Iterated Function Systems Research Articles

Number theoretic considerations related to the scaling spectrum of self-similar measures with consecutive digits

Let p,b≥2 be two integers, and let D=p{0,1,…,b−1}. It is well known that the self-similar measure μpb,D generated by the iterated function system {τd(x)=(pb)−1(x+d)}d∈D is a spectral measure with a spectrum Λ(pb,C)=∑j=0finite(pb)jcj:cj∈C={0,1,…,b−1}.In this paper, we study a class of primitive and composite numbers and their related properties. And we also explore the simple prime numbers and the properties related to their order. Based on these, let r be a positive integer, we give some conditions on the distinct prime numbers t1,t2,…,tr such that the scaling set ∏i=1rtikiΛ(pb,C) is also a spectrum of μpb,D for all ki≥0.

Seismic trends in Indian plate: A study on epicentral characteristics by using a driven iterated function system

This paper studies the trends exhibited by earthquake epicenters around the Indian tectonic Plate with respect to time. For this study, a novel method based on a modified Iterated Function System has been used. The modified Chaos Game Representation plots generated for the study gave characteristic images implying that the occurrence of earthquakes in some specific combinations showed higher preference around the study area. The statistical data on the number of different combinations of earthquake occurrences were identified from these images. The seismicity trends over different periods are compared to the current seismic trend, and their difference is quantified by using a statistical tool called proximity index. The study proves that the patterns exhibited by earthquakes are independent of the number of earthquakes, and the current seismic trends exhibited by the Indian Plate show a high resemblance with the seismic trend during the years 1975–1988.

Invariant Measures for Uncountable Random Interval Homeomorphisms

A necessary and sufficient condition for the iterated function system {f(cdot ,omega ),|,omega in Omega } with probability P to have exactly one invariant measure mu _* with mu _*((0,1))=1 is given. The main novelty lies in the fact that we only require the transformations f(cdot ,omega ) to be increasing homeomorphims, without any smoothness condition, neither we impose conditions on the cardinality of Omega . In particular, positive Lyapunov exponents conditions are replaced with the existence of solutions to some functional inequalities. The stability and strong law of large numbers of the considered system are also proven.

ECG compression technique using fast fractals in the Internet of medical things

AbstractECG signal is widely used in most cardiology e‐health systems. Patients may be monitored continuously for at least 12 h a day. Therefore, the ECG signal size transmitted to a hospital server during continuous monitoring is significant. Furthermore, transmission of the large size ECG signal is a power consuming process. ECG compression is one of the proposed solutions to overcome this problem. In this paper, a new fractal‐based ECG lossy compression technique is proposed. It is clear that fractal can use ECG signal self similarity characteristics efficiently to achieve high compression ratios. The proposed technique is based on developing the fractal model in conjunction with Iterated Function System. Fractal is well known as a time consuming technique, and therefore, new mathematical development is proposed to potentially reduce fractal computations. Experiments have proven the significant performance of fast fractal in comparison with the traditional version. Furthermore, the resultant compression ratios are close to the traditional fractal results and higher than other existing techniques.

Convexity-Preserving Rational Cubic Zipper Fractal Interpolation Curves and Surfaces

A class of zipper fractal functions is more versatile than corresponding classes of traditional and fractal interpolants due to a binary vector called a signature. A zipper fractal function constructed through a zipper iterated function system (IFS) allows one to use negative and positive horizontal scalings. In contrast, a fractal function constructed with an IFS uses positive horizontal scalings only. This article introduces some novel classes of continuously differentiable convexity-preserving zipper fractal interpolation curves and surfaces. First, we construct zipper fractal interpolation curves for the given univariate Hermite interpolation data. Then, we generate zipper fractal interpolation surfaces over a rectangular grid without using any additional knots. These surface interpolants converge uniformly to a continuously differentiable bivariate data-generating function. For a given Hermite bivariate dataset and a fixed choice of scaling and shape parameters, one can obtain a wide variety of zipper fractal surfaces by varying signature vectors in both the x direction and y direction. Some numerical illustrations are given to verify the theoretical convexity results.

Separation Conditions for Iterated Function Systems with Overlaps on Riemannian Manifolds

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau et al. (Monatsch Math 156:325–355, 2009). We also obtain a formula for the Hausdorff dimension of a self-similar set defined by an iterated function system satisfying the finite type condition, generalizing a corresponding result by Jin and Yau (Commun Anal Geom 13:821–843, 2005) and Lau and Ngai (Adv Math 208:647–671, 2007) on Euclidean spaces. Moreover, we obtain a formula for the Hausdorff dimension of a graph self-similar set generated by a graph-directed iterated function system satisfying the graph finite type condition, extending a result by Ngai et al. (Nonlinearity 23:2333–2350, 2010).

Local -fractal interpolation function.

