Narrow Results

In this paper, we construct an Iterated Function System (IFS) on a fuzzy metric space for generating fractal sets in that fuzzy metric space. A new cyclic contraction mapping is defined and utilized in the construction of the IFS. It is shown that the cyclic contraction has a unique fixed point when the space is complete. These contractions automatically do not possess continuity property. A fuzzy variant of the Hausdorff metric is used on the set of compact subsets of the fuzzy metric space. Two illustrative examples are discussed. Further, the fixed point theorem obtained here effectively generalizes a recent result in the literature.

Let the self-affine measure ${\mu }_{M,D}$ be generated by an expanding matrix $M=\mathrm{\text{diag}}\left({\rho }_1^{-1},{\rho }_2^{-1},\dots ,{\rho }_n^{-1}\right)\in M_n(ℝ)$ and a finite integer digit set $D=B\tilde{D}$, where $B\in M_n(\mathbb{Z})$ with $det(B)\ne 0$ and $\tilde{D}\subset {\mathbb{Z}}^n$. In this paper, we show that if $\{x\in [0,1{)}^n:{\sum }_{\tilde{d}\in \tilde{D}}{e}^{2\pi i<\tilde{d},x>}=0\}=\{\frac{1}{s}{(s_1,s_2,\dots ,s_n)}^t:0\le s_1,\dots ,s_n\le s-1\}\setminus \{0\}$ for an integer $s\ge 2$, then $L^2({\mu }_{M,D})$ admits an infinite orthogonal set of exponential functions if and only if there exists ${\rho }_i(1\le i\le n)$ such that $\left|{\rho }_i\right|={(q_i/p_i)}^{1/r_i}$ for some $p_i,q_i,r_i\in \mathbb{N}$ with $gcd(p_i,q_i)=1$ and $gcd(p_i,s)>1$.

Binning the data points of a time series and associating a contraction map with each bin gives rise to a driven IFS representation of the time series. Varying the bins changes the driven IFS, sometimes in complex ways difficult to parse. From the transition matrix for any particular binning we can plot an f(α) curve. Assembling these curves as the bins change gives a surface, which we call the f(α) surface. We use properties of this surface to investigate time series from iterating logistic and tent maps, and also time series of financial data.

The fitting of a given continuous surface defined on a rectangular region in ℝ² is studied by using a fractal interpolation surface, and the error analysis of fitting is made in this paper. The fractal interpolation functions used in surface fitting are generated by a special class of iterated function systems. Some properties of such fractal interpolation functions are discussed. Moreover, the error problems of fitting are investigated by using an operator defined on the space of continuous functions, and the upper estimates of errors are obtained in the sense of two kinds of metrics. Finally, a specific numerical example to illustrate the application of the procedure is also described.

We study numerically the $\alpha$- and $\omega$-limits of the Newton maps of quadratic polynomial transformations of the plane into itself. Our results confirm the conjectures posed in a recent work about the general dynamics of real Newton maps on the plane.

In the present paper, we study chaos in iterated function systems (IFS), namely dynamical systems with several generators. We introduce weak Li–Yorke chaos, chaos in branch, and weak topological chaos to perceive the role of branches to create chaos in an IFS. Moreover, we define another type of chaos, $\sigma$-chaos, on an IFS. Further, we find the necessary conditions to create the chaotic iterated function systems.

Not much is known about the topological structure of a connected self-similar tile whose interior is disconnected, and even less is understood if the interior consists of infinitely many components. We introduce a technique to show that for a large class of self-similar tiles in ℝ², the closure of each component of the interior is homeomorphic to a disk. This allows us to prove such a result for the Eisenstein set, the fundamental domain of a well-known quadratic canonical number system, and some other well-known fractal tiles.

In this paper, we consider a wide class of iterated function systems in R³, and show that their attractors are a class of fractal interpolation surfaces. Based on a refinement equation, we investigate the properties of smoothness of the fractal interpolation functions, and give the results of the smoothness in several cases.

We demonstrate that videofeedback augmented with one or two mirrors can produce stationary fractal patterns obtained by an iterated function system (IFS) with two transformations (one with a reflection, one without), and with four transformations (two with reflections, one without, one with a 180° rotation). Camera placement yielding only a partial image in one or both mirrors can be achieved using IFS with memory. The IFS rules are obtained from the images of three non-collinear points in each of the principal pieces of the image.

The sensitivity analysis for a class of hidden variable fractal interpolation functions (HVFIFs) and their moments is made in the work. Based on a vector valued iterated function system (IFS) determined, we introduce a perturbed IFS and investigate the relations between the two HVFIFs generated by the IFS determined and its perturbed IFS, respectively. An explicit expression for the difference between the two HVFIFs is presented, from which, we show that the HVFIFs are not sensitive to a small perturbation in IFSs. Furthermore, we compute the moment integrals of the HVFIFs and discuss the error of moments of the two HVFIFs. An upper estimate for the error is obtained.

