Showing papers on "Iterated function system published in 2022"

Generalized enriched cyclic contractions with application to generalized iterated function system

Abstract: The purpose of this article is to initiate a new class of cyclic contraction mappings and establish a related fixed point theorem in the framework of a Banach space. To approximate the fixed point, a convergence theorem employing the Krasnoselskij iteration is presented for which a priori and a posterior error estimates are also determined. These results modify and extend several comparable results in the existing literature. As an application, we investigate the iterated function system (IFS) comprised of generalized cyclic contraction mappings. Some examples are also presented to validate the results.

Non-stationary zipper $$\alpha $$-fractal functions and associated fractal operator

Abstract: The present paper aims to introduce a new concept of a non-stationary scheme for the so called fractal functions. Here we work with a sequence of maps for the zipper iterated function systems (IFS). We show that the proposed method generalizes the existing stationary interpolant in the sense of IFS. Further, we study the elementary properties of the proposed interpolant and calculate its box and Hausdorff dimension. Also, we obtain an upper bound of the graph of the fractional integral of the proposed interpolant. We notice that the box dimension of the graph of the proposed interpolant is independent of the signature value for a fixed scale vector. In the end, using the method of fractal perturbation of a given function, we construct the associated fractal operator and study some of its properties.

Big Birkhoff sums in $d$-decaying Gauss like iterated function systems

Abstract: The increasing rate of the Birkhoff sums in infinite iterated function systems with polynomial decay of the derivative (for example the Gauss map) is studied. For different unbounded potential functions, the Hausdorff dimensions of the sets of points whos

Scale-Free Fractal Interpolation

Abstract: An iterated function system that defines a fractal interpolation function, where ordinate scaling is replaced by a nonlinear contraction, is investigated here. In such a manner, fractal interpolation functions associated with Matkowski contractions for finite as well as infinite (countable) sets of data are obtained. Furthermore, we construct an extension of the concept of α-fractal interpolation functions, herein called R-fractal interpolation functions, related to a finite as well as to a countable iterated function system and provide approximation properties of the R-fractal functions. Moreover, we obtain smooth R-fractal interpolation functions and provide results that ensure the existence of differentiable R-fractal interpolation functions both for the finite and the infinite (countable) cases.

Separation Conditions for Iterated Function Systems with Overlaps on Riemannian Manifolds

Abstract: 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).

Critical Intermittency in Random Interval Maps

Abstract: Abstract Critical intermittency stands for a type of intermittent dynamics in iterated function systems, caused by an interplay of a superstable fixed point and a repelling fixed point. We consider critical intermittency for iterated function systems of interval maps and demonstrate the existence of a phase transition when varying probabilities, where the absolutely continuous stationary measure changes between finite and infinite. We discuss further properties of this stationary measure and show that its density is not in $$L^q$$ L q for any $$q>1$$ q > 1 . This provides a theory of critical intermittency alongside the theory for the well studied Manneville–Pomeau maps, where the intermittency is caused by a neutral fixed point.

Iterated function system in ∅- Metric Spaces

Abstract: Fractals have gained great attention from researchers due to their wide applications in engineering and applied sciences. Especially, in several topics of applied sciences, the iterated function systems theory has important roles. As is well known, examples of fractals are derived from the fixed point theory for suitable operators in spaces with complete or compact structures. In this article, a new generalization of Hausdorff distance on , is a class of all nonempty compact subsets of the metric space ( , ). Completeness and compactness of are analogously obtained from its counterparts of ( , ). Furthermore, a fractal is presented under a finite set of generalized -contraction mappings. Also, other special cases are presented.

A fractal interpolation scheme for a possible sizeable set of data

Abstract: In this paper, we propose a new fractal interpolation scheme. More precisely, we consider $a,b\in\mathbb{R}$, $a\

Non-stationary $$\phi$$ ϕ -contractions and associated fractals

Abstract: In this study we provide several significant generalisations of Banach contraction principle where the Lipschitz constant is substituted by real-valued control function that is a comparison function. We study non-stationary variants of fixed-point. In particular, this article looks into “trajectories of maps defined by function systems” which are regarded as generalizations of traditional iterated function system (IFS). The importance of forward and backward trajectories of general sequences of mappings is analyzed. The convergence characteristics of these trajectories determined a non-stationary variant of the traditional fixed-point theory. Unlike the normal fractals which have self-similarity at various scales, the attractors of these trajectories of maps which defined by function systems that may have various structures at various scales. In this literature we also study the sequence of countable IFS having some generalized contractions on a complete metric space.

