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.

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.

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 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.

scholarly journals MULTIPLE REPRESENTATIONS OF REAL NUMBERS ON SELF-SIMILAR SETS WITH OVERLAPS

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950051 ◽  
Author(s):  
KAN JIANG ◽  
XIAOMIN REN ◽  
JIALI ZHU ◽  
LI TIAN
Keyword(s):  
Convex Hull ◽  
Iterated Function System ◽  
Multiple Representations ◽  
Function System ◽  
Real Numbers ◽  
Iterated Function ◽  
Self Similar

Let [Formula: see text] be the attractor of the following iterated function system (IFS) [Formula: see text] where [Formula: see text] and [Formula: see text] is the convex hull of [Formula: see text]. The main results of this paper are as follows: [Formula: see text] if and only if [Formula: see text] where [Formula: see text]. If [Formula: see text], then [Formula: see text]As a consequence, we prove that the following conditions are equivalent:(1) For any [Formula: see text], there are some [Formula: see text] such that [Formula: see text].(2) For any [Formula: see text], there are some [Formula: see text] such that [Formula: see text](3) [Formula: see text].

Hidden Variable Bivariate Fractal Interpolation Surfaces

Fractals ◽  
2003 ◽  
Vol 11 (03) ◽  
pp. 277-288 ◽  
Author(s):  
A. K. B. Chand ◽  
G. P. Kapoor
Keyword(s):  
Iterated Function System ◽  
Degree Of Freedom ◽  
Function System ◽  
Fractal Interpolation ◽  
Fractal Surface ◽  
Hidden Variable ◽  
Iterated Function ◽  
Self Similar ◽  
Vector Valued

We construct hidden variable bivariate fractal interpolation surfaces (FIS). The vector valued iterated function system (IFS) is constructed in ℝ4 and its projection in ℝ3 is taken. The extra degree of freedom coming from ℝ4 provides hidden variable, which is an important factor for flexibility and diversity in the interpolated surface. In the present paper, we construct an IFS that generates both self-similar and non-self-similar FIS simultaneously and show that the hidden variable fractal surface may be self-similar under certain conditions.

THE ENERGY CASCADE OF TURBULENCE WITH QUASI-SELF-SIMILARITY

2009 ◽  
Vol 23 (03) ◽  
pp. 513-516 ◽  
Author(s):  
HAO ZHU ◽  
KEMING CHENG
Keyword(s):  
Energy Spectrum ◽  
Turbulent Flows ◽  
Iterated Function System ◽  
Three Dimensional ◽  
Energy Cascade ◽  
Function System ◽  
Self Similarity ◽  
Iterated Function ◽  
Self Similar ◽  
Nonlinear Disturbance

In this article, we investigate the energy cascade of three-dimensional turbulent flows, in which the break-up process of eddy is quasi-self-similar. Mathematically this kind of turbulence with quasi-self-similar structure eddies can be regarded as cookie-cutter system, and can be generated by self-similar iterated function system (IFS) with added nonlinear disturbance. Using Bowen's result, we can calculate the exponent of dissipative correlated function, dissipated velocity, energy spectrum supported on cookie-cutter system. The present results show that the β-model is feasible for this kind of quasi-self-similar turbulence.

On the group of isometries of planar IFS fractals*

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 445-469
Author(s):  
Qi-Rong Deng ◽  
Yong-Hua Yao
Keyword(s):  
Iterated Function System ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Function System ◽  
Finite Case ◽  
Iterated Function ◽  
Self Similar ◽  
Group Of Isometries ◽  
Necessary And Sufficient ◽  
Affine Set

Abstract For any iterated function system (IFS) on R 2 , let K be the attractor. Consider the group of all isometries on K. If K is a self-similar or self-affine set, it is proven that the group must be finite. If K is a bi-Lipschitz IFS fractal, the necessary and sufficient conditions for the infiniteness (or finiteness) of the group are given. For the finite case, the computation of the size of the group is also discussed.

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.

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