SYMBOLIC AND GEOMETRIC LOCAL DIMENSIONS OF SELF-AFFINE MULTIFRACTAL SIERPINSKI SPONGES IN ℝd

2007 ◽  
Vol 07 (01) ◽  
pp. 37-51 ◽  
Author(s):  
L. OLSEN
Keyword(s):  
Multifractal Analysis ◽  
Multifractal Spectrum ◽  
Separation Condition ◽  
Open Set Condition ◽  
Local Scaling ◽  
Open Set ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Sierpinski Sponges

In this paper we study the multifractal structure of a certain class of self-affine measures known as self-affine multifractal Sierpinski sponges. Multifractal analysis studies the local scaling behaviour of measures. In particular, multifractal analysis studies the so-called local dimension and the multifractal spectrum of measures. The multifractal structure of self-similar measures satisfying the Open Set Condition is by now well understood. However, the multifractal structure of self-affine multifractal Sierpinski sponges is significantly less well understood. The local dimensions and the multifractal spectrum of self-affine multifractal Sierpinski sponges are only known provided a very restrictive separation condition, known as the Very Strong Separation Condition (VSSC), is satisfied. In this paper we investigate the multifractal structure of general self-affine multifractal Sierpinski sponges without assuming any additional conditions (and, in particular, without assuming the VSSC).

Continuity of packing measure functions of self-similar iterated function systems

2011 ◽  
Vol 32 (3) ◽  
pp. 1101-1115 ◽  
Author(s):  
HUA QIU
Keyword(s):  
Iterated Function Systems ◽  
Packing Measure ◽  
Open Set Condition ◽  
Open Set ◽  
Density Theorems ◽  
Iterated Function ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Image Position

AbstractIn this paper, we focus on the packing measures of self-similar sets. Let K be a self-similar set whose Hausdorff dimension and packing dimension equal s. We state that if K satisfies the strong open set condition with an open set 𝒪, then for each open ball B(x,r)⊂𝒪 centred in K, where 𝒫s denotes the s-dimensional packing measure. We use this inequality to obtain some precise density theorems for the packing measures of self-similar sets. These theorems can be used to compute the exact value of the s-dimensional packing measure 𝒫s (K) of K. Moreover, by using the above results, we show the continuity of the packing measure function of the attractors on the space of self-similar iterated function systems satisfying the strong separation condition. This result gives a complete answer to a question posed by Olsen in [15].

GAP SEQUENCES OF GRAPH-DIRECTED SETS

Fractals ◽  
2016 ◽  
Vol 24 (03) ◽  
pp. 1650036
Author(s):  
JUAN DENG ◽  
LIFENG XI
Keyword(s):  
Separation Condition ◽  
Interesting Application ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Directed Sets ◽  
Gap Sequences

This paper studies the gap sequences of graph-directed sets satisfying the strong separation condition. An interesting application is to investigate the gap sequences of self-similar sets with overlaps.

scholarly journals Dimension approximation of attractors of graph directed IFSs by self-similar sets

2018 ◽  
Vol 167 (01) ◽  
pp. 193-207 ◽  
Author(s):  
ÁBEL FARKAS
Keyword(s):  
Iterated Function System ◽  
Black Box ◽  
Function System ◽  
Separation Condition ◽  
Iterated Function ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Distance Sets

AbstractWe show that for the attractor (K1, . . ., Kq) of a graph directed iterated function system, for each 1 ⩽ j ⩽ q and ϵ > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dimHKj − ϵ < dimHK. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property as a ‘black box’ we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets.

Research and Proof of Decidability on Self-Similar Fractals

2014 ◽  
Vol 511-512 ◽  
pp. 1185-1188
Author(s):  
Min Jin
Keyword(s):  
Hausdorff Dimension ◽  
Separation Condition ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Similarity Dimension

Some undecidability on self-affine fractals have been supported. In this paper, we research on the decidability for self-similar fractal of Dubes type. In fact, we prove that the following problems are decidable to test if the Hausdorff dimension of a given Dubes self-similar set is equal to its similarity dimension, and to test if a given Dubes self-similar set satisfies the strong separation condition.

