Comment on the equivalence between fracton/spectral dimensionality,and the dimensionality of recurrence

The fracton/spectral dimensionality ^D fracton is an important characteristic of fractal processes. Until now, it has not been interpreted as the fractal dimensionality of any well-defined fractal set, and it has been claimed that ^D fracton is more intrinsic than the fractal dimensionality. In fact, ^D fractol, is best understood as an originaldynamical reinterpretation of a well-defined previously knownkinetic dimensionality: ^D fracton 's twice the fractal codimensionality of the time instants when a fractal process returns to a point it had previously visited.Presented at theThird Conference on Fractals: Fractals in the Physical Sciences, held at the National Bureau of Standards, Gaithersburg, Maryland, on November 20–23, 1983.While this talk has been widely discussed, its text was not available for inclusion in these proceedings.     

Squig sheets and some other squig fractal constructions

Squig intervals are a class of hierarchically constructed fractals introduced by the author. They can be visualized as the final outcome upon a straight interval of a suitable cascade of local perturbative eddies ruled by two processes called decimation and separation. Their theory is summarized and their scope is extended in several new directions, especially by introducing new forms of separation. Squig intervals are generalized in two dimensions, with fractal dimensions ranging from 1.2886 to 1.589. Squig sheets are constructed in three dimensional space with fractal dimensions ranging from 8/3 up. They should prove useful in modeling the fractal surfaces associated with turbulence and related phenomena. Squig intervals are constructed in three dimensions. Nonsymmetric eddies and the resulting squigs are tackled. Squig trees and intervals are drawn on unconventional lattices, either in the plane or in a prescribed fractal surface. Peyriére'sM systems are mentioned: their study includes the proof that the informal renormalization argument (involving a transfer matrix) is exact for squigs.Presented at theThird Conference on Fractals: Fractals in the Physical Sciences, held at the National Bureau of Standards, Gaithersburg, Maryland, on November 20–23, 1983.The reader's attention should be drawn to the fact that the second and later printings of this book include an update chapter and additional references. Though it should not have been necessary, it may be useful also to mention here that most of the material in this book that concerns physics, e.g., polymers and percolation clusters, wasnot found in either of my two earlier Essays on fractals,Les objects fractals: forme, hasard et dimension (Flammarion,     

Fractals in physics: Squig clusters,diffusions, fractal measures,and the unicity of fractal dimensionality

The three topics discussed in this paper are largely independent. Part 1: Fractal squig clusters are introduced, and it is shown that their properties can match to a remarkable extent those of percolation clusters at criticality. Physics on these new geometric shapes should prove tractable. As background, the author's theories of squig intervals and squig trees are reviewed, and restated in more versatile form. Part 2: The notion of latent fractal dimensionality is introduced and motivated by the desire to simplify the algebra of dimensionality. Scaling noises are touched upon. A common formalism is presented for three forms of anomalous diffusion: the ant in the fractal labyrinth, fractional Brownian motion, and Lévy stable motion. The fractal dimensionalities common to diverse shapes generated by diffusion are given, in Table I, as functions of the latent dimensionalities of the support of the motion and of the diffusion itself. Part 3: It is argued that every fractal point set has a unique fractal dimensionality, but it is pointed out that many fractals involve diverse combinations of many fractal point sets. Such is, in particular, the case for fractal measures and for fractal graphs, often called hierarchical lattices. The fractal measures that the author had introduced in the early 1970s are described, including new developments.     

Multiperiodic multifractal martingale measures

A nonnegative 1-periodic multifractal measure on is obtained as infinite random product of harmonics of a 1-periodic function W(t). Such infinite products are statistically self-affine and generalize certain Riesz products with random phases. They are martingale structures, therefore converge. The criterion on W for nondegeneracy is provided. It differs completely from those for other known random measures constructed as martingale limits of multiplicative processes. In particular, it is very sensitive to small changes in W(t). When these infinite products are interpreted in the framework of thermodynamic formalism for random transformations, logW is a potential function when W>0. For regular enough potentials, in case of degeneracy, the natural normalization makes the sequence of measures converge. Moreover, this normalization is neutral for nondegenerate martingales. The multifractal analysis of the limit martingale measure is performed for a class of potential functions having a dense countable set of jump points.     

Angular gaps in radial diffusion-limited aggregation: two fractal dimensions and nontransient deviations from linear self-similarity

When suitably rescaled, the distribution of the angular gaps between branches of off-lattice radial diffusion-limited aggregation is shown to approach a size-independent limit. The power-law expected from an asymptotic fractal dimension D = 1.71 arises only for very small angular gaps, which occur only for clusters significantly larger than M = 10(6) particles. Intermediate size gaps exhibit an effective dimension around 1.67, even for M--> infinity. They dominate the distribution for clusters with M<10(6). The largest gap approaches a finite limit extremely slowly, with a correction of order M(-0.17).     

Multifractal Power Law Distributions: Negative and Critical Dimensions and Other “Anomalies,” Explained by a Simple Example

Divergence of high moments and dimension of the carrier is the subtitle of Mandelbrot's 1974 seed paper on random multifractals. The key words divergence and dimension met very different fates. Dimension expanded into a multifractal formalism based on an exponent and a function f(). An excellent exposition in Halsey et al. 1986 helped this formalism flourish. But it does not allow divergent high moments and the related inequalities f()<0 and <0. As a result, those possibilities did not flourish. Now their time has come for diverse reasons. The broad 1974 definitions of and f allow <0 and f()<0, but the original presentation demanded to be both developed and simplified. This paper shows that both multifractal anomalies occur in a very simple example, which has been crafted for this purpose. This example predicts the power law distribution. It generalizes and f() beyond their usual roles of being a Hölder exponent and a Hausdorff dimension. The effect is to allow either f or both f and to be negative, and the apparent anomalies are made into sources of new important information. In addition, this paper substantially clarifies the subtle way in which randomness manifests itself in multifractals.