2.4: For each   0 we define the local multifractal decomposition set
or  -level set as:
F  x  spt  dimloc ( x )   
Definition 2.5: We define the fine multifractal spectrum of μ by:
fH ( )  dimH F
This function was first explicitly defined by Halsey et al. in their 1986 paper.
(Olsen, 2000: 5)
Note here that Hausdorff dimension is used instead of box-counting dimension
because often the level sets are dense in spt μ (Falconer, 2003: 284) which
means that, since box-counting dimension is invariant under taking closures we
have:
dimB F  dimB F  dimB (spt  )
and:
dimB F  dimB F  dimB (spt  )
and the multifractal spectrum would be constantly equal to dimB (spt  ) , which
would clearly give us no information about the dynamics of the measure.
It can be shown that   0 : fH ( )  fC ( )  fC ( ) . (Falconer, 2003: 285)
13
In most cases it is computationally awkward to calculate the coarse and fine
multifractal spectrum straight from the above definitions and so we will now
introduce a powerful technique known as the Method of Moments (Falconer,
2003: 283), which will be used throughout this report to give estimates for the fine
and coarse multifractal spectra.
Definition 2.6: For q  r and r > 0 we define the qth power moment sum of μ by:
M r (q )   (C )q
Mr
where Mr is the set of r-mesh cubes C with non-zero μ-measure.
Now, in order to examine the behaviour of Mr(q) as r  0 , we make the following
definition:
 log M r (q ) 
Definition 2.7: For q  R and r > 0:  (q )  lim
 , if this limit exists,
r 0
  log r 
and we see that for small r: M r (q )  r   ( q ) .
We will now take what may initially appear to be a strange diversion to look at
Legendre Transforms. Named after the French mathematician Adrien-Marie
Legendre (1752-1833), the Legendre Transform of a function f is a way of reexpressing f in terms of different variable.
Definition 2.8: Given a convex function f: Rn → R the Legendre Transform
L(f ) : R n  R of f is defined by:
L(f )(a)  supa  x  f ( x )
x
Note that a function f : [a, b]  R is called convex if:
f [x  (1   )y ]  f ( x )  (1   )f ( y ) for all x, y  [a, b] and   (0,1) . (Roberts and
Varberg, 1973: 2)
or, more usefully, a function f : [a, b]  R is convex iff f II ( x )  0 for all x  [a, b]
assuming the second derivative exists on [a, b]   (0,1) . (Roberts and Varberg,
1973: 11)
Also, note that a function f : [a, b]  R is called concave if  f : [a, b]  R is
convex   (0,1) . (Roberts and Varberg, 1973: 2)
14
We can see that since f is a convex function we have that a  x  f (x ) is a
concave function and hence the supremum is well defined. If we now assume
that f is differentiable then we see that the supremum occurs when

a  x  f ( x )  0 , i.e., when a  f . Thus, we see that L(f ) has taken f as a
x
x
f
function of x and re-expressed it as a function of its derivative,
.
x
The Legendre Transform has most of its applications in applied maths. It is often
beneficial to be able to re-express a function in terms of its derivative. For
t
example, let N(t )  N0 e  10 be a model for some radioactive decay. Then the
Legendre transform of N (t ) is L(f )(a)  a  t  N (t ) where a 
N t
N
  0 e 10 and
t
10
we have that:

 10a 
 10a  
  10a  10a1  log 
 
L(f )(a )  a  t  N (t )  10a log 


N
N
0 
0 



and we have re-expressed the decay in terms of a, which in this case is a
measure of the change in decay rate.
The Legendre Transform is often viewed as a way of relating a curve to the set of
all its tangents. This can be demonstrated by the following lemma:
Lemma 2.9: Let f: r → r be a strictly convex function. Then for all t  R the



tangent at t has y-intercept  0, - L(f) f x t   provided the tangent exists.



Proof: Let t  R be a point where f is differentiable. Then the tangent at t exists
f
 x  t  .
and has equation y – f(t) =
x
Now set, x = 0 and we have y  
f
 f


 t  f (t )    t  f (t )   -L(f) f
x

 x

x t



and we have proved the lemma.
Legendre Transforms began to appear in papers on multifractal theory in the
mid-1970’s. Their first appearance was in a paper by Mandelbrot entitled
‘Multiplications aléatoires itérées et distributions invariantes par moyenne
pondérée’1. Mandelbrot indicates that their introduction was due to work he was
1
‘Random iterated multiplications and invariant distributions by weighted average’ [Babel ref]
15
doing in ‘The Cramèr Theory of large deviations’, although he admits that his first
writings on the subject were not stated ‘forcefully enough’ to gain real
recognition. (Mandelbrot, 2004: 67)
Legendre Transforms were considered again by Halsey et al. in their 1986 paper
and their particular insights led to what is now known as the Multifractal
Formalism.
The Multifractal Formalism is a very powerful result in the theory of multifractals.
It states that the coarse and fine multifractal spectra are equal and that they can
be calculated in terms of the Legendre Transform of the aforementioned beta
function (definition 2.7).
For the purposes of this report we will define the Legendre transform of  (q ) in
the following way:
Definition 2.10: fL ( )  inf  (q )  q
qr
Note that, letting  = -a, we see that:
fL ( )  inf  (q )  q  inf  (q )  aq )  inf  (   (q )  aq )
qr
qR
qR
 sup aq   (q )  L(f)(a)
qR
and we have that definition 2.8 and 2.10 are equivalent.
We will refer to fL ( ) as the Legendre spectrum and will now give a fairly simple
proof that it is an upper bound for the multifractal spectra defined in 2.1 and 2.5.
Theorem 2.11: For a finite measure μ on Rn we have that for all   0 :
1
2
3
fH ( )  fC ( )  fC ( )  fL ( )
Proof: adapted from Falconer (2003: 281-282):
Note that we already have inequalities 1 and 2 from above, so it remains to show
that fC ( )  fL ( ) .
We first prove a useful lemma, stated by Falconer (2003: 281):
16
Lemma 2.12: For   0 and q  0 we have: M r (q )  r q N r ( ) and for   0 and
q  0 we have: M r (q )  r q # {r  mesh cubes C with 0  (C)  r  } .
A proof of which is as follows:
Proof:
Let   0 and q  0 . Now we have:
M r (q )    (C ) q 
Mr
  (C )
q

