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) supa 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 10a1 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 qr Note that, letting = -a, we see that: fL ( ) inf (q ) q inf (q ) aq ) inf ( (q ) aq ) qr qR qR sup aq (q ) L(f)(a) qR 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 ) maxdimH 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 3k 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 ... 3k ... 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 q0 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 q0 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 ba 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 ba 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 ba 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 qR inf (q ) (q ) ( q ) q inf ( q ) q inf (q ) q fL ( ) (2) qR qR qR 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 qr 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 qR n log c n log c n q log pi i 1 inf (q ) qR log c 2 n log G 2 q inf (q ) q qR log c log G 2 log AH log AH inf (q ) q qR log c log c log c log G 2 log AH inf (q ) q (q ) ( q ) qR log c log c (1) log AH 2 G inf ( q ) q qR log c 2 log G q q AH inf (q ) q qR 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 Bibliography Bandt, C., Graf, S. & Zähle, M (editors) (1991), Fractal Geometry and Stochastics II. Birkhäuser Verlag Barnsley, Michael F. 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