INTERSECTIONS OF TRANSLATION OF A CLASS OF SIERPINSKI CARPETS

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850034
Author(s):  
JIAN LU ◽  
BO TAN ◽  
YURU ZOU
Keyword(s):  
Iterated Function System ◽  
The Self ◽  
Open Set Condition ◽  
Moran Set ◽  
Open Set ◽  
Iterated Function ◽  
Directed Set ◽  
Self Similar ◽  
Sierpinski Carpets ◽  
Intersection Formula

For [Formula: see text], a middle-[Formula: see text] Sierpinski carpet [Formula: see text] is defined as the self-similar set generated by the iterated function system (IFS) [Formula: see text], where [Formula: see text] is defined by [Formula: see text] Here, [Formula: see text]. In this paper, for [Formula: see text], we investigated the equivalent characterizations of the intersection [Formula: see text] being a generalized Moran set. Furthermore, under some conditions, we show that [Formula: see text] can be represented as a graph-directed set satisfying the open set condition (OSC), and then the Hausdorff dimension can be explicitly calculated.

scholarly journals Lattice self-similar sets on the real line are not Minkowski measurable

2018 ◽  
Vol 40 (1) ◽  
pp. 221-232
Author(s):  
SABRINA KOMBRINK ◽  
STEFFEN WINTER
Keyword(s):  
Real Line ◽  
Iterated Function System ◽  
Function System ◽  
Open Set Condition ◽  
The Real ◽  
Open Set ◽  
Puzzle Piece ◽  
Iterated Function ◽  
Self Similar ◽  
Special Classes

We show that any non-trivial self-similar subset of the real line that is invariant under a lattice iterated function system (IFS) satisfying the open set condition (OSC) is not Minkowski measurable. So far, this has only been known for special classes of such sets. Thus, we provide the last puzzle-piece in proving that under the OSC a non-trivial self-similar subset of the real line is Minkowski measurable if and only if it is invariant under a non-lattice IFS, a 25-year-old conjecture.

scholarly journals Inhomogeneous self-similar sets with overlaps

2017 ◽  
Vol 39 (1) ◽  
pp. 1-18 ◽  
Author(s):  
SIMON BAKER ◽  
JONATHAN M. FRASER ◽  
ANDRÁS MÁTHÉ
Keyword(s):  
Final Section ◽  
Iterated Function System ◽  
Box Dimension ◽  
Open Set Condition ◽  
Open Set ◽  
Bernoulli Convolutions ◽  
Iterated Function ◽  
Different Types ◽  
Self Similar ◽  
Weak Separation

It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and the condensation set. First, we prove that this ‘expected formula’ does not hold in general if there are overlaps in the construction. We demonstrate this via two different types of counterexample: the first is a family of overlapping inhomogeneous self-similar sets based upon Bernoulli convolutions; and the second applies in higher dimensions and makes use of a spectral gap property that holds for certain subgroups of $\text{SO}(d)$ for $d\geq 3$. We also obtain new upper bounds, derived using sumsets, for the upper box dimension of an inhomogeneous self-similar set which hold in general. Moreover, our counterexamples demonstrate that these bounds are optimal. In the final section we show that if the weak separation property is satisfied, that is, the overlaps are controllable, then the ‘expected formula’ does hold.

Empirical multifractal moment measures and moment scaling functions of self-similar multifractals

2002 ◽  
Vol 133 (3) ◽  
pp. 459-485 ◽  
Author(s):  
L. OLSEN
Keyword(s):  
Lipschitz Constant ◽  
The Self ◽  
Probability Vector ◽  
Periodic Functions ◽  
Open Set Condition ◽  
Similar Measure ◽  
Open Set ◽  
Moment Measures ◽  
Self Similar ◽  
Exact Rate Of Convergence

Let Si: ℝd → ℝd for i = 1, …, n be contracting similarities, and let (p1, …, pn) be a probability vector. Let K and μ be the self-similar set and the self-similar measure associated with (Si,pi)i. For q ∈ ℝ and r > 0, define the qth covering moment and the qth packing moment of μ by[formula here]where the infimum is taken over all r-spanning subsets E of K, and the supremum is taken over all r-separated subsets F of K. If the Open Set Condition (OSC) is satisfied then it is well known that[formula here]where β(q) is defined by [sum ]ipqirβi(q) = 1 (here ri denotes the Lipschitz constant of Si). Assuming the OSC, we determine the exact rate of convergence in (*): there exist multiplicatively periodic functions πq, Πq: (0,∞) → ℝ such that[formula here]where ε(r) → 0 as r[searr ]0. As an application of (**) we show that the empirical multi-fractal moment measures converges weakly:[formula here]where, for each positive r, Er is a (suitable) minimal r-spanning subset of K and Fr is a (suitable) maximal r-separated subset of K, and [Hscr ]q,β(q)μ and [Pscr ]q,β(q)μ are the multifractal Hausdorff measure and the multifractal packing measure, respectively.