C M r
 ( C ) r 
r 
q
C M r
 ( C ) r 
 r q # {r  mesh cubes C with  (C)  r  }  r q N r ( )
Now let   0 and q  0 and we have:
M r (q )    (C )q 
Mr
  (C )
C M r
0   ( C ) r 
q

r 
q
C M r
0   ( C ) r 
 r q # {r  mesh cubes C with 0   (C)  r  }
and we have proved the lemma.
Now let   0 and   0 satisfying 0     .
Now, by lemma 2.2 there exists an ε1 satisfying 0 < ε1 < η and an r1 > 0 such that
r < r1 and ε < ε1 imply that:
r  fC ( )  Nr (   )  Nr (   )
Now, as is common when proving results in multifractal theory, we will consider
the case where q  0 separately from the case where q < 0.
Case 1: let q  0.
Hence, by lemma 2.12 we have that:
M r (q )  r q (  ) N r (   )  r q (  ) N r (   )  N r (   )  r q (  ) r  fC ( ) (1)
17
Case 2: let q < 0
Hence, by lemma 2.12 we have that:
M r (q )  r q (  ) # {r  mesh cubes C with 0   (C)  r   }
 r q (  )  M r  N r (   )  r q (  ) N r (   )  N r (   )  r q (  ) r  fC ( ) (2)
and hence, from (1) and (2), we have for all q  R:

 log r q (  ) r  fC ( )
 log M r (q ) 

lim


r 0
 log r
  log r  r 0 
 (q )  lim
  q(   )    f ( ) lim log r

C


r 0  log r


 q(   )    fC ( )
So, we have that   0 :   (0, ) : fC ( )   (q )  (   )q  
and hence deduce that for all   0 : fC ( )   (q )  q
and that fC ( )  fL ( ) and we have proved theorem 2.11.
The lower bound case, and hence the completion of the multifractal formalism, is
usually considerably more difficult to prove and in fact does not always hold, as
in the following example:
18
Example 2.13:
Consider two measures  and  with disjoint supports, F and G, respectively.
Now suppose that the fine multifractal spectra of  and are given by:
Figure 2.13.1
Figure 2.13.1 shows the fine multifractal spectra of μ and
ν
Note that these measures can be easily constructed. For example, let  be the
measure described in section 3 below and let  be a similar measure, supported
by the middle third Cantor set shifted onto [2, 3] with the measure distributed in
the ratio 19 : 8 9 at each stage of construction.
Now, consider the measure  defined by      (note that  is a measure
since sums of measures are measures). Now, recall that the fine multifractal
spectrum of  is defined by fH ( )  dimH F where F are the levels sets of 
and since, for arbitrary sets E and F, we have that:
dimH (E  F )  maxdimH E, dimH F (Falconer, 2003: 32)
we can deduce that:
19
F  x  spt  dimloc  ( x )   
x  spt  dimloc  ( x )     x  spt  dimloc  ( x )     F  F
1
Note that 1 relies on the fact that spt   spt   spt  is a disjoint union.
Now we have that:





fH ( )  dimH F  dimH F  F  max dimH F , dimH F  max fH ( ), fH ( )

and hence, the fine multifractal spectrum of  can be calculated by taking the
maximum of the fine multifractal spectrum of  and  respectively.
Figure 2.13.2
Figure 2.13.2 shows the fine multifractal spectrum fH ( ) of

Observe that fH ( ) is not a concave function, but the Legendre Transform
of  (q ) defined for  is always concave (Olsen, 2000: 13) and we deduce that
fH ( )  fL ( ) and that the Multifractal Formalism does not hold. Note, however,
that the Legendre Transform of  (q ) would be a concave upper bound for the
fine multifractal spectra shown above.
20
Verifying the Multifractal Formalism for specific classes of measures has become
a topic of great interest in current research. Much work has been done on the
subject and many types of multifractal are now well understood. For example, it
can be shown that the Multifractal Formalism always holds for self-similar
measures. Where a self-similar measure is one which is made up of scaled
down copies of itself. For a proof of this, see Falconer (2003: 294-295).
For an in depth analysis of many classes of measure and a detailed approach to
the multifractal formalism see the seminal paper by Lars Olsen entitled “A
Multifractal Formalism” published in 1995.
21
3 - The Middle Third Cantor Set
The following few chapters of this report will be dedicated to examining measures
supported by a fractal subset of [0, 1] = E0 known as the middle third Cantor set.
The set itself, named after mathematician Georg Cantor (1845-1918), is
constructed iteratively by first taking the unit interval and then successively
removing the “middle thirds” of intervals left intact at the kth level of recursion Ek,
with the middle third Cantor set F being defined to be what remains after an

