Iterated Function Systems Research Articles

The Invariant Measure for a Countable Generalized Iterated Function System

The aim of this paper is to answer one of the open questions raised in Strobin [Qual. Theory Dyn. Syst. 19, 85 (2020)] of whether there exists an invariant (Hutchinson) measure for generalized iterated function systems of any order, consisting of a countably infinite number of maps. Our results likewise strengthen those obtained in Secelean [Mediterr. J. Math. 11, 361–372 (2014)], where the existence of the invariant measure is ascertained only for the case of generalized iterated function systems of order 2, consisting of functions which satisfy a particular contractive condition.

On bivariate fractal interpolation for countable data and associated nonlinear fractal operator

Abstract Fractal interpolation has been conventionally treated as a method to construct a univariate continuous function interpolating a given finite data set with the distinguishing property that the graph of the interpolating function is the attractor of a suitable iterated function system. On the one hand, attempts have been made to extend the univariate fractal interpolation from a finite data set to a countably infinite set. On the other hand, fractal interpolation in higher dimensions, particularly the theory of fractal interpolation surfaces (FISs), has received increasing attention for more than a quarter century. This article targets a two-fold extension of the notion of fractal interpolation by providing a general framework to construct FISs for a prescribed set consisting of countably infinite data on a rectangular grid. By using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operators are established, thereby allowing the interaction of the notion of fractal interpolation with the theory of nonlinear operators.

Fixed Point Results for Fuzzy Enriched Contraction in Fuzzy Banach Spaces with Applications to Fractals and Dynamic Market Equillibrium

We introduce fuzzy enriched contraction, which extends the classical notion of fuzzy Banach contraction and encompasses specific fuzzy non-expansive mappings. Our investigation establishes both the presence and uniqueness of fixed points considering this broad category of operators using a Krasnoselskij iterative scheme for their approximation. We also show the graphical representation of fuzzy enriched contraction and analyze its graph for different values of beta. The implications of these findings extend to significant results within fuzzy fixed-point theory, enriching the understanding of iterative processes in fuzzy metric spaces. To demonstrate the versatility of our innovative concepts and the associated fixed-point theorems, we provide illustrative examples that showcase their applicability across diverse domains, including the generation of fractals. This demonstrates the relevance of fuzzy enriched contraction to iterated function systems, enabling the study of fractal structures under various contractive conditions. Additionally, we explore practical applications of fuzzy enriched contraction in dynamic market equilibrium, offering new insights into stability and convergence in economic models. Through this unified framework, we open new avenues for both theoretical advancements and real world applications in fuzzy systems.

Generalized invariant measures for non-autonomous conformal iterated function systems

Generalized invariant measures for non-autonomous conformal iterated function systems

A φ-Contractivity Fixed Point Theory and Associated φ-Invariant Self-Similar Sets

In this paper, we apply a generalized variant of the concept of fixed point theory due to contraction mappings on metric spaces to construct a general class of iterated function systems relative to the so-called φ-contraction mappings on a metric space. In particular, we give a general framework to the Hutchinson method of constructing self-similar sets as fixed points of suitable mappings issued from the φ-contractions on the metric space. The results may open a new axis in the generalization of self-similar sets and associated self-similar functions. Moreover, our results may be extended to general metric spaces with suitable assumptions. The theoretical results are applied for the computation of the fractal dimension of a concrete example of the new self-similar sets.

Dimension of Planar Non-conformal Attractors with Triangular Derivative Matrices

We study the dimension of the attractor and quasi-Bernoulli measures of parametrized families of iterated function systems of non-conformal and non-affine maps. We introduce a transversality condition under which, relying on a weak Ledrappier-Young formula, we show that the dimensions equal to the root of the subadditive pressure and the Lyapunov dimension, respectively, for almost every choice of parameters. We also exhibit concrete examples satisfying the transversality condition with respect to the translation parameters.

Invariant idempotent ⁎-measures generated by iterated function systems

It is known that every continuous t-norm ⁎ generates a functor of the so-called ⁎-measures in the category of compact Hausdorff spaces. Similarly to the case of the hyperspace functor and the probability measure functors one can define the notion of invariant ⁎-measure for iterated function systems of contractions on compact metric spaces.We provide a simple proof of existence and uniqueness of invariant ⁎-measures. Some examples of invariant ⁎-measures, for different t-norms ⁎, are presented.

Construction of an Iterated Function System on ℝ2 whose Attractor is a QR Code

A QR code is a two-dimensional bar code that encodes a message into a finite collection of black and white squares which are arranged in a square region with a specific order. Considering a QR code as a compact subset of R 2 , we construct an Iterated Function System (IFS) on R 2 whose attractor is the given QR code.

On nonlinear iterated function systems with overlaps

We construct an example of an iterated function system on the line, consisting of linear fractional transformations, such that two of the maps share a fixed points, but the dimension of the attractor equals the conformal dimension, so that there is no “dimension drop”.

Projections of four corner Cantor set: Total self-similarity, spectrum and unique codings

Given ρ∈(0,1/4], the four corner Cantor set E⊂R2 is a self-similar set generated by the iterated function system {(ρx,ρy),(ρx,ρy+1−ρ),(ρx+1−ρ,ρy),(ρx+1−ρ,ρy+1−ρ)}. For θ∈[0,π) let Eθ be the orthogonal projection of E onto a line with an angle θ to the x-axis. In principle, Eθ is a self-similar set having overlaps. In this paper we give a complete characterization on which the projection Eθ is totally self-similar. We also study the spectrum of Eθ, which turns out that the spectrum achieves its maximum value if and only if Eθ is totally self-similar. Furthermore, when Eθ is totally self-similar, we calculate its Hausdorff dimension and study the subset Uθ which consists of all x∈Eθ having a unique coding. In particular, we show that dimHUθ=dimHEθ for Lebesgue almost every θ∈[0,π). Finally, for ρ=1/4 we prove that the possibility for Eθ to contain an interval is strictly smaller than that for Eθ to have an exact overlap.