Gibbs measures on self-affine Sierpiński carpets and their singularity spectrum

2007 ◽  
Vol 27 (5) ◽  
pp. 1419-1443 ◽  
Author(s):  
JULIEN BARRAL ◽  
MOUNIR MENSI
Keyword(s):  
Multifractal Analysis ◽  
Geometrical Structure ◽  
Gibbs Measures ◽  
Separation Condition ◽  
Singularity Spectrum ◽  
Strong Separation ◽  
Probability Spaces ◽  
Strong Separation Condition ◽  
Multifractal Nature ◽  
Sierpinski Carpets

AbstractWe consider a class of Gibbs measures on self-affine Sierpiński carpets and perform the multifractal analysis of its elements. These deterministic measures are Gibbs measures associated with bundle random dynamical systems defined on probability spaces whose geometrical structure plays a central role. A special subclass of these measures is the class of multinomial measures on Sierpiński carpets. Our result improves the already known result concerning the multifractal nature of the elements of this subclass by considerably weakening and in some cases even eliminating a strong separation condition of geometrical nature.

scholarly journals Computability of the packing measure of totally disconnected self-similar sets

2015 ◽  
Vol 36 (5) ◽  
pp. 1534-1556 ◽  
Author(s):  
MARTA LLORENTE ◽  
MANUEL MORÁN
Keyword(s):  
Sierpinski Gasket ◽  
Separation Condition ◽  
Packing Measure ◽  
Sierpiński Gasket ◽  
Additional Information ◽  
Strong Separation ◽  
Self Similar ◽  
Contraction Factor ◽  
Strong Separation Condition ◽  
Totally Disconnected

We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called Strong Separation Condition (SSC) and prove its convergence to the value of the packing measure. We also test the algorithm with examples that show both the accuracy of the algorithm for the most regular cases and the possibility of using the additional information provided by it to obtain formulas for the packing measure of certain self-similar sets. For example, we are able to obtain a formula for the packing measure of any Sierpinski gasket with contraction factor in the interval $(0,\frac{1}{3}]$.

scholarly journals Interval projections of self-similar sets

2018 ◽  
Vol 40 (1) ◽  
pp. 194-212
Author(s):  
ÁBEL FARKAS
Keyword(s):  
Separation Condition ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition

We show if $K$ is a self-similar $1$-set that either satisfies the strong separation condition or is defined via homotheties then there are at most finitely many lines through the origin such that the projection of $K$ onto them is an interval.

Open set condition for graph directed self-similar structure

2013 ◽  
Vol 276 (1-2) ◽  
pp. 243-260 ◽  
Author(s):  
Tian-jia Ni ◽  
Zhi-ying Wen
Keyword(s):  
Open Set Condition ◽  
Open Set ◽  
Self Similar

scholarly journals SPACE-FILLING CURVES OF SELF-SIMILAR SETS (III): SKELETONS

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050028
Author(s):  
HUI RAO ◽  
SHU-QIN ZHANG
Keyword(s):  
Finite Type ◽  
Type Condition ◽  
Space Filling ◽  
Open Set Condition ◽  
Substitution Rule ◽  
Space Filling Curves ◽  
Neighbor Graph ◽  
Open Set ◽  
Self Similar

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. In a previous paper by Dai and the authors [Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule, Nonlinearity 32(5) (2019) 1772–1809] it was shown that for all the connected self-similar sets with a skeleton satisfying the open set condition, space-filling curves can be constructed. In this paper, we give a criterion of existence of skeletons by using the so-called neighbor graph of a self-similar set. In particular, we show that a connected self-similar set satisfying the finite-type condition always possesses skeletons: an algorithm is obtained here.

Regularity of Hausdorff measure function for conformal dynamical systems

2016 ◽  
Vol 160 (3) ◽  
pp. 537-563 ◽  
Author(s):  
MARIUSZ URBAŃSKI ◽  
ANNA ZDUNIK
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Measure ◽  
Hölder Continuity ◽  
Iterated Function Systems ◽  
Separation Condition ◽  
Holder Continuity ◽  
Measure Function ◽  
Iterated Function ◽  
Strong Separation ◽  
Strong Separation Condition

AbstractWe deal with the question of continuity of numerical values of Hausdorff measures in parametrised families of linear (similarity) and conformal dynamical systems by developing the pioneering work of Lars Olsen and the work [SUZ]. We prove Hölder continuity of the function ascribing to a parameter the numerical value of the Hausdorff measure of either the corresponding limit set or the corresponding Julia set. We consider three cases. Firstly, we consider the case of parametrised families of conformal iterated function systems in $\mathbb{R}$k with k ⩾ 3. Secondly, we consider all linear iterated function systems consisting of similarities in $\mathbb{R}$k with k ⩾ 1. In either of these two cases, the strong separation condition is assumed. In the latter case the Hölder exponent obtained is equal to 1/2. Thirdly, we prove such Hölder continuity for analytic families of conformal expanding repellers in the complex plane $\mathbb{C}$. Furthermore, we prove the Hausdorff measure function to be piecewise real–analytic for families of naturally parametrised linear IFSs in $\mathbb{R}$ satisfying the strong separation condition. On the other hand, we also give an example of a family of linear IFSs in $\mathbb{R}$ for which this function is not even differentiable at some parameters.

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