Borel complexity of the family of attractors for weak IFSs

Abstract: This paper is an attempt to measure the difference between the family of iterated function systems attractors and a broader family, the set of attractors for weak iterated function systems. We discuss Borel complexity of the set wIFS\(^d\) of attractors for weak iterated function systems acting on \([0,1]^d\) (as a subset of the hyperspace \(K([0,1]^d)\) of all compact subsets of \([0,1]^d\) equipped with the Hausdorff metric). We prove that wIFS\(^d\) is \(G_{\delta\sigma}\)-hard in \(K([0,1]^d)\), for all \({d\in\mathbb{N}}\). In particular,wIFS\(^d\) is not \(F_{\sigma\delta}\) (in contrast to the family IFS\(^d\) of attractors for classical iterated function systems acting on \([0,1]^d\), which is \(F_{\sigma}\)). Moreover, we show that in the one-dimensional case, wIFS\(^1\) is an analytic subset of \(K([0,1])\).

Fractal Curves on Banach Algebras

Abstract: Most of the fractal functions studied so far run through numerical values. Usually they are supported on sets of real numbers or in a complex field. This paper is devoted to the construction of fractal curves with values in abstract settings such as Banach spaces and algebras, with minimal conditions and structures, transcending in this way the numerical underlying scenario. This is performed via fixed point of an operator defined on a b-metric space of Banach-valued functions with domain on a real interval. The sets of images may provide uniparametric fractal collections of measures, operators or matrices, for instance. The defining operator is linked to a collection of maps (or iterated function system, and the conditions on these mappings determine the properties of the fractal function. In particular, it is possible to define continuous curves and fractal functions belonging to Bochner spaces of Banach-valued integrable functions. As residual result, we prove the existence of fractal functions coming from non-contractive operators as well. We provide new constructions of bases for Banach-valued maps, with a particular mention of spanning systems of functions valued on C*-algebras.

"A characterization of fuzzy fractals generated by an orbital fuzzy iterated function system"

Abstract: "Orbital fuzzy iterated function systems are obtained as a combination of the concepts of iterated fuzzy set system and orbital iterated function system. It turns out that, for such a system, the corresponding fuzzy operator is weakly Picard, its fixed points being called fuzzy fractals. In this paper we present a structure result concerning fuzzy fractals associated to an orbital fuzzy iterated function system by proving that such an object is perfectly determined by the action of the initial term of the Picard iteration sequence on the closure of the orbits of certain elements."

Comparison of the sets of attractors for systems of contractions and weak contractions

Abstract: For $n,d\in\mathbb{N}$ we consider the families: - $L_n^d$ of attractors for iterated function systems (IFS) consisting of $n$ contractions acting on $[0,1]^d$, - $wL_n^d$ of attractors for weak iterated function systems (wIFS) consisting of $n$ weak contractions acting on $[0,1]^d$. We study closures of the above families as subsets of the hyperspace $K([0,1]^d)$ of all compact subsets of $[0,1]^d$ equipped in the Hausdorff metric. In particular, we show that $\overline{L_n^d}=\overline{wL_n^d}$ and $L_{n+1}^d\setminus\overline{L_n^d} eq\emptyset$, for all $n,d\in\mathbb{N}$. What is more, we construct a compact set belonging to $\overline{L_2^d}$ which is not an attractor for any wIFS. We present a diagram summarizing our considerations.

Geometry of Measures in Random Systems with Complete Connections

Abstract: We study new relations between countable iterated function systems (IFS) with overlaps, Smale endomorphisms and random systems with complete connections. We prove that stationary measures for countable conformal IFS with overlaps and place-dependent probabilities, are exact dimensional; moreover we determine their Hausdorff dimension. Next, we construct a family of fractals in the limit set of a countable IFS with overlaps $$\mathcal S$$ , and study the dimension for certain measures supported on these subfractals. In particular, we obtain families of measures on these subfractals which are related to the geometry of the system $$\mathcal S$$ .

Zipper Fractal Functions with Variable Scalings

Abstract: Zipper fractal interpolation function (ZFIF) is a generalization of fractal interpolation function through an improved version of iterated function system by using a binary parameter called a signature. The signature allows the horizontal scalings to be negative. ZFIFs have a complex geometric structure, and they can be non-differentiable on a dense subset of an interval I. In this paper, we construct k-times continuously differentiable ZFIFs with variable scaling functions on I. Some properties like the positivity, monotonicity, and convexity of a zipper fractal function and the one-sided approximation for a continuous function by a zipper fractal function are studied. The existence of Schauder basis of zipper fractal functions for the space of k-times continuously differentiable functions and the space of p-integrable functions for p ∈ [1,∞) are studied. We introduce the zipper versions of full Müntz theorem for continuous function and p-integrable functions on I for p ∈ [1,∞).

Average geodesic distance of sierpiński-type networks

Abstract: The well-known Sierpiński square is a fractal generated by iterated function system (IFS). In this paper, we focus on a class of fractal networks created by IFS. We show a universal approach to solve the average geodesic distance of these fractal networks.

Invariant measures for iterated function systems with inverses

Abstract: We consider iterated function systems that contain inverses in the overlapping case. We focus on the parameterized families of iterated function systems with inverses, satisfying the transversality condition. We show that the invariant measure is absolutely continuous for a.e. parameter when the random walk entropy is greater than the Lyapunov exponent. We also show that if the random walk entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value.