infinite number of iterations, i.e., F   E k .
k 0
Figure 3.1
Figure 3.1 shows the first 5 stages in the construction of the middle third Cantor set
For convenience we label the intervals in the construction as is shown in figure
3.2 below.
22
Figure 3.2
Figure 3.2 demonstrates the labelling of the intervals in the construction of F
Although a simple example, the middle third Cantor set displays many of the
geometrical features we would expect to see in a fractal. For this reason it
provides an excellent basis for an introduction to multifractal analysis.
The set F consists of all points in [0,1] with a base 3 decimal expansion
consisting of only the digits 0 and 2. Hence, we can define a bijection from F to
[0,1] by mapping a number with base 3 decimal expansion 0.m1m2m3...  F to an
number with base 2 expansion 0.n1n2n3… where all 2’s are replaced with 1’s.
Thus F is seen to be an uncountable set. Despite being ‘large’ in the sense of
cardinality, F is ‘small’ in measure theoretic terms as it has zero Lebesgue
measure. To see this note that the kth stage of construction Ek of F is made up
k
of 2k intervals each of length 3-k and so has Lebesgue measure 2k3-k = 2 3   0
as k→  .
F also has a fractional dimension. It can also be shown that:
dimB F  dimH F 
Note that
log 2
. (Falconer, 2003: 34-35 and 47)
log 3
log 2
 0 which is the topological dimension of F.
log 3
23
We will now consider several examples of probability measures supported by F.
Probability measures can be constructed by taking a unit of measure and
distributing in some way at each stage in the above construction. For example, if
at the kth stage in the construction of F we divide the mass up in the ratio 13 : 2 3 ,
i.e., we take each interval present at the (k-1)th level and split the measure by
sending 13 to the left and 2 3 to the right to form the kth level, we form a
multifractal measure. For a full treatment of this and other similar examples see
Falconer (2003: 278-285).
Falconer uses the method of moments to show that the Legendre spectrum for
this measure is given by:

log(  13   2 3  ) q ( 13 )q log( 13 )  ( 2 3 )q log( 2 3 )

Theorem 3.3: fL ( ) 
log 3
( 13 )q  ( 2 3 )q log 3
q
q



where fL ( ) is parameterised by q.
Note that the Multifractal Formalism holds in this case since the measure
constructed above is self-similar, and so the coarse and fine multifractal spectra
are indeed equal to the Legendre spectrum.
Figure 3.4
Figure 3.4 shows the graph of the multifractal spectra of measure defined above
24
It is important to note that the maximum of the multifractal spectra shown above
log 2
is
 dimH (spt  ) and that this maximum corresponds to  q 0 .
log 3
In the following four chapters we will consider different measures constructed in a
similar way, where the measure is distributed in the ratio p1(k):p2(k) but where the
distribution functions p1(k) and p2(k) are functions of k, i.e., we divide the
measure up in different ratios depending on the stage of construction k.
The only constraint on p1(k) and p2(k) is that for all k we must maintain the
relationship p1(k) + p2(k) = 1. Otherwise we would not construct a probability
measure.
25
4 - Evaluating β(q) for General Distribution Functions
Before we consider some specific classes of measures constructed using
distribution functions we will first prove some useful results in general. We shall
see that the qth power moment sum can be calculated quite easily and hence we
can express β(q) as a limit as in definition 2.7. The difficulties usually lie in firstly
showing that this limit is well defined and then evaluating it.
First let μ be a probability measure, supported by F, and constructed by dividing
the mass in the ratio p1(k ) : p2 (k ) at the kth level of construction where p1(k )
and p2 (k ) are arbitrary distribution functions satisfying p1(k )  p2 (k )  1 for all k.
Theorem 4.1: For the measure μ described above and for k  0 the qth power
moment sum is given by:
M 3  k (q ) 
 p (i )  p
k
q
1
( i )
q
2
i 1

Proof:
Let k  N and let M 3  k be the set of 3-k-mesh cubes C for which μ(C)>0. Now we
have:
M 3  k (q ) 
  (C )
M
q
3k
Note that:
 (I i ... i 1 )q   (I i ... i 2 )q   (I i ... i )  p1(k )   (I i ... i )  p2 (k )
q
1
k
1
k

  (I i1 ... i k )q p1(k )q  p2 (k )q
1

Using this observation we have that:
26
k
q
1
k
  (C )
M
q




 p1 (k ) q  p2 (k )q  p1 (k  1) q  p2 (k  1) q  p1 (k  2) q  p2 (k  2)q  ...
3k


...  p1 (1) q  p2 (1) q   ([0,1])





k
 p1 (k ) q  p2 (k )q  ...  p1 (1)q  p2 (1)q   p1 (i ) q  p2 (i ) q

i 1
So we have: M 3 k (q ) 
 p (i )  p
k
q
1
( i )
q
2
i 1
 and we have proved the theorem.
We can now evaluate β(q) by noting that it is enough to let r  0 through the
discrete set of values 3  k k as k   and so we have:
 

k

q
q

log
p1 (i )  p2 (i )


 log M 3  k (q ) 
 log M r (q ) 

i 1
 (q )  lim
 lim 
 lim 



k
r 0
k   log( 3
k 

log
r
k log 3
)










if this limit exists.
We now have the following theorem:
Theorem 4.2: For arbitrary distribution functions p1(k ) and p2 (k ) we have that:
 (q ) 