We present a novel approach to approximate the smallest disc to enclose an affine IFS attractor at any accuracy. The approach is efficient and characterized by very low memory requirements. It is based on a concept of spanning points we introduce to describe the extent of an IFS attractor. We give the formal definition of these points and show how to construct them efficiently in practice. A proof that the smallest disc to enclose the spanning points includes the smallest enclosing disc of the IFS attractor is given. We also show that the former disc approximates the latter with an error depending on parameters relevant to the construction of the points. The main benefit of using the spanning points is that their number is usually small even for high approximation accuracies. Thus, we can apply with success one of the known, smallest enclosing disc algorithms to the spanning points. The enclosed experimental results obtained for popular IFS fractals confirm the theoretical assertions of the paper, by showing that our method performs well in practice.

The aim of our work is to elaborate a method to build parametric shapes (curves, surfaces, …) with a non uniform local aspect: every point is assigned a "geometric texture" that evolves continuously from a smooth aspect to a rough aspect.

We rely on previous work that enables us to represent both smooth and fractal free form curves to propose a formalism based on finite families of iterated function systems that generalizes this previous approach. The principle is to blend shapes with uniform aspects to define a shape with a variable aspect.

In 1969, Kannan¹ gave the definition of a new mapping which had presented a condition which is more lenient than contraction condition. The purpose of this note is to introduce K-Iterated Function System using Kannan mapping which will cover a larger range of mappings. We also prove the Collage theorem for the K-Iterated Function System.

By applying a result from the theory of subshifts of finite type,¹ we generalize the result of Frame and Lanski² to IFS with multistep memory. Specifically, we show that for an IFS with m-step memory, there is an IFS with 1-step memory (though in general with many more transformations than ) having the same attractor as .

We continue the program of Bedient et al.¹ by investigating some of the ways of embedding IFS with 1-step memory into IFS with 2-step memory, and 1- and 2-step memory into IFS with 3-step memory. This reveals a hierarchy of attractors of m-step memory IFS as subsets of attractors of n-step memory IFS.

This study discussed the application of a fractal interpolation method in satellite image data reconstruction. It used low-resolution images as the source data for fractal interpolation reconstruction. Using this approach, a high-resolution image can be reconstructed when there is only a low-resolution source image available.

The results showed that the high-resolution image data from fractal interpolation can effectively enhance the sharpness of the border contours. Implementing fractal interpolation on an insufficient image resolution image can avoid jagged edges and mosaic when enlarging the image, as well as improve the visibility of object features in the region of interest. The proposed approach can thus be a useful tool in land classification by satellite images.

In this paper, we investigate the fractals generated by the iterated function system of fuzzy contractions in the fuzzy metric spaces by generalizing the Hutchinson-Barnsley theory. We prove some existence and uniqueness theorems of fractals in the standard fuzzy metric spaces by using the fuzzy Banach contraction theorem. In addition to that, we discuss some results on fuzzy fractals such as Collage Theorem and Falling Leaves Theorem in the standard fuzzy metric spaces with respect to the standard Hausdorff fuzzy metrics.

Fractal interpolation functions provide a new light to the natural deterministic approximation and modeling of complex phenomena. The present paper proposes construction of natural cubic spline coalescence hidden variable fractal interpolation surfaces (CHFISs) over a rectangular grid through the tensor product of univariate bases of cardinal natural cubic spline coalescence hidden variable fractal interpolation functions (CHFIFs). Natural cubic CHFISs are self-affine or non-self-affine in nature depending on the IFS parameters of univariate natural cubic spline CHFIFs. An upper bound of the error between the natural cubic spline blended coalescence fractal interpolant and the original function is deduced. Convergence of the natural cubic CHFIS to the original function , and their derivatives are deduced. The effects free variables, constrained free variables and hidden variables are discussed on the natural cubic spline CHFIS with suitably chosen examples.

In this paper, first we equip the automorphism group of the p-ary rooted tree X* with a natural metric and define a family of contractions on Aut(X*). Then, we construct an iterated function system (IFS) whose attractor is the closure of the adding machine group on Aut(X*). Finally, we show that this group is a strong self-similar group in the sense of IFS.

Riemann–Liouville fractional calculus of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is studied in this paper. It is shown in this paper that fractional integral of order $\nu$ of a CHFIF defined on any interval $[a,b]$ is also a CHFIF albeit passing through different interpolation points. Further, conditions for fractional derivative of order $\nu$ of a CHFIF is derived in this paper. It is shown that under these conditions on free parameters, fractional derivative of order $\nu$ of a CHFIF defined on any interval $[a,b]$ is also a CHFIF.