Topological prevalence of variable speed of convergence in the deterministic chaos game

Let A be the attractor of a Banach contractive iterated function system (IFS) on a complete space. We prove that the orbit generated by a typical (in the sense of Baire category) driver recovers A with every possible speed. Our result extends the one from the paper: Leśniak et al. (Chaos 32(1):013110, 2022). We also show that our result is optimal from a certain point of view.

Remarkable results obtained when studying a question concerning the invariant set of an IFS

Some years ago, a question was raised about whether an iterated function system consisting of a finite number of Reich (or Kannan) contractions (generally discontinuous contractive mappings) has a unique compact invariant set. In this article, we give remarkable results obtained by studying previous research articles that tried to answer this question positively and previous counterexamples that demonstrate that this question is negatively answered. Our results strongly encourage the development of fractal theory because the existence, uniqueness, compactness and boundedness of invariant sets with respect to functional families are one of the main research topics in the theory of fractals.

Enriched Z-Contractions and Fixed-Point Results with Applications to IFS

In this manuscript, we initiate a large class of enriched (d,Z)-Z-contractions defined on Banach spaces and prove the existence and uniqueness of the fixed point of these contractions. We also provide an example to support our results and give an existence condition for the uniqueness of the solution to the integral equation. The results provided in the manuscript extend, generalize, and modify the existence results. Our research introduces novel fixed-point results under various contractive conditions. Furthermore, we discuss the iterated function system associated with enriched (d,Z)-Z-contractions in Banach spaces and define the enriched Z-Hutchinson operator. A result regarding the convergence of Krasnoselskii’s iteration method and the uniqueness of the attractor via enriched (d,Z)-Z-contractions is also established. Our discoveries not only confirm but also significantly build upon and broaden several established findings in the current body of literature.

A self-similar set with non-locally connected components

Luo, Rao and Xiong [Topol. Appl. 322 (2022), 108271] conjectured that if a planar self-similar iterated function system with the open set condition does not involve rotations or reflections, then every connected component of the attractor is locally connected. We create a homogeneous counterexample of Lalley–Gatzouras type, which disproves this conjecture.

Packing measure and dimension of the limit sets of IFSs of generalized complex continued fractions

We consider a family of conformal iterated function systems (for short, CIFSs) of generalized complex continued fractions. Note that in our previous paper we showed that the proper-dimensional Hausdorff measure of the limit set of each CIFS is zero and the packing measure of the limit set with respect to the Hausdorff dimension is positive. In this paper, we show that the packing dimension and the Hausdorff dimension of the limit set of each CIFS in the family are equal, and the proper-dimensional packing measure of the limit set is finite.

Approximation with fractal radial basis functions

The article reports on the construction of a general class of fractal radial basis functions (RBFs) in the literature. The fractal RBFs is defined through fractal perturbation of a RBF through suitable choice of iterated function system (IFS). A fractal RBF may be smooth depending on the choice of the germ function and the IFS parameters. Characterizations of conditionally strictly positive definite and strictly positive definite fractal functions are studied using the definition of k-times monotonicity. Furthermore, error estimates and shape-preserving properties for the approximants Pfα defined through linear combination of cardinal fractal RBFs are investigated. Several examples are presented to illustrate the convergence of the operator Pfα across various parameters, highlighting the advantages of the fractal approximant Pfα over the corresponding classical operator Pf. Finally, estimates for the box dimension of the graphs of approximants derived from fractal radial basis functions are given.

The packing measure of the linear Gauss system

For every k \in \mathbb{N} , let f_{k}:[\frac{1}{k+1}, \frac{1}{k}] \to [0,1] be decreasing, linear functions such that f_{k}(\frac{1}{k+1}) = 1 and f_{k}(\frac{1}{k}) = 0 , k = 1, 2, \dots . We define the iterated function system S_{n} by limiting the collection of functions f_{k} to first n, meaning S_{n} = \{f_{k} \}_{k=1}^{n} . Let J_{n} denote the limit set of S_{n} . Then, \lim\limits_{n\to \infty}\mathcal{P}_{h_n}(J_{n}) = 2 , where h_{n} is the packing dimension of J_{n} and \mathcal{P}_{h_n} is the corresponding packing measure.

Purely periodic continued fractions and graph-directed iterated function systems

We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P1R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{P}\\,}}^1\\mathbb {R}$$\\end{document}, determines a dual pair of graph-directed iterated function systems, whose attractors contain intervals and constitute the domains of a dual pair of Gauss-type maps. Our framework covers many continued fraction algorithms (such as Farey fractions, Ceiling, Even and Odd, Nearest Integer, …\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ldots $$\\end{document}) and provides explicit dual algorithms and characterizations of those quadratic irrationals having a purely periodic expansion.

Entropy of Non-autonomous Iterated Function Systems

Entropy of Non-autonomous Iterated Function Systems

L q -spectra of graph-directed planar non-conformal measures

We estimate the upper and lower Lq -spectra of measures generated by a class of graph-directed planar non-conformal iterated function systems. These systems are composed of C1+ϵ contractions with lower triangular pointwise Jacobian. The estimation provides a criterion for the existence of the Lq -spectra in certain case. As an application, we derive the box-counting dimension of planar self-affine sets generated by lower triangular matrices, whose geometric construction satisfies a finite overlapping type condition.