1 k
1
q
q 
lim   log p1(i )  p2 (i ) 
log 3 k   k i 1

if this limit exists.
We will use theorem 4.2 in the following few chapters to evaluate the Legendre
spectrum for some specific classes of measures.
27
5 - Convergent Distribution Functions
In this chapter we will consider convergent distribution functions where p1(k) and
p2(k) are defined in such a way that they both converge as k→  .
We let F  [0,1] be the usual middle third Cantor set and label the intervals in the
usual way. Now let μ be a probability measure, supported by F, constructed by
dividing the mass in the ratio p1(k ) : p2 (k ) at the kth level of construction where
p1(k )  p2 (k )  1 and p1(k)→ p1  (0,1) and p2(k)→ p2  (0,1) as k→  .
By theorem 4.2 we have that:
 (q ) 


1 k
1
q
q 
lim   log p1(i )  p2 (i )  (*)
k


log 3
 k i 1

assuming this limit exists.
We will first state and prove a classic result from analysis.
Lemma 5.1: If (ak)k is a sequence of real numbers such that ak  a  R  {}
as k   then:
1 k
 ai  a as k   .
k i 1
Proof:
Case 1: a  R
Let   0 and choose N  N such that for all k > N we have ak  a  
Now let k > N and we have:
1 k
1 N
1 k
1 N
1 k
1 N
k N
a

a

a

a

(
a


)

ai 
(a   )






i
i
i
i
k i 1
k i 1
k i N 1
k i 1
k i N 1
k i 1
k
 a   as k   (1)
and we also have:
28
1 k
1 N
1 k
1 N
1 k
1 N
k N
a i   a i   a i   a i   (a   )   a i 
(a   )

k i 1
k i 1
k i N 1
k i 1
k i N 1
k i 1
k
 a   as k   (2)
Hence, from (1) and (2) we have that for all   0 :
1 k

1 k

a    lim   ai   lim   ai   a  
k


k  k i 1

 k i 1 
1 k

and we deduce that lim   ai  exists and equals a.
k  k
 i 1 
Case 2: a  
Let M  R and choose N  N such that for all k > N we have ak  M
Now let k > N and we have that:
1 k
1 N
1 k
1 N
1 k
1 N
k N
a

a

a

a

M

ai 
M






i
i
i
i
k i 1
k i 1
k i N 1
k i 1
k i N 1
k i 1
k
 M as k  
Hence, we have that for all M  R :
1 k

lim   ai   M
k  k i 1

1 k

and we can deduce that lim   ai    and we have hence proved the lemma.
k  k
 i 1 
And now, by lemma 5.1, we have that the limit (*) exists and that:
 

log( p1  p2 )
1
q
q
 (q ) 
lim log p1(k )  p2 (k ) 
k


log 3
log 3
q
q
This will be a convex function and thus the Legendre Transform can be
calculated easily as in (Falconer) to give:
29

log( p1  p2 ) q p1 log p1  p2 log p2
fL ( ) 

q
q
log 3
p1  p2 log 3
q
q
q

q


Note that this Legendre Transform will be the same as the multifractal spectrum
obtained for the measure described by Falconer (2003: 278-285) where the
measure was distributed in the ratio p1 : p2 at each stage of construction. So it
appears that the dynamics of a measure constructed on the middle third Cantor
set depend only on the limiting behavior of the distribution functions and not on
the functions themselves. We believe that it would be straightforward to extend
this hypothesis to all measures constructed using convergent distribution
functions supported by self similar sets satisfying the strong separation condition.
We will now consider the exceptional case where p1(k)→ 0 and p2(k)→ 1 as
k→  . Using theorem 5.1 we can see that all cases where p1(k)→ 0 will be the
same and so we will demonstrate this situation via an example.
Let μ be a probability measure supported by F constructed by dividing the mass
in the ratio 1/( k  1) : k /( k  1) at the kth level of construction, i.e., we define the
distribution functions to be p1(k) = 1/(k+1) and p2(k)=k/(k+1).
Note that
1
k
k 1


 1 and that p1(k)→0 and p2(k)→ 1 as k→  .
k 1 k 1 k 1
Figure 5.2
Figure 5.2 shows the distribution of measure in the construction of μ after 4 stages of
recursion
30
q0
 


