Iterated Function Systems Research Articles

On separation properties for iterated function systems of similitudes

On separation properties for iterated function systems of similitudes

Strongly-Fibred Iterated Function Systems and the Barnsley–Vince triangle

Strongly-Fibred Iterated Function Systems and the Barnsley–Vince triangle

On the Fractal interpolation functions associated with Matkowski contractions

<abstract><p>In this paper we investigate an iterated function system that defines a fractal interpolation function, where ordinate scaling, that is Lipschitz constant in Banach contraction principle is substituted by real-valued control function. In such a manner, fractal interpolation functions associated with Matkowski contractions are obtained and provide a new framework of approximating experimental data. Furthermore, given a data generating function $ f $, we study a new class of fractal interpolation functions which converge to $ f $.</p></abstract>

Activation Functions Effect on Fractal Coding Using Neural Networks

Activation functions play an essential role in converting the output of the artificial neural network into nonlinear results, since without this nonlinearity, the results of the network will be less accurate. Nonlinearity is the mission of all nonlinear functions, except for polynomials. The activation function must be differentiable for backpropagation learning. This study’s objective is to determine the best activation functions for the approximation of each fractal image. Different results have been attained using Matlab and Visual Basic programs, which indicate that the bounded function is more helpful than other functions. The non-linearity of the activation function is important when using neural networks for coding fractal images because the coefficients of the Iterated Function System are different according to the different types of fractals. The most commonly chosen activation function is the sigmoidal function, which produces a positive value. Other functions, such as tansh or arctan, whose values can be positive or negative depending on the network input, tend to train neural networks faster. The coding speed of the fractal image is different depending on the appropriate activation function chosen for each fractal shape. In this paper, we have provided the appropriate activation functions for each type of system of iterated functions that help the network to identify the transactions of the system.

Semifractals from multifunctions on product spaces and generalized iterated function systems

Semifractals from multifunctions on product spaces and generalized iterated function systems

ON THE CLASSICAL INTEGRAL OF FRACTAL FUNCTIONS

In this paper, the integral of classical fractal interpolation function (FIF) and A-fractal function is explored for both the cases of constant and variable scaling factors. The definite integral for the classical FIF in the closed interval of [Formula: see text] is estimated. The novel notion of affine-quadratic FIF is introduced and integrated for both constant and variable scaling factors. It is demonstrated that its integral is not an affine-quadratic FIF, however it is a FIF. Similarly, by choosing the vertical scaling factors as constants and variables, A-fractal function is integrated. Further, by assuming certain condition on the block matrix, it is shown that like the original A-fractal function its integral is also an attractor for the iterated function system.

Zipper Fractal Functions with Variable Scalings

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,∞).

On the code space and Hutchinson measure for countable iterated function system consisting of cyclic [formula omitted]-contractions

The notion of Iterated Function System (IFS) plays a vital role in the theory of fractals. The investigation of the code space, code map, and invariant measure associated with an IFS is an integral part of the IFS theory. During the last few decades, there has been significant interest in studying various generalizations and extensions of the classical IFS theory. A countable IFS consisting of cyclic φ-contractions (Pasupathi et al. 2020) is a recent addition to the IFS theory. However, the aforementioned study confines to itself the existence of a set attractor (fractal) associated with the IFS consisting of cyclic φ-contractions. This work aims to supplement the deliberations on the aforementioned IFS. We study the code space associated with the countable cyclic φ-contraction IFS and prove that the results are analogous to the classical IFS setting. Our main finding is that a unique invariant measure (Hutchinson measure) corresponding to the countable cyclic φ-contraction IFS with probabilities exists.

Thermodynamic Formalism for General Iterated Function Systems with Measures.

This paper introduces a theory of Thermodynamic Formalism for Iterated Function Systems with Measures (IFSm). We study the spectral properties of the Transfer and Markov operators associated to a IFSm. We introduce variational formulations for the topological entropy of holonomic measures and the topological pressure of IFSm given by a potential. A definition of equilibrium state is then natural and we prove its existence for any continuous potential. We show, in this setting, a uniqueness result for the equilibrium state requiring only the Gâteaux differentiability of the pressure functional.

Fractals via Controlled Fisher Iterated Function System

This paper explores the generalization of the fixed-point theorem for Fisher contraction on controlled metric space. The controlled metric space and Fisher contractions are playing a very crucial role in this research. The Fisher contraction on the controlled metric space is used in this paper to generate a new type of fractal set called controlled Fisher fractals (CF-Fractals) by constructing a system named the controlled Fisher iterated function system (CF-IFS). Furthermore, the interesting results and consequences of the controlled Fisher iterated function system and controlled Fisher fractals are demonstrated. In addition, the collage theorem on controlled Fisher fractals is established as well. The newly developing IFS and fractal set in the controlled metric space can provide the novel directions in the fractal theory.

