Iterated Function Systems Research Articles

Iterated Function Systems Enriched with Symmetry

Iterated Function Systems Enriched with Symmetry

Linear Fractional Galton–Watson Processes in Random Environment and Perpetuities

Abstract Linear fractional Galton–Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, as the forward iterations of an iterated function system, has an a.s. convergent counterpart in the positive recurrent case given by the corresponding backward iterations. The present expository article aims to provide an explicit view at how these aspects of random difference equations and their stationary limits, called perpetuities, enter into the results and the analysis, especially in quenched regime. Although most of the results presented here are known, we hope that the offered perspective will be welcomed by some readers.

A Study of the International Stock Market Behavior During COVID-19 Pandemic Using a Driven Iterated Function System.

We propose a novel approach to visualize and compare financial markets across the globe using chaos game representation (CGR) of iterated function systems (IFS). We modified a fractal method, widely used in life sciences, and applied it to study the effect of COVID-19 on global financial markets. This modified driven IFS approach is used to generate compact fractal portraits of the financial markets in form of percentage CGR (PC) plots and subtraction percentage (SP) plots. The markets over different periods are compared and the difference is quantified through a parameter called the proximity (Pr) index. The reaction of the financial market across the globe and volatility to the current pandemic of COVID-19 is studied and modeled successfully. The imminent bearish and a surprise bullish pattern of the financial markets across the world is revealed by this fractal method and provides a new tool to study financial markets.

Ledrappier\u2013Young formulae for a family of nonlinear attractors

We study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show that these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier–Young formula.

Diameter diminishing to zero IFSs

In this paper we introduce the notion of diameter diminishing to zero iterated function system, study its properties and provide alternative characterizations of it.

Uniformly perfect and hereditarily non uniformly perfect analytic and conformal non-autonomous attractor sets

Conditions are given which imply that certain non-autonomous analytic iterated function systems (NIFSs) in the complex plane have uniformly perfect attractor sets, while other conditions imply the attractor is pointwise thin and thus hereditarily non uniformly perfect. Examples are given to illustrate the main theorems, as well as to indicate how they generalize other results. Examples are also given to illustrate how possible generalizations of corresponding results for autonomous IFS's do not hold in general in this more flexible setting. Further, applications to non-autonomous Julia sets are given. Lastly, since our definition of NIFS is in some ways more general than others found in the literature, a careful analysis is given to show when certain familiar relationships still hold, along with detailed examples showing when other relationships do not hold.

New equilibria of non-autonomous discrete dynamical systems

In the framework of non-autonomous discrete dynamical systems in metric spaces, we propose new equilibrium points, called quasi-fixed points, and prove that they play a role similar to that of fixed points in autonomous discrete dynamical systems. In this way some sufficient conditions for the convergence of iterative schemes of type xk+1=Tkxk in metric spaces are presented, where the maps Tk are contractivities with different fixed points. The results include any reordering of the maps, even with repetitions, and forward and backward directions. We also prove generalizations of the Banach fixed point theorems when the self-map is substituted by a sequence of contractivities with different fixed points. The theory presented links the field of dynamical systems with the theory of iterated function systems. We prove that in some cases the set of quasi-fixed points is an invariant fractal set. The hypotheses relax the usual conditions on the underlying space for the existence of invariant sets in countable iterated function systems.

Dimension estimates for iterated function systems and repellers. Part II

AbstractThis is the second part of our study on the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on $\mathbb {R}^d$ . In the first part [D.-J. Feng and K. Simon. Dimension estimates for $C^1$ iterated function systems and repellers. Part I. Preprint, 2020, arXiv:2007.15320], we proved that the upper box-counting dimension of the attractor of every $C^1$ IFS on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of every ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Here we introduce a generalized transversality condition (GTC) for parameterized families of $C^1$ IFSs, and show that if the GTC is satisfied, then the dimensions of the IFS attractor and of the ergodic invariant measures are given by these upper bounds for almost every (in an appropriate sense) parameter. Moreover, we verify the GTC for some parameterized families of $C^1$ IFSs on ${\Bbb R}^d$ .