 log 2 q  0
Claim:  (q )  
 log 3

q 0
 0
Proof:
We have that:
k

1 i q
 log 
q

i 1 (1  i )
 (q )  lim 
k 
k log 3




1 k
1
1 i q

 lim   log

k  k
(1  i )q
 i 1
 log 3




 q  0

1 k
exists and by lemma 5.1, since

 2 q  0 , we have that:
(1  k )q

 0 q  0
q
q0
 



1
1  k q   log 2 q  0

 (q ) 
 lim  log
log 3 k 
(1  k )q   log 3

q 0
 0
and we have proved our claim.
31
Figure 5.3
Figure 5.3 shows a plot of β(q)
We can see that  (q ) is not a convex function and so we have to be careful
about interpreting the resulting Legendre Transform. Although we can see μ is
not a multifractal measure, it still displays some interesting dynamics.
Although μ was constructed in a similar way to the measure (constructed by
Falconer) above, it does not exhibit multifractal characteristics. The problem lies
in the fact that p1(k)→0. The measure piles up at the end points of the intervals
and behaves similarly to a finite collection of point measures. We conjecture that
the fine multifractal spectrum will be constantly equal to zero.
Given the above results, the next natural questions are: what will happen if the
distribution functions do not converge? Will we get a multifractal distribution?
Will we have to deal with upper and lower limits if the multifractal spectra defined
in 2.1 and 2.5 do not converge?
These questions will be addressed for some specific classes of measures in the
following sections.
32
6 - Periodic Non-Convergent Distribution Functions
In this chapter we will consider a measure  , constructed in the usual way,
where p1(k) and p2(k) are periodic non-convergent distribution functions, i.e.,
p1(k) and p2(k) do not converge as k→  but they are both periodic in that:
n  N : k  N : p1(k )  p1(k  n) and p2 (k )  p2 (k  n)
It turns out that in this case  is a self-similar measure.
Consider the following alternative construction for the middle third Cantor set:
Begin with the unit interval [0,1]. Then, let the 1st stage of construction E1I be
equal to En, where En is the nth stage in the usual construction. And, in general
let EkI = Ekn.
It is easily seen that F   E k  E kI .
k
k
Now we see that  can be constructed as a self-similar
above alternative construction for the middle third Cantor
measure in 2kn different ways at the kth level of
pi1 (1)  pi 2 (2)  pi 3 (3)  ...  pi n (n ) to I i1i 2i 3 ...i n where i1, i2, i3,…, in
measure by using the
set and distributing the
iteration, by sending
 1,2
The relevant multifractal spectra can then be easily calculated using the methods
described by Falconer.
We will now consider the more interesting case where p1(k) and p2(k) are nonperiodic and non-convergent.
33
7 - Non-Periodic Non-Convergent Distribution Functions
The situation where the distribution functions are non-convergent and nonperiodic is considerably more difficult to handle in general. However, due to
some important results in the theory of ‘uniformly distributed sequences’ we can
say a great deal about a specific class of distribution functions which we will call
quasi-periodic non-convergent distribution functions. Here, we view p1(k) and
p2(k) as continuous functions of x rather than discrete functions of k and term
p1(x) and p2(x) quasi-periodic if they are periodic with an irrational period  .
Note that if the period was rational then we would be in the case of periodic
distribution functions described above.
We will first state some results from Kuipers and Niederreiter (1974) and Brin and
Stuck (2002) that will be crucial to the following arguments. Uniform Distribution
of Sequences is a seminal work by L. Kuipers and H. Niederreitter which studies
sequences which are uniformly distributed. Roughly speaking a sequence is said
to be uniformly distributed on some base set X if, asymptotically, it is spread as
evenly as possible across X. Kuipers and Niederreiter (1974: 1) give a more
formal definition:
Definition 7.1: A sequence (xn)n of real numbers is said to be uniformly
distributed modulo 1 if for every half open interval [a, b )  [0,1] we have:

 x n  [a, b) 1  n  N 

lim 
  ba
N 
N




The concept of uniform distribution modulo 1 can be extended to uniform
distribution modulo  for a real number  in a natural way. We make the
following definition:
Definition 7.2: A sequence (xn)n of real numbers is said to be uniformly
distributed modulo  if for every half open interval [a, b )  [0, ] we have:

 x n  [a, b)1  n  N 
 ba
lim 

N 
N





Note that if, for some   0 , we have that ( xn  )n is uniformly distributed mod 1
then :
34
xn
a
b


 x n  [a, b) 1  n  N 

    [  ,  ) 1  n  N 
 b a ba
lim 
 lim 

  
N 
N 
N
N
 









and we see that (xn)n is uniformly distributed modulo  .
Now, by Weyl’s Theorem, the sequence n  n is uniformly distributed mod 1 if
and only if  is irrational (Brin and Stuck, 2002: 89). Hence, by the above, the
sequence (n)n is uniformly distributed mod  if and only if  is irrational.
Now we note a key theorem from Kuipers and Niederreiter (1974: 2).
Theorem 7.3: The sequence (xn)n of real numbers is uniformly distributed mod 1
if and only if for every real valued piecewise continuous function f defined on [0,1]
we have:
1 N

lim   f ({ x n })   f ( x )dx
N  N
 n 1
 0
1
Proof: Omitted (Kuipers and Niederreiter, 1974: 2-3).
Note that Kuipers and Niederreiter only give a proof for continuous f, but this
could easily be extended to include piecewise continuous functions.
We can extend this result using definition 7.2 to give the following theorem:
Theorem 7.4: If a sequence (xn)n of real numbers is uniformly distributed mod 
then for every real valued piecewise continuous function f defined on [0, ] we
have:

1 N

lim   f ({ x n })  1  f ( x )dx
N  N
 n 1

0
Proof: Let (xn)n be a sequence of real numbers that is uniformly distributed mod
 . Then ( xn  )n is uniformly distributed mod 1 and we have:
u 
( u   x )
1 N

1 N

du
lim   f ({ x n })  lim   f ({  xn  })   f (  x )dx   f (u )

N  N

 n 1
 N   N n 1
 0
u 0
1


1

 f ( x )dx
0
35

1

 f (u )du
0
Now, returning to our measure, we note that, by theorem 4.2, we have:
 (q ) 


1 k
1
q
q 
lim   log p1(i )  p2 (i ) 
k


log 3
 k i 1

Now since p1(x) and p2(x) are periodic with period  we have that:
 (q ) 



