Can randomness alone tune the fractal dimension?

ANNALS OF PURE AND APPLIED LOGIC, 2014

The (constructive Hausdorff) dimension of a point x in Euclidean space is the algorithmic information density of x. Roughly speaking, this is the least real number dim(x) such that r×dim(x) bits suffices to specify x on a general-purpose computer with arbitrarily high precisions 2 −r. The dimension spectrum of a set X in Euclidean space is the subset of [0, n] consisting of the dimensions of all points in X. The dimensions of points have been shown to be geometrically meaningful (Lutz 2003, Hitchcock 2003), and the dimensions of points in self-similar fractals have been completely analyzed (Lutz and Mayordomo 2008). Here we begin the more challenging task of analyzing the dimensions of points in random fractals. We focus on fractals that are randomly selected subfractals of a given self-similar fractal. We formulate the specification of a point in such a subfractal as the outcome of an infinite two-player game between a selector that selects the subfractal and a coder that selects a point within the subfractal. Our selectors are algorithmically random with respect to various probability measures, so our selector-coder games are, from the coder's point of view, games against nature. We determine the dimension spectra of a wide class of such randomly selected subfractals. We show that each such fractal has a dimension spectrum that is a closed interval whose endpoints can be computed or approximated from the parameters of the fractal. In general, the maximum of the spectrum is determined by the degree to which the coder can reinforce the randomness in the selector, while the minimum is determined by the degree to which the coder can cancel randomness in the selector. This constructive and destructive interference between the players' randomnesses is somewhat subtle, even in the simplest cases. Our proof techniques include van Lambalgen's theorem on independent random sequences, measure preserving transformations, an application of network flow theory, a Kolmogorov complexity lower bound argument, and a nonconstructive proof that this bound is tight.

American Journal of Mathematics, 2020

We characterize the existence of certain geometric configurations in the fractal percolation limit set A in terms of the almost sure dimension of A. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemerédi theorems for random discrete sets, we also consider the corresponding problem for sets of positive ν-measure, where ν is the natural measure on A. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating the intersections of the products of m independent realizations of A with transversal planes and, more generally, algebraic varieties, and extend some well known features of independent percolation on trees to a setting with long-range dependencies.

In the present paper we define statistically self-similar sets, and, using a modification of method described K.J.Falconer find a Hausdorff dimension of a statistically self-similar set.

1999

We consider a Cantor-like set as a geometric projection of a Bernoulli process. P. Billingsley (1960) and C. Dai and S.J. Taylor (1994) introduced dimension-like indices in the probability space of a stochastic process. Under suitable regularity conditions we find closed formulae linking the Hausdorff, box and packing metric dimensions of the subsets of the Cantor–like set, to the corresponding Billingsley dimensions associated with a suitable Gibbs measure. In particular, these formulae imply that computing dimensions in a number of well-known fractal spaces boils down to computing dimensions in the unit interval endowed with a suitable metric. We use these results to generalize density theorems in Cantor–like spaces. We also give some examples to illustrate the application of our results.

1997

We show that when the standard techniques for calculating fractal dimensions in empirical data (such as the box counting) are applied on uniformly random structures, apparent fractal behavior is observed in a range between physically relevant cutoffs. This range, spanning between one and two decades for densities of 0.1 and lower, is in good agreement with the typical range observed in experiments. The dimensions are not universal and depend on density. Our observations are applicable to spatial, temporal and spectral random structures, all with non-zero measure. Fat fractal analysis does not seem to add information over routine fractal analysis procedures. Most significantly, we find that this apparent fractal behavior is robust even to the presence of moderate correlations. We thus propose that apparent fractal behavior observed experimentally over a limited range in some systems, may often have its origin in underlying randomness.

Probability Theory and Related Fields, 1987

Birkhäuser Basel eBooks, 2000

In the first section fractal dimensions of certain measures and sets are studied. Lars Olsen describes the different multifractal formalisms for measures and investigates their behaviour for constructions like forming products or intersections of measures. The contribution of Yuval Peres, Wilhelm Schlag, and Boris Solomyak centers around the absolute continuity or singularity of Bernoulli convolutions.

Fractals in Physics, 1986

The fractal dimensions of various types of Jntersection sets of random fractals are discussed. ffris frÀipi a better understanding oÍ'issues concernlng-upper critical dimensionality' F'lory upp"oiimátions or universality oi critical behaviour w'lth respect to different excluded volume màttranisms. Diffusion on sel?-intersecting fractals, and on fractals with local bridges' is also iónjiaereA in the light of the above discussion, and accurate results for the relative exponents' obtained from ana'lysis of Monte-Carlo generated enumeratlons, are presented.

We investigate, both analytically and numerically, the kinetic and stochastic counterpart of the triadic Cantor set. The generator that divides an interval either into three equal pieces or into three pieces randomly and remove the middle third is applied to only one interval, picked with probability proportional to its size, at each generation step in the kinetic and stochastic Cantor set respectively. We show that the fractal dimension of the kinetic Cantor set coincides with that of its classical counterpart despite the apparent differences in the spatial distribution of the intervals. For the stochastic Cantor set, however, we find that the resulting set has fractal dimension d f = 0.56155 which is less than its classical value d f = ln 2 ln 3 . Nonetheless, in all three cases we show that the sum of the d f th power, d f being the fractal dimension of the respective set, of all the intervals at all time is equal to one or the size of the initiator [0, 1] regardless of whether it is recursive, kinetic or stochastic Cantor set. Besides, we propose exact algorithms for both the variants which can capture the complete dynamics described by the rate equation used to solve the respective model analytically. The perfect agreement between our analytical and numerical simulation is a clear testament to that.

Forum Mathematicum, 2000

V-variable fractals, for V = 1, 2, 3,. .. , interpolate between random homogeneous fractals and random recursive fractals. We compute the almost sure Hausdorff dimension of V-variable fractals satisfying the uniform open set condition. Important roles are played by the notion of a neck, leading to spatial homogeneity at various levels of magnification, and a variant of the Furstenberg Kesten theorem for products of certain random V × V matrices.