About Some Monge–Kantovorich Type Norm and Their Applications to the Theory of Fractals

If X is a Hilbert space, one can consider the space cabv(X) of X valued measures defined on the Borel sets of a compact metric space, having a bounded variation. On this vector measures space was already introduced a Monge–Kantorovich type norm. Our first goal was to introduce a Monge–Kantorovich type norm on cabv(X), where X is a Banach space, but not necessarily a Hilbert space. Thus, we introduced here the Monge–Kantorovich type norm on cabvLq([0,1]),(1<q<∞). We obtained some properties of this norm and provided some examples. Then, we used the Monge–Kantorovich norm on cabvKn(K being R or C) to obtain convergence properties for sequences of fractal sets and fractal vector measures associated to a sequence of iterated function systems.

Existence of invariant idempotent measures by contractivity of idempotent Markov operators

We prove that the idempotent Markov operator generated by contractive max plus normalized iterated function system (IFS) is also a contractive map w.r.t. natural metrics on the space of idempotent measures. This gives alternative proofs of the existence of invariant idempotent measures for such IFSs.

Upper Bounds for the Hausdorff Dimension of Weierstrass Curves

We produce an upper bound for the Hausdorff dimension of the graph of a Weierstrass-type function. Whilst strictly weaker than existing results, it has the advantage of being directly computable from the theory of hyperbolic iterated function systems (IFS).

Fractal Curves on Banach Algebras

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.

Generating Sierpinski gasket from matrix calculus in Dempster–Shafer theory

Graphics with fractal features are usually generated using the Iterated Function System (IFS). IFS can generate the entire family of Sierpinski gaskets by performing different operations on the attractors. As the most classical graphic, Sierpinski gasket can also be generated using mod(n,2). Dempster–Shafer Theory (DST), as a mathematical theory about evidence, models information on the all possible combination states (power set), which relates to 2n. In this paper, we explore the relationship between the Sierpinski gasket and matrix calculus in DST, which is the first time to connect fractal theory and DST from the perspective of geometry. In addition, based on the generation process of the matrices, we propose a method to generate the Sierpinski Gasket using the Kronecker product.

Existence of a fractal of iterated function systems containing condensing functions for the degree of nondensifiability

From a fixed point theorists’ view, Hutchinson considered a fractal set as a fixed point problem and applied the Banach contraction principle to prove its existence. In this paper, we present a result about the existence of fractal for a finite iterated condensing function using the degree of nondensifiability.

Approximation by Quantum Meyer-König-Zeller Fractal Functions

In this paper, a novel class of quantum fractal functions is introduced based on the Meyer-König-Zeller operator Mq,n. These quantum Meyer-König-Zeller (MKZ) fractal functions employ Mq,nf as the base function in the iterated function system for α-fractal functions. For f∈C(I), I closed interval in R, it is shown that a sequence of quantum MKZ fractal functions {fn(qn,α)}n=0∞ exists which converges uniformly to f without altering the scaling function α. The shape of fn(qn,α) depends on q as well as the other iterated function system parameters. For f,g∈C(I), f≥g>0, we show that a sequence {fn(qn,α)}n=0∞ exists with fn(qn,α)≥g>0 converging to f. Quantum MKZ fractal versions of some classical Müntz theorems are also presented. For q=1, the box dimension and some approximation-theoretic results of MKZ α-fractal functions are investigated in C(I). Finally, MKZ α-fractal functions are studied in Lp spaces with p≥1.

Chain Mixing and Multi-transitivities of Iterated Function Systems

Chain Mixing and Multi-transitivities of Iterated Function Systems

On a Certain Class of IFSs and Their Attractors

Our purpose in this paper is to consider a new class of iterated function systems (IFS) based on the concept of orbital condition introduced by Miculescu et al. (Analele Universităţii de Vest Timişoara Seria Matematică-Informatică LVI 2:71–80, 2018). On the given IFS there are imposed some sufficient conditions guaranteeing the existence of an attractor. There are also established some further results which describe the nature of the attractors of the considered type. The introduced theory is supported by some examples of IFSs for which the attractors are also depicted.

Multiple-Function Systems Based on Regular Subdivision

Self-similar fractals can be generated using subdivision and the subdivision curves/surfaces are actually attractors. Such a connection has been studied between fractals and an extended family of subdivision including stationary and non-stationary schemes. This paper aims to move one step further on such a connection and introduce multiple-function systems, which has a set of function systems and choose one for each step of iteration. These multiple-function systems can be obtained by deriving the iterated function systems based on the subdivision operators and applying some modifications, including deleting some transformations, to them. Such multiple-function systems can be arranged in a tree structure and can generate different attractors along different paths in the tree. Several examples are presented to illustrate the performance of these multiple-function systems.