Constructions of the (global) fractal interpolation functions on standard function spaces got a lot of attention in the last centuries. Motivated by the newly introduced local fractal functions corresponding to a local iterated functions system which is the generalization of the traditional iterated functions system we construct the local non-affine - fractal functions in this article. A few examples of the graphs of these functions are provided. A fractal operator which takes the classical function to its local fractal counterpart is defined and some of its properties are also studied.

Vector-valued Ruelle operator for weakly contractive IFS and Dini matrix potentials

The (scalar) Ruelle operator theory is well-known both in fractal geometry and dynamical systems. In this paper we consider vector-valued Ruelle operators for weakly contractive iterative function systems associated with Dini matrix potentials. We generalize the result in the paper (J. Math. Anal. Appl. 299 341–56) to weakly contractive iterated function system. More exactly, our main theorem gives a sufficient condition for the vector-valued Ruelle operator with the Perron–Frobenius property.

Existence Results for Wardoski-Type Convex Contractions and the Theory of Iterated Function Systems

The purpose of this paper is to define the notion of extended convex ℱ contraction by imposing less conditions on the function ℱ satisfying certain contractive conditions. We prove the existence of fixed points for these types of mappings in the setting of b-metric spaces. In addition, some illustrative examples are provided to show the usability of the obtained results. Lastly, we use the obtained fixed-point results to find the fractals with respect to the iterated function systems in the framework of b-metric spaces. Furthermore, the variables involved in the b-metric space are symmetric, and symmetry plays an important role in solving the nonlinear problems defined in operator theory.

The structure of fuzzy fractals generated by an orbital fuzzy iterated function system

Abstract In this article, we present a structure result concerning fuzzy fractals generated by an orbital fuzzy iterated function system ( ( X , d ) , ( f i ) i ∈ I , ( ρ i ) i ∈ I ) \left(\left(X,d),{({f}_{i})}_{i\in I},{\left({\rho }_{i})}_{i\in I}) . Our result involves the following two main ingredients: (a) the fuzzy fractal associated with the canonical iterated fuzzy function system ( ( I N , d Λ ) , ( τ i ) i ∈ I , ( ρ i ) i ∈ I ) \left(\left({I}^{{\mathbb{N}}},{d}_{\Lambda }),{\left({\tau }_{i})}_{i\in I},{\left({\rho }_{i})}_{i\in I}) , where d Λ {d}_{\Lambda } is Baire’s metric on the code space I N {I}^{{\mathbb{N}}} and τ i : I N → I N {\tau }_{i}:{I}^{{\mathbb{N}}}\to {I}^{{\mathbb{N}}} is given by τ i ( ( ω 1 , ω 2 , … ) ) ≔ ( i , ω 1 , ω 2 , … ) {\tau }_{i}\left(\left({\omega }_{1},{\omega }_{2},\ldots )):= \left(i,{\omega }_{1},{\omega }_{2},\ldots ) for every ( ω 1 , ω 2 , … ) ∈ I N \left({\omega }_{1},{\omega }_{2},\ldots )\in {I}^{{\mathbb{N}}} and every i ∈ I i\in I ; (b) the canonical projections of certain iterated function systems associated with the fuzzy fractal under consideration.

A characterization method for the physical features of honed texture surface of engine cylinder liner

A machine vision method is proposed to characterize the physical features of the honed surface texture on the cylinder liner of internal combustion engines. According to the gray image information of the surface texture of the cylinder liner, the curvelet transform method is employed to extract the physical features of the multi-scale anisotropic surface topography image of the cylinder liner, and the partitioned iterated function system (PIFS) is constructed to calculate fractal dimension to describe the physical features of the surface texture of the cylinder liner. The proposed PIFS-based method is applied to the honed surface and dimple-textured surface of the cylinder liner, the effectiveness and the accuracy are verified by comparing with the other two methods.

Hausdorff and box dimensions of self-affine sets in a locally compact non-Archimedean field

We consider affine iterated function systems in a locally compact non-Archimedean field . We establish the theory of singular value decomposition in and compute the box and Hausdorff dimensions of self-affine sets in , in generic sense, which is an analogy of Falconer’s result for the real case. In , the box and Hausdorff dimensions of self-affine sets can be obtained only when the norms of linear parts of affine transformations are strictly less than . However, in a locally compact non-Archimedean field, the same result can be obtained without the restriction of the norms.

Exponential control of the trajectories of iterated function systems with application to semi-strong GARCH models

AbstractWe establish new results on the strictly stationary solution to an iterated function system. When the driving sequence is stationary and ergodic, though not independent, the strictly stationary solution may admit no moment but we show an exponential control of the trajectories. We exploit these results to prove, under mild conditions, the consistency of the quasi-maximum likelihood estimator of GARCH(p,q) models with non-independent innovations.

