Fractal dimension for fractal structures: A Hausdorff approach (original) (raw)

Fractal dimension for fractal structures: A Hausdorff approach revisited

In this paper, we use fractal structures to study a new approach to the Hausdorff dimension from both continuous and discrete points of view. We show that it is possible to generalize the Hausdorff dimension in the context of Euclidean spaces equipped with their natural fractal structure. To do this, we provide three definitions of fractal dimension for a fractal structure and study their relationships and mathematical properties. One of these definitions is in terms of finite coverings by elements of the fractal structure. We prove that this dimension is equal to the Hausdorff dimension for compact subsets of Euclidean spaces. This may be the key for the creation of new algorithms to calculate the Hausdorff dimension of these kinds of space.

A survey on fractal dimension for fractal structures

Along the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Carathéodory, Sierpi´nski, and Besicovitch, to quote some of them. They were some of the pioneers exploring objects having self-similar patterns or showing anomalous properties with respect to standard analytic attributes. Among the new tools developed to deal with this kind of objects, fractal dimension has become one of the most applied since it constitutes a single quantity which throws useful information concerning fractal patterns on sets. Several years later, fractal structures were introduced from Asymmetric Topology to characterize self-similar symbolic spaces. Our aim in this survey is to collect several results involving distinct definitions of fractal dimension we proved jointly with Prof.M.A. Sánchez-Granero in the context of fractal structures.

Fractal dimension for fractal structures

The main goal of this paper is to provide a generalized definition of fractal dimension for any space equipped with a fractal structure. This novel theory generalizes the classical box-counting dimension theory on the more general context of GF-spaces. In this way, if we select the so-called natural fractal structure on any Euclidean space, then the box-counting dimension becomes just a particular case. This idea allows to consider a wide range of fractal structures to calculate the effective fractal dimension for any subset of this space. Unlike it happens with the classical theory of fractal dimension, the new definitions we provide may be calculated in contexts where the box-counting one can have no sense or cannot be calculated. Nevertheless, the new models can be computed for any space admitting a fractal structure, just as easy as the box-counting dimension in empirical applications.

Theoretical properties of fractal dimensions for fractal structures

Hausdorff dimension, which is the oldest and also the most accurate model for fractal dimension, constitutes the main reference for any fractal dimension definition that could be provided. In fact, its definition is quite general, and is based on a measure, which makes the Hausdorff model pretty desirable from a theoretical point of view. On the other hand, it turns out that fractal structures provide a perfect context where a new definition of fractal dimension could be proposed. Further, it has been already shown that both Hausdorff and box dimensions can be generalized by some definitions of fractal dimension formulated in terms of fractal structures. Given this, and being mirrored in some of the properties satisfied by Hausdorff dimension, in this paper we explore which ones are satisfied by the fractal dimension definitions for a fractal structure, that are explored along this work.

The exact Hausdorff dimension for a class of fractal functions

Journal of Mathematical Analysis and Applications, 1992

  1. have considered a class of real functions whose graphs are, in general, fractal sets in R2. In this paper we give sufficient conditions for the fractal and Hausdorff dimensions to be equal for a certain subclass of fractal functions. The sets we consider are examples of self-affine fractals generated using iterated function systems (i.f.s.). Falconer [S] has shown that for almost all such sets the fractal and Hausdorff dimensions are equal and he gives a formula for the common dimension, due originally to Moran [S]. These results, however, give no information about individual fractal functions, In this paper we extend Moran's original method and show that if certain conditions on the i.f.s. are satisfied, then the two dimensions are equal. Kono [ 111 and Bedford [ 121 considered special cases of the subclass of fractal functions that we will introduce. Bedford and Urbanski [13] use a nonlinear setting to present conditions for the equality of Hausdorff and fractal dimension. However, their criteria are based on measure-theoretic characterizations and the use of the concept of generalized pressure. Our criterion on the other hand is based on the underlying geometry of the attractor and is easier to verify. We will show this on two specific examples which are more general than the self-ahine functions presented in [13].

Counterexamples in theory of fractal dimension for fractal structures

Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. For this purpose, it is always used the box dimension, since it is easy to calculate, though the Hausdorff dimension, which is the oldest and also the most accurate fractal dimension, presents the best analytical properties. Additionally, fractal structures provide an appropriate topological context where new models of fractal dimension for a fractal structure could be developed in order to generalize the classical models of fractal dimension. In this survey, we gather different definitions and counterexamples regarding these new models of fractal dimension in order to show the reader how they behave mathematically with respect to the classical models, and also to point out which features of such models can be exploited to powerful effect in applications.

How to calculate the Hausdorff dimension using fractal structures

In this paper, we provide the first known overall algorithm to calculate the Hausdorff dimension of any compact Euclidean subset. This novel approach is based on both a new discrete model of fractal dimension for a fractal structure which considers finite coverings and a theoretical result that the authors contributed previously in [14]. This new procedure combines fractal techniques with tools from Machine Learning Theory. In particular, we use a support vector machine to decide the value of the Hausdorff dimension. In addition to that, we artifi- cially generate a wide collection of examples that allows us to train our algorithm and to test its performance by external proof. Some analyses about the accuracy of this approach are also provided.

Generalizations of the Hausdorff dimension of fractal measures

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.

Dimensions of Points in Self-Similar Fractals

SIAM Journal on Computing, 2008

Self-similar fractals arise as the unique attractors of iterated function systems (IFSs) consisting of finitely many contracting similarities satisfying an open set condition. Each point x in such a fractal F arising from an IFS S is naturally regarded as the "outcome" of an infinite coding sequence T (which need not be unique) over the alphabet Σ k = {0,. .. , k − 1}, where k is the number of contracting similarities in S. A classical theorem of Moran (1946) and Falconer (1989) states that the Hausdorff and packing dimensions of a self-similar fractal coincide with its similarity dimension, which depends only on the contraction ratios of the similarities. The theory of computing has recently been used to provide a meaningful notion of the dimensions of individual points in Euclidean space. In this paper, we use (and extend) this theory to analyze the dimensions of individual points in fractals that are computably self-similar, meaning that they are unique attractors of IFSs that are computable and satisfy the open set condition. Our main theorem states that, if F ⊆ R n is any computably self-similar fractal and S is any IFS testifying to this fact, then the dimension identities dim(x) = sdim(F) dim π S (T)