1 k
1
q
q 
lim   log p1 (i )  p2 (i ) 
log 3 k   k i 1



1 k
1
q
q 
lim   log p1 (i mod  )  p2 (i mod  ) 
log 3 k   k i 1

and from above we know that (i mod  )i is uniformly distributed on [0,  ], since
 is taken to be irrational, and so we have that:
 (q ) 



1
log p1( x )q  p2 ( x )q dx (7.5)

 log 3 0
Despite this apparently simple form for  (q ) this integral is very difficult to
evaluate explicitly for all q. It is possible, however, to evaluate it numerically for
specific q, as in the following example:
Example 7.6:
In order to make calculations as simple as possible we choose p1(x) and p2(x) to
be piecewise linear.
Let p1(x) = x
2
(mod 2 ) and p2(x) = 1 - p1(x).
36
Figure 7.6.1
Figure 7.6.1 shows a plot of p1(x)
Note that p1(x) and p2(x) are indeed quasi-periodic, with period
convergent.
2 , and non-
We now have (by 7.5) that:
 (q ) 
 log 
2 log 3
1
2
q
x
2
 (1 
x
2

)q dx
0
This integral can now be evaluated using Maple for specific values of q:
The following Maple code was used to evaluate  (q ) from q = -5 to 5 by intervals
of 0.01:
f:=ln((x/sqrt(2))^q+(1-x/sqrt(2))^q);
B:=(1/(sqrt(2)*ln(3)))*int(f,x=0..sqrt(2));
L:={};
for i from -5 to 5 by 0.01 do
L:={op(L),[i,evalf(subs(q=i,B))]}:
end do:
plot(L,style=point,symbolsize=1,color=black);
37
Figure 7.6.2
Figure 7.6.2 shows a pointwise plot of
β(q)
Using a similar method, we can use Maple to plot the Legendre Transform,

, parameterized by q.
fL ( )   (q )  q 
q
38
Figure 7.1.3
Figure 7.1.3 shows a pointwise plot of
fL ( )
This numerical method could be used to accurately plot  (q ) and fL ( ) for many
other quasi-periodic distribution functions. The non-quasi periodic case remains
unsolved, however, and it is a possibility for future research to investigate further
strategies for evaluating  (q ) for non-convergent non-periodic non-quasi-periodic
distribution functions.
One interesting geometrical observation we can make from figure 7.1.3 is that
despite the irregularity of the distribution of mass in the construction of the
measure, fL ( ) appears to be symmetrical around q = 0. The following section
will investigate symmetry of the Legendre spectrum in general and highlight
some key geometrical features of the supporting set which influence the
dynamics of the measure.
39
8 - Symmetry of Spectra
When we began to examine measures constructed with irregular distribution
functions it may have been expected that the Legendre spectrum would mirror
this irregularity, and, in some specific examples, be non-symmetrical. So far this
has not been the case as every Legendre spectrum we have evaluated has been
symmetrical. We will show in this chapter that this is no coincidence.
We will now state and prove our main theorem.
Theorem 8.1: Every measure constructed using distribution functions and
supported by the middle third Cantor set has a symmetric Legendre spectrum,
assuming  (q ) exists for all q.
Proof:
First let F be the middle third Cantor set and  be a measure constructed with
arbitrary distribution functions p1(k) and p2(k). We have from theorem 4.2 that:
 (q ) 


1 k
1
q
q 
lim   log p1(i )  p2 (i ) 
k


log 3
 k i 1

(†)
assuming this limit exists,
and
fL ( )   (q )  q 
where  is given by:   

q

(page 15).
q
Hence, we have:
 


k
 1


q
q 

lim 
log p1 (i )  p2 (i ) 

k


q
q
 k log 3 i 1



k
 1

q
q 
  lim 
log p1(i )  p2 (i ) 

k  k log 3
i 1 q


 1 k p1(i )q log p1(i )  p2 (i )q log p2 (i ) 
1

lim  

log 3 k   k i 1
p1(i )q  p2 (i )q

40
and fL ( ) has a maximum at

q 0


 1 k log p1 (i )  log p2 (i ) 
1
lim  

log 3 k   k i 1
1 1

1 k

1
lim   log p1 (i )p2 (i )
2 log 3 k   k i 1

Now in proving the theorem it suffices to show that


1
1 k
fL  
lim  log p1(i )p2 (i )     fL ( )
k


k i 1
 log 3

First note that:
 (q )   ( q )


 




1 k
1 k
1
1
q
q 
q
q 
lim   log p1 (i )  p2 (i )  
lim   log p1 (i )  p2 (i ) 
log 3 k   k i 1
 log 3 k   k i 1


1 k
1
q
q
q
q 
lim   log p1 (i )  p2 (i )  log p1 (i )  p2 (i ) 
k


log 3
 k i 1



1 k
p1 (i )q  p2 (i )q 
1 k
1
1
q
q

lim   log

lim

  log p1 (i ) p2 (i ) 
q
q
k


k


log 3
p1 (i )  p2 (i )  log 3
 k i 1

 k i 1

1 k

q
lim   log p1 (i )p2 (i )
log 3 k   k i 1

(2)
Now we have that:




1
1 k
q
1 k
fL  
lim  log p1 (i )p2 (i )     inf  (q ) 
lim  log p1 (i )p2 (i )  q 
log 3 k  k i 1
 log 3 k  k i 1
 qR 

 inf  (q )   (q )   ( q )  q  inf  ( q )  q  inf  (q )  q  fL ( )
(2)
qR
qR
qR
41
and, thus, we have proved the theorem.
Note that the limit (†) exists for all measures considered in chapters 5, 6 and 7
and will exist in a number of more general cases. If the limit does not exist we
can define  (q) and the upper Legendre spectrum as:
 (q ) 