The Lq$L^q$ spectrum of self‐affine measures on sponges

AbstractIn this paper, a sponge in is the attractor of an iterated function system consisting of finitely many strictly contracting affine maps whose linear part is a diagonal matrix. A suitable separation condition is introduced under which a variational formula is proved for the spectrum of any self‐affine measure defined on a sponge for all . Apart from some special cases, even the existence of their box dimension was not proved before. Under certain conditions, the formula has a closed form which in general is an upper bound. The Frostman and box dimension of these measures is also determined. The approach unifies several existing results and extends them to arbitrary dimensions. The key ingredient is the introduction of a novel pressure function which aims to capture the growth rate of box counting quantities on sponges. We show that this pressure satisfies a variational principle which resembles the Ledrappier–Young formula for Hausdorff dimension.

Set-Valued $$\alpha $$-Fractal Functions

In this paper, we introduce the concept of the $$\alpha $$ -fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal functions. Further, we estimate the perturbation error between the given continuous function and its $$\alpha $$ -fractal function. Additionally, we define a new graph of a set-valued function different from the standard graph introduced in the literature and establish some bounds on the fractal dimension of the newly defined graph of some special classes of set-valued functions. Also, we explain the need to define this new graph with examples. In the sequel, we prove that this new graph of an $$\alpha $$ -fractal function is an attractor of an iterated function system.

On the Minkowski content of self-similar random homogeneous iterated function systems

The Minkowski content of a compact set is a fine measure of its geometric scaling. For Lebesgue null sets it measures the decay of the Lebesgue measure of epsilon neighbourhoods of the set. It is well known that self-similar sets, satisfying reasonable separation conditions and non-log commensurable contraction ratios, have a well-defined Minkowski content. When dropping the contraction conditions, the more general notion of average Minkowski content still exists. For random recursive self-similar sets the Minkowski content also exists almost surely, whereas for random homogeneous self-similar sets it was recently shown by Zähle that the Minkowski content exists in expectation.
 In this short note we show that the upper Minkowski content, as well as the upper average Minkowski content of random homogeneous self-similar sets is infinite, almost surely, answering a conjecture posed by Zähle. Additionally, we show that in the random homogeneous equicontractive self-similar setting the lower Minkowski content is zero and the lower average Minkowski content is also infinite. These results are in stark contrast to the random recursive model or the mean behaviour of random homogeneous attractors.

Nonlinear fractal interpolation functions on the Koch curve

Nonlinear fractal interpolation functions defined on the Koch curve by replacing the usual Banach contractions used to define (linear) fractal interpolation functions with Rakotch contractions are considered. This allows to take care of the nonlinear term in the associated Read–Bajraktarević operator, respectively, the underlying iterated function system. Moreover, we present an extremely explicit example to demonstrate the effectiveness of the obtained results.

Notes on the Transversality Method for Iterated Function Systems—A Survey

This is a brief survey of selected results obtained using the “transversality method” developed for studying parametrized families of fractal sets and measures. We mostly focus on the early development of the theory, restricting ourselves to self-similar and self-conformal iterated function systems.

A Generalized Multiple Criteria Data-Fitting Model With Sparsity and Entropy With Application to Growth Forecasting

In this article, we present an extended data-fitting model which involves different and conflicting criteria, and we propose an algorithm based on a scalarization technique to solve it. Our model integrates in a unique framework three different criteria, namely, a data-fitting term, and the entropy and the sparsity of the set of unknown parameters. This model can be analyzed by means of multiple criteria decision-making techniques. We then validate the proposed modified algorithm using two computational experiments: We analyze the problem of handwritten digit recognition using a logistic regression model and a deep neural network model, respectively. In the final part of the article, we employ this methodology to forecasting instead. Given the importance of forecasting techniques to predict the future, which in turn can lead to positive impacts on firm performance, we propose two numerical experiments focusing on the forecast of the US GDP. In the first one, we proceed by means of a modified iterated function system with grayscale maps-type fractal operator, and, in the second one, we implement a modified neural network-based model.

Lossy compression of general random variables

AbstractThis paper is concerned with the lossy compression of general random variables, specifically with rate-distortion theory and quantization of random variables taking values in general measurable spaces such as, e.g. manifolds and fractal sets. Manifold structures are prevalent in data science, e.g. in compressed sensing, machine learning, image processing and handwritten digit recognition. Fractal sets find application in image compression and in the modeling of Ethernet traffic. Our main contributions are bounds on the rate-distortion function and the quantization error. These bounds are very general and essentially only require the existence of reference measures satisfying certain regularity conditions in terms of small ball probabilities. To illustrate the wide applicability of our results, we particularize them to random variables taking values in (i) manifolds, namely, hyperspheres and Grassmannians and (ii) self-similar sets characterized by iterated function systems satisfying the weak separation property.