Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces (original) (raw)

The effect of projections on fractal sets and measures in Banach spaces

Ergodic Theory and Dynamical Systems, 2006

We study the extent to which the Hausdorff dimension of a compact subset of an infinite-dimensional Banach space is affected by a typical mapping into a finitedimensional space. It is possible that the dimension drops under all such mappings, but the amount by which it typically drops is controlled by the 'thickness exponent' of the set, which was defined by Hunt and Kaloshin (Nonlinearity 12 (1999), 1263-1275). More precisely, let X be a compact subset of a Banach space B with thickness exponent τ and Hausdorff dimension d. Let M be any subspace of the (locally) Lipschitz functions from B to R m that contains the space of bounded linear functions. We prove that for almost every (in the sense of prevalence) function f ∈ M, the Hausdorff dimension of f (X) is at least min{m, d/(1 + τ)}. We also prove an analogous result for a certain part of the dimension spectra of Borel probability measures supported on X. The factor 1/(1 + τ) can be improved to 1/(1 + τ/2) if B is a Hilbert space. Since dimension cannot increase under a (locally) Lipschitz function, these theorems become dimension preservation results when τ = 0. We conjecture that many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero. We also discuss the sharpness of our results in the case τ > 0.

How projections affect the dimension spectrum of fractal measures

1997

We introduce a new potential-theoretic definition of the dimension spectrum D q of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if 1 < q 2 and µ is a Borel probability measure with compact support in R n , then under almost every linear transformation from R n to R m , the q-dimension of the image of µ is min(m, D q (µ)); in particular, the q-dimension of µ is preserved provided m D q (µ). We also present results on the preservation of information dimension D 1 and pointwise dimension. Finally, for 0 q < 1 and q > 2 we give examples for which D q is not preserved by any linear transformation into R m . All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.

Fractal dimension for fractal structures: A Hausdorff approach

This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a suitable discretization of the Hausdorff theory of fractal dimension. We also find some connections between our definition and the classical ones and also with fractal dimensions I & II (see M.A. Sánchez-Granero and M. Fernández-Martínez (2010) [16]). Therefore, we generalize them and obtain an easy method in order to calculate the fractal dimension of strict self-similar sets which are not required to verify the open set condition.

On Hausdorff Dimension of Random Fractals

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.

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.

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].

Fractal dimensions for inclusion hyperspaces and non-additive measures

Matematychni Studii

Analogues of Hausdorff dimension, upper and lower box dimensions for the inclusion hyperspaces and non-additive regular measures (capacities) on metric compacta are introduced. Their relations to the respective dimensions of sets and additive measures are investigated. Methods for finding and estimating fractal dimensions of self-similar inclusion hyperspaces and self-similar non-additive measures are presented.

Ergodic fractal measures and dimension conservation

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.

Projective properties of fractal sets

Chaos, Solitons & Fractals, 2008

In this paper, it is shown that a bound on the box dimension of a set in 3D can be established by estimating the box dimension of the discrete image of the projected set i.e. from an image in 2D. It is possible to impose limits on the Hausdorff dimension of the set by estimating the box dimension of the projected set. Furthermore, it is shown how a realistic X-ray projection can be viewed as equivalent to an ideal projection when regarding estimates of fractal dimensions.

Small sets of reals through the prism of fractal dimensions

A separable metric space X is an H-null set if any uniformly continuous image of X has Hausdorff dimension zero. H-null, P − → -null and P-null sets are defined likewise, with other fractal dimensions in place of Hausdorff dimension. We investigate these sets and show that in 2 ω they coincide, respectively, with strongly null, meager-additive, (T ) and null-additive sets. Some consequences: A subset of 2 ω is meager-additive if and only if it is E-additive; if f : 2 ω → 2 ω is continuous and X is meager-additive, then so is f (X), and likewise for null-additive and (T ) sets.