1 k
1
q
q 
lim   log p1(i )  p2 (i ) 
log 3 k   k i 1

and

fL ( )  inf  (q )  q
qr

respectively, with  (q ) and the lower Legendre spectrum defined analogously.
Then a similar argument can be used to prove that the upper and lower Legendre
spectra are both symmetrical.
So it turns out that all the measures we have considered so far necessarily have
symmetrical Legendre spectra. It is natural to ask if this result can be extended
to include measures supported on other fractal subsets of [0, 1]. This question
will be addressed in the following chapter.
42
9 - Some Geometry of the Legendre Spectra
“Unfortunately, the pedagogical excellence of the heuristic approach of Halsey et
al. misled many users by creating the widespread impression that the function
f(  ) always has a graph shaped like the mathematical symbol  ”
-Benoit Mandelbrot (2004: 223).
The theorems stated and proved in this chapter will aim to demonstrate the
motivation behind Mandelbrot’s assertion.
We will first consider a more general fractal construction. Similar to the
construction of the middle third Cantor set we begin with the unit interval [0,1] but
this time instead of leaving two intervals of length 13 after the first iteration we
leave n intervals of length 1 and remove n-1 intervals of equal length. Note
c
that c  2n  1 .
Figure 9.1
Figure 9.1 shows the construction of a generalized Cantor set with n = 3
We will now consider measures constructed on the generalized Cantor set with
constant distribution functions p1(k) = p1, p2(k) = p2 and p3(k) = p3 for all k, where
p1 + p2 + p3 = 1.
Theorem 9.2: A measure  constructed with constant distribution functions on
a generalized Cantor set is asymmetric if and only if n > 2.
43
Proof:
Using a similar argument to that used in the proof of theorem 4.2 we can show
that:
n
 (q ) 
log  pi
q
i 1
log c
and we also have that:
n
p
q
logpi

i 1
 
 n


q 
  pi  log c
 i 1

i
Now we define A, G and H to be the arithmetic, geometric and harmonic means
q
of pi i respectively.
 
 n q
G    pi 
 i 1

1 n q
A   pi
n i 1
1
n
H
n
n

i 1
1
pi
q
Now we note that:
n
 (q )   ( q ) 
log  pi
n
q
i 1
log c

log 
1
i 1
pi
q
log c

log c
and that fL ( ) has a maximum at
n
n

q 0

 log pi
i 1
 n 
  1 log c
 i 1 
44
 H   log AH
log( nA)  log n

log  pi
i 1
n log c
log c
(1)
Now, in a similar vein to the method used in the proof of theorem 8.1, we
n


 2 log  pi


i 1
   , but in this case show that it is necessarily different
consider fL 


n log c




from fL ( ) for some value of q.
We have:
n
n




 2 log  pi

2
q
log
pi



i 1
i 1
fL  
    inf  (q ) 
 q 

 qR
n log c
n log c










 n q
log  pi 


 i 1

 inf  (q ) 
qR
log c



2
n




log G 2

 q   inf  (q ) 
 q 
qR
log c






log G 2
log AH log AH 
 inf  (q ) 
 q 


qR
log c
log c
log c 


log G 2
log AH 
 inf  (q ) 
 q   (q )   ( q ) 

qR
log c
log c 

(1)

log AH 2

G

 inf  ( q )  q 
qR
log c


2

 
 
log G
q  q
AH




 

  inf  (q )  q 

qR
log c






(2)
At this point we can see that (2) = fL ( ) and we will hence have symmetry if
G 2  AH for all q which will be the case for n = 2 or p1  p2   pn .
The infimum in (2) will occur when:
45
2


log G


AH  

  (q )  q 
0
q 
log c



i.e. when
 
So we have that (2) =  (q )  q
   log G 2  log AH 

 .

q q 
log c


  log G 2  log AH  log G 2  log AH
 
 q 
q
q 
log c
log c

Now we define d(q) by:
n


 2 log  pi


i 1
d (q )  fL ( )  fL 
 


n log c




So we have that:
d (q )   (q )  q
q
 

  log G 2  log AH  log G 2  log AH 

 
   (q )  q
q

q 
q
q 
log c
log c


  log G 2  log AH  log G 2  log AH

 
q 
log c
log c

Now by noting that:

