Topology and separation of self-similar fractals in the plane

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.

Probability Theory and Related Fields, 1987

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)

2010

M. Lapidus and C. Pomerance (1990-1993) and K.J. Falconer (1995) proved that a self-similar fractal in $\mathbb{R}$ is Minkowski-measurable iff it is of non-lattice type. D. Gatzouras (1999) proved that a self-similar fractal in $\mathbb{R}^d$ is Minkowski measurable if it is of non-lattice type (though the actual computation of the content is intractable with his approach) and conjectured that it is not Minkowski measurable if it is of lattice type. Under mild conditions we prove this conjecture and in the non-lattice case we improve his result in the sense that we express the content of the fractal in terms of the residue of the associated $\zeta$-function at the Minkowski-dimension.

Topology and its Applications, 1999

Let K be a self-similar set in R n which has similarity dimension n and nonempty interior. In this paper it is shown that the topological boundary of K has Hausdorff dimension less than n. Examples are given to show that although the dimension of the boundary is strictly less than n, it may be arbitrarily close to n.

Journal of Contemporary Technology and Applied Engineering

In various mathematical sciences, sets and functions in topology have been extensively developed and exploited. Some novel separation axioms have been discovered through studying generalizations of closures duo to closed sets, ∧-sets, and V-sets. Self similar fractals have important role in some real life problems as physics and engineering. In this paper, the topology described by the family of-∧ and-V-sets in topological spaces is defined and studied in terms of ∧-Sets and ∨-Sets. Also, the topological space (◻, ∧) is defined and studied. Additionally, several features of these sets are presented, as well as some associated new separation axioms. Finally, we improved theorems 4.3 [5] and 6.5 [6] for Caldas et al. We approximate self similar fractals through graph theory to topological spaces. Some topological properties such as separation axioms are studied. Finally, the kernel of topological approximations of fractals are calculated in terms of their connecting points.

Fundamenta Mathematicae, 2002

The notion of NST domain and the closed related notion of ball condition, both topological in nature and quite useful within the theory of function spaces, are compared with each other (and with the older concept of porosity) and also with other notions of interest, like the concepts of d-set and of interior regular domain, which have a measure-theoretical nature. Also, after extending the idea of NST (not so terrible) to a larger class of sets, such property is studied in the context of anisotropic self-affine fractals.

Proceedings of the American Mathematical Society, 2008

For a very simple family of self-similar sets with two pieces we prove, using a technique of Solomyak, that the intersection of the pieces can be a Cantor set with any dimension in [0, 0.2] as well as a finite set of any cardinality 2 m . The main point is that the open set condition is fulfilled for all these examples.

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.

Bulletin of The London Mathematical Society, 1999

A deterministic self-similar Cantor set F defines a natural probability space, the elements of which are neighbourhoods of various sizes of all points in F. Considering random variables on this probability space, some porosity and density parameters are explicitly calculated.