Dimension estimates for $C^1$ iterated function systems and repellers. Part I

Abstract: Abstract This is the first paper in a two-part series containing some results on dimension estimates for $C^1$ iterated function systems and repellers. In this part, we prove that the upper box-counting dimension of the attractor of any $C^1$ iterated function system (IFS) on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Similar results are obtained for the repellers for $C^1$ expanding maps on Riemannian manifolds.

Iterated Function System in Fingerprint Images

Abstract: Biometric identification is the process of determining a person’s identity. Everyday people want to protect their personal belongings and have easy accessibility to it at a fair cost. Biometric-based identification is more secure than any other technique since it binds an identity to a specific person rather than a password or a code that anybody could use. Other security measures, such as smart ID cards and chips are frequently paired. One of the important biometric systems is fingerprint images. Iterated function systems are self-similarity-based method for creating fractals. In this paper, it obtains an iterated function system for fingerprint images which leads a good representation of the given images.

Construction of a class of weighted bivariate fractal interpolation functions

Abstract: A new type of iterated function systems is constructed based on different weights, and it is proved that this type of iterated function systems generates a class of bivariate continuous functions whose graphs are the fractal interpolation surfaces passing through the given interpolation points. Considering the influences of the fractal interpolation functions on different weights and basic functions, we give the corresponding error estimation formula. Finally, the calculation formula for the integral moments of this class of bivariate fractal interpolation functions is given.

Animation design using virtual reality modeling and fractal morphing technology

Abstract: The fractal Iterated Function System (IFS) algorithm is studied to simulate swinging plants’ animation effect under natural conditions. First, the collage method and the IFS attractor parameter morphing method render the fractal plant. Second, the IFS attractor parameter morphing is added on this basis. Third, the swinging plant’s animation morphing is achieved by combining the physics and adjusting the morphing parameters. Finally, the effects of leaves transforming into trees and swinging in the wind are simulated. The simulation results verify the effectiveness of the proposed algorithm. Compared with traditional methods, it reduces the time required to render images and overcomes interactive control limitations. The ability of attractors with parameter variables to control images is very stable. Continuous adjustment of parameters can effectively solve image distortion caused by matrix decomposition during fractal morphing. The proposed algorithm provides some references for applying fractal morphing technology in animation design and computer graphics.

Transition phenomena for the attractor of an iterated function system

Abstract: Iterated function systems (IFSs) and their attractors have been central in fractal geometry. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Two natural questions concerning contractivity arise. First, whether an IFS needs to be contractive to admit an attractor? Second, what occurs to the attractor at the boundary between contractivity and expansion of an IFS? The first question is addressed in the paper by providing examples of highly noncontractive IFSs with attractors. The second question leads to the study of two types of transition phenomena associated with an IFS family that depend on a real parameter. These are called lower and upper transition attractors. Their existence and properties are the main topic of this paper. Lower transition attractors are related to the semiattractors, introduced by Lasota and Myjak in 1990s. Upper transition attractors are related to the problem of continuous dependence of an attractor upon the IFS. A main result states that, for a wide class of IFS families, there is a threshold such that the IFSs in the one-parameter family have an attractor for parameters below the threshold and they have no attractor for parameters above the threshold. At the threshold there exists a unique upper transition attractor.

On properties of similarity boundary of attractors in product dynamical systems

Abstract: <p style='text-indent:20px;'><i>Fractals</i> in higher dimensional <i>dynamical systems</i> have significant roles in physics and other applied sciences. In this paper, one of the key property of fractals, called <i>self similarity</i> in product systems, is studied using the concept of <i>similarity boundary</i>. The relationship between similarity boundary of an attractor in a product space to one of its projection spaces is discussed. The impact of <i>inverse invariance</i> of similarity boundary on its coordinate iterated function system is analyzed. Fractals satisfying the <i>strong open set condition</i>, restricted to attractors in product spaces, are characterized. The relationship between similarity boundary of attractors in product spaces and their overlapping sets is also obtained. The equivalency of the restricted open set condition (ROSC) and the strong open set condition in product spaces, is proved. Self similarity of an attractor in a product system is characterized using the Hausdorff measure of its similarity boundary. Also, the Hausdorff dimensions of the overlapping set and similarity boundary of attractors for different types of iterated function systems are obtained.</p>

Dimensions of slices through the Sierpiński gasket

Abstract: We discuss the intersections of the d-dimensional Sierpiński gasket with -dimensional hyperplanes in a particular fixed direction. We give the values of the Hausdorff dimension, the lower box dimension, the upper box dimension, and the packing dimension of each slice and determine the criterion for which the Hausdorff dimension of the slice takes Marstrand's value. In order to investigate the slices, we improve the theory of non-autonomous conformal iterated function systems.