Pietronero / Tosatti Fractals in Physics

1. Auflage 2012
ISBN: 978-0-444-59841-7
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 489 Seiten

Fractals in Physics

Autoren/Hrsg.

Pietronero, L.

Tosatti, E.

Leseproben

SELF-AFFINE FRACTAL SETS, I: THE BASIC FRACTAL DIMENSIONS

Benoit B. MANDELBROT,     Physics Department, IBM Research Center; Mathematics Department, Harvard University, Cambridge, MA 02138 USA*

The notion of -fractal dimension is explored -for various -fractal curves or dusts that are not self -similar, but are diagonally self - affine. A diagonal self -affinity stretches the coordinates in different ratios. It is showed that, in contrast to the unique -fractal dimension of strictly self-similar sets, one needs in general several distinct notions. Most important are the concepts of dimension obtained via the mass in a sphere and via covering by uniform boxes. One -finds it does not matter which definition is taken, but it matters greatly whether one interpolates or extrapolates. Thus, one obtains two sharply distinct dimensions: a local one, valid on scales well below, and a global one, valid on scales well above, a certain crossover scale.

1 INTRODUCTION

This paper examines, on three levels corresponding to three parts, what happens to diverse alternative definitions of fractal dimension when they are generalized from self-similar fractals to certain self-affine fractals. The substance of my Scripta paper1 is incorporated. Self-similar fractals were the original objects on which diverse fractal dimensions had been tested in detail, and their values were found to coincide.2 When a method works well in one case, it is tempting to apply it under increasingly wide conditions. The more general context of self-affine fractals now deserves systematic attention.

I have coined “self-affine” and “self-similar” in 1964 (the latter is so accepted now, that its age has become hard to believe), but “affine” goes back to Euler. In this paper, no specific knowledge of affine geometry will be required, but it is amusing to quote a characterization of that field by E. Snapper and R.J. Troyer: “Roughly speaking, affine geometry is what remains after practically all ability to measure length, area, angles, etc… has been removed from Euclidean geometry. One might think that affine geometry is a poverty-stricken subject. On the contrary, it is quite rich”. I hope to convince the reader that self-affine fractals also prove to be a very surprisingly rich topic.

One well-known but very special example of self-affine fractal is the record of Wiener’s scalar Brownian motion, which is the random process with independent and stationary Gaussian increments. This record has a well-known invariance property: setting B(O) = O, the processes B(t) and b-1/2B(bt) are identical in distribution for every ratio b > O. One observes that the rescaling ratios of t and of B are different, hence the transformation from B(t) to b-1/2 B(bt) is not a similitude but a more general “affinity.” This is why B(t) was called “statistically self-affine” on page 350 of my book2.

While a similarity is a linear transformation that shrinks or expands all the vectors implicit in a geometric figure in the same ratio, an affinity is a linear transformation that shrinks different vectors differently according to their directions. More precisely, B(t) is unchanged statistically under “diagonal affinity”, a notion explained in Section 2.

In this paper, diagonally self-affine fractal curves, dusts, and other sets, are examined in detail on several successive levels of generality, first in the plane: records of functions (random or not) analogous to B(t), and two levels of more general sets, and then in space. In each case, the fractal-dimensional properties are shown to exhibit new and surprising complications. The different roles of the single “all-purpose” fractal dimension of a fully self-similar set are now shared by a multitude of different “special purpose” fractal dimensions.

This part covers the mass-box dimension, and the gap dimension. Conceptually and mathematically, the most original finding is that a self-affine fractal’s mass-box dimension has distinct local and global variants. For example, in a recursive case of middle generality, the single base b is replaced by two bases b' and b'', and the classical expression logbN is replaced by the combination of logb,(Nb'/b'') (local) and logb?(Nb''/b') (global). In addition, lying between the above, we often have a distinct “gap dimension”, with a single value extending over all scales, namely .

Part II covers the dimensions obtained by walking a divider along a curve or “triangulating” a surface. The values one obtains are “doubly anomalous”, namely: distinct from those obtained in Part I.

Part III starts with recent mathematical findings3,4,5 concerning certain cases of the Hausdorff-Besicovitch dimension DHB, extends them, and discusses the meaning of the “double anomaly” found in certain cases.

The richness and complexity of this study are not purely mathematical, but reflect the richness and complexity of nature. Again, increasingly complex fractals are considered, the continuing refinement of our description of their structure demands a continuingly increasing number of fractal dimensions.

2 THE NOTIONS OF AFFINITY, DIAGONAL AFFINITY, AND SELF-AFFINITY

2.1 Background to diagonal affinity

Let us think of the Brownian record B(t) again. In Wiener’s original interpretation, t is time and B is a physical particle’s location on a spatial axis. The two coordinates play sharply different roles, and the units of B and t (cm. and second?) can be chosen independently. Rotation is not allowable, because it leads to sets that are no longer records of functions.

Forming the expression B(t)-dt (another application of affinity) introduces a function called “Brownian motion with a drift,” which is a very different process -from B(t). In an even earlier interpretation by Louis Bachelier (in 1900), t is time and B is a price in francs. The same remarks apply. However, an interpretation that I advanced later is substantially different: B(t) describes the vertical section of one of my Brown landscapes (my book2, chapter 28); the coordinates still play different roles, since gravity defines the vertical direction, makes overhangs an exception and makes it is useful to represent the relief by a (single-valued) function. However, both B and t are lengths in this example, and their units can no longer be chosen independently. For reasons that will transpire later, the best is to choose as unit of both B and t to be the tc such that |B(t+tc) - B(t)| ~ |tc|. This tc will be called the “crossover scale.” This unit’s counterpart in the original Brownian motion interpretation depends upon the units that happen to be selected for t and B; therefore, the crossover is in general not intrinsic.

The local and global dimensions that are introduced examined below are separated by this crossover. Without it, the local versus global distinction could not be clearcut.

2.2 Diagonal affinities

Section 2.1. shows that in the case of the record of B(t) - and the same will hold for the other fractals in this paper - a special role is played by affinities whose invariant set is made of straight lines that are parallel to the coordinate axes. Such an affinity, which I propose to call diagonal, operates in the E-dimensional affine space AE. Each member of a collection of affinities is specified by giving a fixed point of coordinates and an array of reduction ratios rm (0< m< E - 1), and considering the map

The ratios rm need not be positive. And they must not all be equal, because otherwise the transformation would fall be a similitude. The inverses 1/|rm| = bm, called bases, are integers in the simplest examples that are constructed recursively.

Most of the examples will be sets in the affine plane A2 (E=2). We shall write b'=max bm, b''=min bm, and H=log b''/log b'. This H, called affinity exponent, will satisfy 0

Formally, a linear transformation is the sequence of a translation and a multiplication by a matrix; we only tackle the cases where the matrix is diagonal and its diagonal terms are not identical. The product of two diagonal affinities is a diagonal affinity. Thus, a collection of diagonal affini ties can be used as the basis for a group.

The issues to be addressed involve the meaning of “square”, “distance”, and “circle” in affine geometry. (See second paragraph of Section 1). These notions remain meaningful for relief cross-sections, but for records of noise or of price, the units along the t axis and along the B axis are set up independently of each other. There being no meaning to equal height and width, a square cannot be defined. Similarly, a circle cannot be defined, because its square radius R2 = ?t2 + ?B2 would have to combine the units along both axes. Furthermore, one cannot “walk a divider” along a self-affine...