Fractal Dimensions and Random Transformations

2001

We study random recursive constructions with finite "memory" in complete metric spaces and the Hausdorff dimension of the generated random fractals. With each such construction and any positive number β we associate a linear operator V (β) in a finite dimensional space. We prove that under some conditions on the random construction the Hausdorff dimension of the fractal coincides with the value of the parameter β for which the spectral radius of V (β) equals 1.

Bulletin of the Brazilian Mathematical Society, New Series, 2021

We investigate in this work some situations where it is possible to estimate or determine the upper and the lower q-generalized fractal dimensions D ± µ (q), q ∈ R, of invariant measures associated with continuous transformations over compact metric spaces. In particular, we present an alternative proof of Young's Theorem [31] for the generalized fractal dimensions of the Bowen-Margulis measure associated with a C 1+α-Axiom A system over a two-dimensional compact Riemannian manifold M. We also present estimates for the generalized fractal dimensions of an ergodic measure for which Brin-Katok's Theorem is satisfied punctually, in terms of its metric entropy. Furthermore, for expansive homeomorphisms (like C 1-Axiom A systems), we show that the set of invariant measures such that D + µ (q) = 0 (q ≥ 1), under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each s ∈ [0, 1), D + µ (s) is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric. Finally, we show that, for some dynamical systems, the metric entropy of an invariant measure is typically zero, settling a conjecture posed by Sigmund in [25] for Lipschitz transformations which satisfy the specification property. Key words and phrases. Expansive homeomorphisms, Hausdorff dimension, packing dimension, invariant measures, generalized fractal dimensions, dynamical systems with specification * Work partially supported by CIENCIACTIVA C.G. 176-2015 † Work partially supported by FAPEMIG (a Brazilian government agency; Universal Project 001/17/CEX-APQ-00352-17) popular of all, the Hausdorff dimension, introduced in 1919 by Hausdorff, which gives a notion of size useful for distinguishing between sets of zero Lebesgue measure. Unfortunately, the Hausdorff dimension of relatively simple sets can be very hard to calculate; besides, the notion of Hausdorff dimension is not completely adapted to the dynamics per se (for instance, if Z is a periodic orbit, then its Hausdorff dimension is zero, regardless to whether the orbit is stable, unstable, or neutral). This fact led to the introduction of other characteristics for which it is possible to estimate the size of irregular sets. For this reason, some of these quantities were also branded as "dimensions" (although some of them lack some basic properties satisfied by Hausdorff dimension, such as σ-stability; see [12]). Several good candidates were proposed, such as the correlation, information, box counting and entropy dimensions, among others. Thus, in order to obtain relevant information about the dynamics, one should consider not only the geometry of the measurable set Z ⊂ X (where X is some Borel measurable space), but also the distribution of points on Z under f (which is assumed to be a measurable transformation). That is, one should be interested in how often a given point x ∈ Z visits a fixed subset Y ⊂ Z under f. If µ is an ergodic measure for which µ(Y) > 0, then for a typical point x ∈ Z, the average number of visits is equal to µ(Y). Thus, the orbit distribution is completely determined by the measure µ. On the other hand, the measure µ is completely specified by the distribution of a typical orbit. This fact is widely used in the numerical study of dynamical systems where the distributions are, in general, non-uniform and have a clearly visible fine-scaled interwoven structure of hot and cold spots, that is, regions where the frequency of visitations is either much greater than average or much less than average respectively.

Ergodic Theory and Dynamical Systems, 2016

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and Mandelbrot percolation. In each setting we compute either thealmost sureor theBaire typicalAssouad dimension and consider some illustrative examples. Our results reveal a phenomenon common to each of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure-theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.

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.

Ergodic Theory and Dynamical Systems, 2008

A linear map from one Euclidean space to another may map a compact set bijectively to a set of smaller Hausdorff dimension. For 'homogeneous' fractals (to be defined), there is a phenomenon of 'dimension conservation'. In proving this we shall introduce dynamical systems whose states represent compactly supported measures in which progression in time corresponds to progressively increasing magnification. Application of the ergodic theorem will show that, generically, dimension conservation is valid. This 'almost everywhere' result implies a non-probabilistic statement for homogeneous fractals.

Physics Letters A, 1985

Several definitions of generahzed fractal dimensions are reviewed, generalized, and interconnected. They concern (i) different ways of averaging when treating fractal measures (instead of sets); (ii) "'partial dimensions" measuring the fraetality in different directions, and adding up to the generalized dimensions discussed before.

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.

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.

Physica A: Statistical Mechanics and its Applications, 2002

We present a generalized stochastic Cantor set by means of a simple cut and delete process and discuss the self-similar properties of the arising geometric structure. To increase the flexibility of the model, two free parameters, m and b, are introduced which tune the relative strength of the two processes and the degree of randomness respectively. In doing so, we have identified a new set with a wide spectrum of subsets produced by tuning either m or b. Measuring the size of the resulting set in terms of fractal dimension, we show that the fractal dimension increases with increasing order and reaches its maximum value when the randomness is completely ceased.

Forum Mathematicum

The families of V -variable fractals for V D 1; 2; 3; : : : , together with their natural probability distributions, interpolate between the corresponding families of random homogeneous fractals and of random recursive fractals. We investigate certain random V V matrices associated with these fractals and use them to compute the almost sure Hausdorff dimension of V -variable fractals satisfying the uniform open set condition.