   n q 
q
log G 2  q
log  pi 
q
q   i 1




2
n
n


  q   q log p 
 i 

q 
 i 1 


 log G 2
we can deduce that:
d (q ) 
log AH
  log AH 


q
log c
q  log c 
46
2
n
n

  q log p 
 i 

 i 1 

2
n
and hence, we can see that d(q) will be non-zero for at least some range of q,
provided n > 2, and we can deduce that fL ( ) is not symmetric about q = 0 and
we have proved theorem 9.2.
Example 9.3:
Consider the case when n = 3 and p1  0.15, p2  0.35 and p3  0.5 .
Figure 9.3.1
Figure 9.3.1 shows a plot of fL(  ) as a function of q (in red), a plot of d(q) (in black) and a
plot of fL(  ) –d(q) for q > 0 (in blue)
So it turns out that symmetry of fL ( ) is a rather special property and in general
does not occur. It is natural to ask if fL ( ) is symmetrical for measures
constructed on Cantor-like sets if and only if each interval remaining at the kth
stage of construction is divided into two to form the (k+1)th stage as in the middle
third Cantor set described above. We shall demonstrate in the following chapter
that this is not true by considering lop-sided Cantor sets.
47
10 - A Note on Lop-sided Supporting Sets
We have seen that symmetry of fL ( ) occurs only in the special case of n = 2 in
theorem 9.2. We will now show that this symmetry not only relies on n but on the
symmetry of the supporting set. To do this we will consider self similar measures
(with constant distribution functions p1 and p2) supported by a lop-sided Cantor
set, i.e., an asymmetric Cantor set where the number of intervals remaining at
the kth stage of construction is 2k, but the lengths of the intervals vary.
Figure 10.4
Figure 10.4 shows the construction of a lop-sided Cantor set
In this case it can be shown that  (q ) satisfies:
q
p1 c1
 (q )
 p2 c 2
q
 (q )
 1. (Falconer, 2003: 287)
This equation cannot be solved to give an explicit formula for  (q ) in general but
if we choose c2 such that c2= c12 then we have a quadratic in c 1
hence find  (q ) explicitly.
48
 (q )
and can
We have:
q
 (q )
 c1
 (q )
p1 c1
 p 2 (c 1 )  ( q )  1  p 2 (c 1
q
2
 p1  p1
q

q
2q
2 p2
 4 p2
 (q )
) 2  p1 c1
q
 (q )
1 0
q
q
  p q  p 2q  4 p q
1
1
2
log
q
2 p2

  (q ) 
log c1


 log p12q  4 p2 q  p1q   log( 2p2 q )



log c1
Note that we take the positive square root to guarantee the argument of the
logarithm is positive.
We can now calculate fL ( ) explicitly and plot it as a function of  . This can be
done for specific examples easily using Maple.
The following Maple procedure, multi, requires an input of I = p1, r=p2 and c=c1
and returns a plot of fL ( ) as a function of  as required along with the line y  
and the constant line y  dimH spt :
multi:=proc(l,r,c)
local B, a, q, M, m;
global L;
B:=((ln((l^(2*q)+4*r^q)^(1/2)-l^q)-ln(2)-q*ln(r))/ln(c)):
M:=ln(sqrt(5)/2-1/2)/ln(c):
plot(B,q=-10..10);
a:=-diff(B,q):
m:=subs(q=0,a):
L:=B+a*q:
q:=solve(alpha=a,q): L:
return
plot([L,alpha,M],alpha=0..2*m,y=0..1.1*M,color=[black]);
end proc:
49
For example, multi(1/3, 2/3, 1/2) returns:
Figure 10.5
Figure 10.5 shows a plot of fL(  ) for p1 =
1
3
, p2 =
2
3
, c1 =
1
2
and c2 =
1
4
We can clearly see from the above graph that fL ( ) is not symmetrical.
This example gives a further indication of what Mandelbrot was talking about in
the quotation given at the beginning of chapter 9. Symmetry of the Legendre
spectrum is, in actual fact, a rather rarified property and it appears to require both
symmetry of the supporting set and the identity G2 = AH, which only holds for all
q if and only if the support is constructed by splitting intervals in two or trivially if
p1 = p2 = p3 = … = pn.
50
11 - Conclusion and Summary of Results
In chapters 5, 6 and 7 we studied multifractal measures constructed using
distribution functions and supported by the middle third Cantor set. When the
distribution functions are convergent or periodic we can evaluate the Legendre
spectrum quite easily, and, by drawing on results from the theory of uniformly
distributed sequences, we can extend these results to the quasi-periodic case.
We believe that these results could be generalised in a natural way to include
measures constructed on other fractal sets. For example, if we considered the
Sierpinski triangle, we would require three distribution functions to construct
analogous measures. Due to the increase in the number of distribution functions,
analysis of these measures would require more cases. For example, if p1(k) was
convergent and p2(k) was quasi-periodic, then the behaviour of p3(k) = 1 – p1(k) –
p2(k) would follow a new type of behaviour.
As noted in chapter 7 it would be another possibility for future research to
investigate measures constructed on the middle third Cantor set using
distribution functions which are non-convergent, non-periodic and non-quasiperiodic. In this case we believe that it would be possible to choose p1(k) and
p2(k) such that  (q ) is not well defined and we would have to work with  (q )
and  (q ) as defined at the end of chapter 8.
As for the symmetry properties of fL ( ) it would be interesting to extend theorem
9.2 to consider the case when the distribution functions are not constant. We
conjecture that fL ( ) would remain non-symmetric in all cases for n > 2. It would
also be worth looking at some properties of d (q ) , defined in chapter 9, in more
detail. For example, it was initially thought that d (q ) could not have any
crossovers, i.e. could not equal zero for any value of q other than 0 and  . This,
however, is not the case as the following plot shows.
51
Figure 11.1
Figure 11.1 shows a plot of d(q) for n = 5 and p1(k)= 120 , p2(k)= 110 , p3(k)= 15 , p4(k)=
(k)= 9
p5
1
5
and
20
It is clear from the above diagram that d (q ) has a crossover. Note, however,
that we used n = 5 in order to find this example. We conjecture that d (q ) cannot
have a crossover for n = 3. It would be a possibility for future work to investigate
these crossovers more thoroughly. More specifically by addressing the following
two questions: Precisely when can crossovers occur? Can there ever be more
than one?
52
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