scholarly journals The mutual singularity of the relative multifractal measures

2021 ◽  
Vol 8 (1) ◽  
pp. 18-26
Author(s):  
Zied Douzi ◽  
Bilel Selmi
Keyword(s):  
General Framework ◽  
The Self ◽  
Open Set Condition ◽  
Open Set ◽  
Self Similar ◽  
Bernoulli Measures ◽  
Multifractal Measures

Abstract M. Das proved that the relative multifractal measures are mutually singular for the self-similar measures satisfying the significantly weaker open set condition. The aim of this paper is to show that these measures are mutually singular in a more general framework. As examples, we apply our main results to quasi-Bernoulli measures.

scholarly journals The quasi-Assouad dimension of stochastically self-similar sets

2019 ◽  
Vol 150 (1) ◽  
pp. 261-275 ◽  
Author(s):  
Sascha Troscheit
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function Systems ◽  
Random Sets ◽  
Open Set Condition ◽  
Fractal Percolation ◽  
Open Set ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar

AbstractThe class of stochastically self-similar sets contains many famous examples of random sets, for example, Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and V -variable sets and the removal of overlap conditions.

Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps

2011 ◽  
Vol 63 (3) ◽  
pp. 648-688 ◽  
Author(s):  
Sze-Man Ngai
Keyword(s):  
Golden Ratio ◽  
Iterated Function Systems ◽  
Spectral Dimension ◽  
Open Set Condition ◽  
Bernoulli Convolution ◽  
One Dimensional ◽  
Open Set ◽  
Iterated Function ◽  
Self Similar ◽  
Set Up

AbstractWe set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

scholarly journals Hölder Parameterization of Iterated Function Systems and a Self-Aflne Phenomenon

2021 ◽  
Vol 9 (1) ◽  
pp. 90-119
Author(s):  
Matthew Badger ◽  
Vyron Vellis
Keyword(s):  
Metric Spaces ◽  
Optimal Parameter ◽  
Complete Metric Space ◽  
Iterated Function Systems ◽  
Open Set Condition ◽  
Interesting Phenomenon ◽  
Open Set ◽  
Iterated Function ◽  
Self Similar ◽  
Path Connected

Abstract We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/s)-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α > s. At the endpoint, α = s, a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/s)-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s-dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝ n is always at most the ambient dimension n, the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n.

The intersections of self-similar and self-affine sets with their perturbations under the weak separation condition

2016 ◽  
Vol 38 (4) ◽  
pp. 1353-1368 ◽  
Author(s):  
QI-RONG DENG ◽  
XIANG-YANG WANG
Keyword(s):  
Iterated Function System ◽  
The Self ◽  
Function System ◽  
Separation Condition ◽  
Weak Separation Condition ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Affine Set ◽  
Totally Disconnected

For a self-similar or self-affine iterated function system (IFS), let$\unicode[STIX]{x1D707}$be the self-similar or self-affine measure and$K$be the self-similar or self-affine set. Assume that the IFS satisfies the weak separation condition and$K$is totally disconnected; then, by using the technique of neighborhood decomposition, we prove that there is a neighborhood$\unicode[STIX]{x1D6FA}$of the identity map Id such that$\sup \{\unicode[STIX]{x1D707}(g(K)\cap K):g\in \unicode[STIX]{x1D6FA}\setminus \{\text{Id}\}\}<1$.

scholarly journals A strongly irreducible affine iterated function system with two invariant measures of maximal dimension

2020 ◽  
pp. 1-22
Author(s):  
IAN D. MORRIS ◽  
CAGRI SERT
Keyword(s):  
Invariant Measures ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Finite Union ◽  
Function System ◽  
Maximal Dimension ◽  
Similar Measure ◽  
Open Set ◽  
Iterated Function ◽  
Self Similar

Abstract A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb {R}^{d}$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor, which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of $\mathbb {R}^{d}$ . In this paper we give an example where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.

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