An estimate of the Hausdorff dimension of a weak self-similar set

We propose an estimate to quantitatively evaluate the Hausdorff dimension of a self-similar set based on a system of weak contractions each of whose contraction coefficient is not a constant but a function of a parameter. Using the estimate, we investigate the topological structures specific to this weak self-similar set.     

Hausdorff measure of uniform self-similar fractals

Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on R ^d . Denote by D the Hausdorff dimension, by H ^D (E) the Hausdorff measure and by diam(E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and H ^D (E) = diam(E) ^D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover, we present a parametrised UIFS, where ω depends on c and show the inequatily H ^D (E) < diam(E) ^D , if c is small enough.     

Symmetry and enumeration of self-similar fractals

A general method is presented for enumerating the distinct self-similarsets that arise as attractors of certain families of iteratedfunction systems, using group theory to analyse the symmetriesof the attractors. A range of examples are investigated in thisway.     

On the open set condition for self-similar fractals

For self-similar sets, the existence of a feasible open set is a natural separation condition which expresses geometric as well as measure-theoretic properties. We give a constructive approach by defining a central open set and characterizing those points which do not belong to feasible open sets.

     

Dimension spectra of random subfractals of self-similar fractals

The dimension of a point x   in Euclidean space (meaning the constructive Hausdorff dimension of the singleton set {x}$\{x\}$) is the algorithmic information density of x  . Roughly speaking, this is the least real number dim(x)$\dim (x)$ such that r×dim(x)$r\times \dim (x)$ bits suffice to specify x   on a general-purpose computer with arbitrarily high precision 2^−r$2^{-r}$. The dimension spectrum of a set X   in Euclidean space is the subset of [0,n]$[0,n]$ consisting of the dimensions of all points in X.     

The distributional dimension of fractals

In the book [1] H.Triebel introduces the distributional dimension of fractals in an analytical form and proves that： for Г as a non-empty set in R^n with Lebesgue measure ｜Г｜ = 0, one has dimH Г = dimD Г, where dimD Г and dimH Г are the Hausdorff dimension and distributional dimension, respectively. Thus we might say that the distributional dimension is an analytical definition for Hausdorff dimension. Therefore we can study Hausdorff dimension through the distributional dimension analytically. By discussing the distributional dimension, this paper intends to set up a criterion for estimating the upper and lower bounds of Hausdorff dimension analytically. Examples illustrating the criterion are included in the end.     

Quantization dimension for infinite self-similar probabilities

The quantization dimension function for a probability measure induced by a set of infinite contractive similarity mappings and a given probability vector is determined. A relationship between the quantization dimension function and the temperature function of the thermodynamic formalism arising in multifractal analysis is established. The result in this paper is an infinite extension of Graf and Luschgy [S. Graf, H. Luschgy, The quantization dimension of self-similar probabilities, Math. Nachr. 241 (2002) 103-109].     

Existence of self-similar energies on finitely ramified fractals

A self-similar energy on finitely ramified fractals can be constructed starting from an eigenform, i.e., an eigenvector of a special operator defined on the fractal. In this paper, we prove two existence results for regular eigenforms that consequently are existence results for self-similar energies on finitely ramified fractals. The first result proves the existence of a regular eigenform for suitable weights on fractals, assuming only that the boundary cells are separated and the union of the interior cells is connected. This result improves previous results and applies to many finitely ramified fractals usually considered. The second result proves the existence of a regular eigenform in the general case of finitely ramified fractals in a setting similar to that of P.C.F. self-similar sets considered, for example, by R. Strichartz in [11]. In this general case, however, the eigenform is not necessarily on the given structure, but is rather on only a suitable power of it. Nevertheless, as the fractal generated is the same as the original fractal, the result provides a regular self-similar energy on the given fractal.     

Topological properties of self-similar fractals with one parameter

In this paper, we study two classes of planar self-similar fractals $T_{\epsilon }$ with a shifting parameter ε. The first one is a class of self-similar tiles by shifting x-coordinates of some digits. We give a detailed discussion on the disk-likeness (i.e., the property of being a topological disk) in terms of ε. We also prove that $T_{\epsilon }$ determines a quasi-periodic tiling if and only if ε is rational. The second one is a class of self-similar sets by shifting diagonal digits. We give a necessary and sufficient condition for $T_{\epsilon }$ to be connected.     

