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].

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 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.

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.

scholarly journals Approximate convex hull of affine iterated function system attractors

2012 ◽  
Vol 45 (11) ◽  
pp. 1444-1451 ◽  
Author(s):  
Anton Mishkinis ◽  
Christian Gentil ◽  
Sandrine Lanquetin ◽  
Dmitry Sokolov
Keyword(s):  
Convex Hull ◽  
Iterated Function System ◽  
Function System ◽  
Iterated Function

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.

scholarly journals On fractal properties of Weierstrass-type functions

2019 ◽  
Vol 12 (2) ◽  
Author(s):  
Claire David
Keyword(s):  
Real Number ◽  
Iterated Function System ◽  
Function System ◽  
Box Dimension ◽  
Intrinsic Properties ◽  
Weierstrass Function ◽  
Real Numbers ◽  
New Class ◽  
Fractal Properties ◽  
Iterated Function

In the sequel, starting from the classical Weierstrass function defined, for any real number $x$, by $ {\mathcal W}(x)=\displaystyle \sum_{n=0}^{+\infty} \lambda^n\,\cos \left(2\, \pi\,N_b^n\,x \right)$, where $\lambda$ and $N_b$ are two real numbers such that~\mbox{$0 <\lambda<1$},~\mbox{$ N_b\,\in\,\N$} and $ \lambda\,N_b > 1 $, we highlight intrinsic properties of curious maps which happen to constitute a new class of iterated function system. Those properties are all the more interesting, in so far as they can be directly linked to the computation of the box dimension of the curve, and to the proof of the non-differentiabilty of Weierstrass type functions.

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 Dimension spectrum of infinite self-affine iterated function systems

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Natalia Jurga
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Real Numbers ◽  
Dimension Spectrum ◽  
Iterated Function ◽  
Isolated Points ◽  
First Time

AbstractGiven an infinite iterated function system (IFS) $${\mathcal {F}}$$ F , we define its dimension spectrum $$D({\mathcal {F}})$$ D ( F ) to be the set of real numbers which can be realised as the dimension of some subsystem of $${\mathcal {F}}$$ F . In the case where $${\mathcal {F}}$$ F is a conformal IFS, the properties of the dimension spectrum have been studied by several authors. In this paper we investigate for the first time the properties of the dimension spectrum when $${\mathcal {F}}$$ F is a non-conformal IFS. In particular, unlike dimension spectra of conformal IFS which are always compact and perfect (by a result of Chousionis, Leykekhman and Urbański, Selecta 2019), we construct examples to show that $$D({\mathcal {F}})$$ D ( F ) need not be compact and may contain isolated points.

scholarly journals PEMANFAATAN ITERATED FUNCTION SYSTEM (IFS) UNTUK MEMBANGKITKAN MOTIF ANYAMAN UKURAN n x n

2021 ◽  
Vol 21 (1) ◽  
pp. 25
Author(s):  
Ingka Maris ◽  
Kosala Dwidja Purnomo ◽  
Bagus Juliyanto
Keyword(s):  
Computer Technology ◽  
Iterated Function System ◽  
Grid Size ◽  
Function System ◽  
Two Dimensional ◽  
Basic Pattern ◽  
Iterated Function ◽  
Calculation Process ◽  
Self Similar ◽  
Mathematical Calculation

Woven is one of art thats very close to life. Woven has a pattern consisting of  two-dimensional (2D) and has a basic pattern. Along with the development times, technology is also growing including computer technology. Computer can be used for mathematical calculation process, one of them is fractal. Fractal Sierpinski carpet is formed from a square that use Iterated Function System (IFS) method. This method is exact self-similar resulting in the same fractal with the original constituent object. The writer want to get woven pattern using computer technology, that is GUI application in Matlab that utilizes the IFS method on fractal. Woven patterns formed from woven that have a grid size of  n x n and are given a few iterations. So, that it can make it easier for craftsmen to make woven pattern that are interesting and varied.

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