SELF-SIMILARITY OF SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS WITH RESPECT TO A FRACTAL MEASURE

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950014
Author(s):  
H. KUNZE ◽  
D. LA TORRE ◽  
F. MENDIVIL ◽  
E. R. VRSCAY
Keyword(s):  
Integral Equation ◽  
Iterated Function System ◽  
Differential Forms ◽  
Self Similarity ◽  
Similar Measure ◽  
Picard Operator ◽  
Convergence Results ◽  
Iterated Function ◽  
Classical Integral ◽  
Self Similar

In this paper, we study solutions of a variation of a classical integral equation (based on the Picard operator) in which Lebesgue measure is replaced by a self-similar measure [Formula: see text]. Our main interest is in the fractal nature of the solutions and we use Iterated Function System (IFS) tools to investigate the behavior and self-similarity of these solutions. Both the integral and differential forms of the equation are discussed since each brings useful insights. Several convergence results are provided along with illustrative examples that show the applications of the theory when the underlying fractal object is the celebrated Cantor set. Additionally we show that the solution to our integral equation inherits self-similarity from the defining measure [Formula: see text].

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 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.

SPECTRALITY AND NON-SPECTRALITY OF PLANAR SELF-SIMILAR MEASURES WITH FOUR-ELEMENT DIGIT SETS

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050130
Author(s):  
SI CHEN ◽  
MIN-WEI TANG
Keyword(s):  
Orthonormal Basis ◽  
Iterated Function System ◽  
The Self ◽  
Function System ◽  
Similar Measure ◽  
Unit Matrix ◽  
Iterated Function ◽  
Self Similar

Let [Formula: see text] be the unit matrix and [Formula: see text]. In this paper, we consider the self-similar measure [Formula: see text] on [Formula: see text] generated by the iterated function system [Formula: see text] where [Formula: see text]. We prove that there exists [Formula: see text] such that [Formula: see text] is an orthonormal basis for [Formula: see text] if and only if [Formula: see text] for some integer [Formula: see text].

scholarly journals Per-tiling, rep-tiling and Penrose tiling: a notion to edge cordial and cordial labeling

2015 ◽  
Vol 4 (2) ◽  
pp. 308
Author(s):  
Sathakathulla A. A.
Keyword(s):  
Iterated Function System ◽  
Function System ◽  
Penrose Tiling ◽  
Entire Space ◽  
Self Similarity ◽  
Geometric Figure ◽  
Iterated Function

<p>A fractal is a complex geometric figure that continues to display self-similarity when viewed on all scales. A simple, yet unifying method is provided for the construction of tiling by tiles obtained from the attractor of an iterated function system (IFS). This tiling can be used to extend a fractal transformation on the entire space upon which the IFS acts. There are many in this family of tiling fractals curves but for my study, I have considered each one from the above family of tiling fractals. These fractals have been considered as a graph and the same has been viewed under the scope of cordial and edge cordial labeling to apply this concept for further study.</p>

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 &gt; 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dimHKj − ϵ &lt; 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.

ECCENTRIC DISTANCE SUM OF SIERPIŃSKI GASKET AND SIERPIŃSKI NETWORK

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950016 ◽  
Author(s):  
JIN CHEN ◽  
LONG HE ◽  
QIN WANG
Keyword(s):  
Asymptotic Formula ◽  
Complex Networks ◽  
Sierpinski Gasket ◽  
The Self ◽  
Self Similarity ◽  
Sierpiński Gasket ◽  
Similar Measure ◽  
Self Similar ◽  
Eccentric Distance

The eccentric distance sum is concerned with complex networks. To obtain the asymptotic formula of eccentric distance sum on growing Sierpiński networks, we study some nonlinear integral in terms of self-similar measure on the Sierpiński gasket and use the self-similarity of distance and measure to obtain the exact value of this integral.

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.

Generalized vector multiplicative cascades

2001 ◽  
Vol 33 (4) ◽  
pp. 874-895 ◽  
Author(s):  
Julien Barral
Keyword(s):  
Banach Algebras ◽  
Branching Random Walk ◽  
Similar Measure ◽  
Algebra Of Functions ◽  
Convergence Results ◽  
Abstract Approach ◽  
Self Similar ◽  
Convergence Almost Surely ◽  
Varying Environment ◽  
Multiplicative Processes

We define the extension of the so-called ‘martingales in the branching random walk’ in R or C to some Banach algebras B of infinite dimension and give conditions for their convergence, almost surely and in the Lp norm. This abstract approach gives conditions for the simultaneous convergence of uncountable families of such martingales constructed simultaneously in C, the idea being to consider such a family as a function-valued martingale in a Banach algebra of functions. The approach is an alternative to those of Biggins (1989), (1992) and Barral (2000), and it applies to a class of families to which the previous approach did not. We also give a result on the continuity of these multiplicative processes. Our results extend to a varying environment version of the usual construction: instead of attaching i.i.d. copies of a given random vector to the nodes of the tree ∪n≥0N+n, the distribution of the vector depends on the node in the multiplicative cascade. In this context, when B=R and in the nonnegative case, we generalize the measure on the boundary of the tree usually related to the construction; then we evaluate the dimension of this nonstatistically self-similar measure. In the self-similar case, our convergence results make it possible to simultaneously define uncountable families of such measures, and then to estimate their dimension simultaneously.

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