Beurling dimension and self-similar measures

In this paper the authors study the Beurling dimension of Bessel sets and frame spectra of some self-similar measures on ${\mathbb{R}}^d$ and obtain their exact upper bound of the dimensions, which is the same given by Dutkay et al. (2011) [8]. The upper bound is attained in usual cases and some examples are given to explain our theory.     

Quantization dimension of random self-similar measures

In this paper, we study the quantization dimension of a random self-similar measure μ supported on the random self-similar set K(ω). We establish a relationship between the quantization dimension of μ and its distribution. At last we give a simple example to show that how to use the formula of the quantization dimension.     

Embedded continued fractals and their Hausdorff dimension

A continued fractal is a curve which is associated to a real number[0, 1]. Properties of the continued fraction expansion of appear as geometrical properties ofQ . It is shown how number theoretic properties of affect topological and geometric properties ofQ such as existence, continuity, Hausdorff dimension, and embeddedness.Communicated by Michael F. Barnsley.     

Hausdorff dimension of a class of non-self-similar fractals

Translated from Matematicheskie Zametki, Vol. 50, No. 5, pp. 129–133, November, 1991.     

Quantization dimension and temperature function for recurrent self-similar measures

We determine the quantization dimension function for the image measure supported on a recurrent self-similar set of an ergodic Markov measure, and its relationship with the temperature function of the thermodynamic formalism arising in multifractal analysis is established.     

On-diagonal oscillation of the heat kernels on post-critically finite self-similar fractals

For the canonical heat kernels p _t (x, y) associated with Dirichlet forms on post-critically finite self-similar fractals, e.g. the transition densities (heat kernels) of Brownian motion on affine nested fractals, the non-existence of the limit ${\lim_{t\downarrow 0}t^{d_{s}/2}p_{t}(x,x)}$ is established for a “generic” (in particular, almost every) point x, where d _s denotes the spectral dimension. Furthermore the same is proved for any point x in the case of the d-dimensional standard Sierpinski gasket with d ≥ 2 and the N-polygasket with N ≥ 3 odd, e.g. the pentagasket (N = 5) and the heptagasket (N = 7).     

Hausdorff dimension of self-similar sets with overlaps

We provide a simple formula to compute the Hausdorff dimension of the attractor of an overlapping iterated function system of contractive similarities satisfying a certain collection of assumptions. This formula is obtained by associating a non-overlapping infinite iterated function system to an iterated function system satisfying our assumptions and using the results of Moran to compute the Hausdorff dimension of the attractor of this infinite iterated function system, thus showing that the Hausforff dimension of the attractor of this infinite iterated function system agrees with that of the attractor of the original iterated function system. Our methods are applicable to some iterated function systems that do not satisfy the finite type condition recently introduced by Ngai and Wang.      

Divergence points of self-similar measures and packing dimension

Let μ be a self-similar measure in Rd. A point x∈Rd for which the limit does not exist is called a divergence point. Very recently there has been an enormous interest in investigating the fractal structure of various sets of divergence points. However, all previous work has focused exclusively on the study of the Hausdorff dimension of sets of divergence points and nothing is known about the packing dimension of sets of divergence points. In this paper we will give a systematic and detailed account of the problem of determining the packing dimensions of sets of divergence points of self-similar measures. An interesting and surprising consequence of our results is that, except for certain trivial cases, many natural sets of divergence points have distinct Hausdorff and packing dimensions.     

Hausdorff dimension in a family of self-similar groups

For every prime p and every monic polynomial f, invertible over p, we define a group G _{p, f} of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group . We show that the constructed groups are self-similar, regular branch groups. This enables us to calculate the Hausdorff dimension of their closures, providing concrete examples (not using random methods) of topologically finitely generated closed subgroups of the group of p-adic automorphisms with Hausdorff dimension arbitrarily close to 1. We provide a characterization of finitely constrained groups in terms of the branching property, and as a corollary conclude that all defined groups are finitely constrained. In addition, we show that all infinite, finitely constrained groups of p-adic automorphisms have positive and rational Hausdorff dimension and we provide a general formula for Hausdorff dimension of finitely constrained groups. Further “finiteness” properties are also discussed (amenability, torsion and intermediate growth). Partially supported by NSF grant DMS-0600975.     

平面上一类自相似集的共形维数

本文证明了平面上一类自相似集的共形维数为1.此外还证明了这些自相似集与任何Hausdorff维数为1的度量空间都不是拟对称等价的.这表明,对于这些自相似集而言,共形维数定义中的下确界不能达到.