On the contraction ratio of iterated function systems whose attractors are Sierpinski n-gons

• The creation of a perfectly symmetric figure by a completely random process. • There is a particular ratio that turns the attractor of the dynamical system into a fractal. • The ratio that leads to a fractal depends solely on the number of sides in the original polygon. In this paper we apply the chaos game to n -sided regular polygons to generate fractals that are similar to the Sierpinski gasket. We show that for each n -gon, there is an exact ratio that will yield a perfect gasket. We then find a formula for this ratio that depends only on the angle π / n .

The Asymptotic Average Shadowing Property of Iterated Function Systems

The Asymptotic Average Shadowing Property of Iterated Function Systems

Generalized modular fractal spaces and fixed point theorems

In this article, we introduce a new concept of Hausdorff distance through generalized modular metric on nonempty compact subsets and study some topological properties of it. This concept with contraction theory and the iterated function system (IFS) helps us to define a generalized modular fractal space.

An extensive survey on fractal structures using iterated function system in patch antennas

SummaryOver the last few years, fractal structure has been used in several engineering fields such as surface physics, telecommunication, fluid mechanics, and so forth. In telecommunication, often there is a demand for miniaturized patch antennas with fractal structures because it is capable of handling very high frequencies with its diminutive structure. The space filling capability of the fractal structures is used to effectively distribute the surface current all over the radiating material. Fractal structures have a better size reduction capability which aids a miniaturized antenna design. This article provides a generalized and detailed survey on fractal structures in a Euclidean space using iterated function system (IFS) in patch antennas. There are various techniques to design a fractal structure, namely, IFSs, escape‐time fractals, strange attractors, L‐systems, random fractals, and so forth. IFSs are widely used technique to construct a finite set of construction mapping on a complete metric space. The fractal geometries designed using the technique are Sierpinski carpet and gasket, Koch snowflake, deterministic tree/binary tree, Gosper island, Minkowski island, Cantor set, T‐square, and some space filling fractals. This paper deals with the above given fractal structures with their design expressions and shapes in recent developments and applications mentioned. The fractal geometries are used in MIMO antennas to abridge the mutually coupled radiators and improve the gain and efficiency of the antenna. This paper also analyzes the effects of fractal structures on MIMO antennas and discusses the ascendancy fractal structures by cross‐referencing the significant antenna parameters.

Partitioned iterated function systems by regression models for head pose estimation

Head pose estimation represents an important computer vision technique in different contexts where image acquisition cannot be controlled by an operator, making face recognition of unknown subjects more accurate and efficient. In this work, starting from partitioned iterated function systems to identify the pose, different regression models are adopted to predict the angular value errors (yaw, pitch and roll axes, respectively). This method combines the fractal image compression characteristics, such as self-similar structures in order to identify similar head rotation, with regression analysis prediction. The experimental evaluation is performed on widely used benchmark datasets, i.e., Biwi and AFLW2000, and the results are compared with many existing state-of-the-art methods, demonstrating the robustness of the proposed fusion approach and excellent performance.

Bernoulli randomness and Bernoulli normality

AbstractOne can consider μ‐Martin‐Löf randomness for a probability measure μ on 2ω, such as the Bernoulli measure given . We study Bernoulli randomness of sequences in with parameters , and we reintroduce Bernoulli normality, where the uniform distribution of digits is replaced with a Bernoulli distribution. We prove the equivalence of three characterizations of Bernoulli normality. We show that every Bernoulli random real is Bernoulli normal, and this has the corollary that the set of Bernoulli normal reals has full Bernoulli measure in . We give an algorithm for computing Bernoulli normal sequences from normal sequences so that we can give explicit examples of Bernoulli normal reals. We investigate an application of randomness to iterated function systems. Finally, we list a few further questions relating to Bernoulli randomness and Bernoulli normality.

GENERALIZED KOCH CURVES AND THUE–MORSE SEQUENCES

Let [Formula: see text] be the well-known [Formula: see text] Thue–Morse sequence [Formula: see text] Since the 1982–1983 work of Coquet and Dekking, it is known that [Formula: see text] is strongly related to the famous Koch curve. As a natural generalization, for [Formula: see text], we use [Formula: see text] to define the generalized Koch curve, where [Formula: see text] is the generalized Thue–Morse sequence defined to be the unique fixed point of the morphism [Formula: see text] [Formula: see text] beginning with [Formula: see text] and [Formula: see text], and we prove that generalized Koch curves are the attractors of the corresponding iterated function systems. For the case that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the open set condition holds, and then the corresponding generalized Koch curve has Hausdorff, packing and box dimension [Formula: see text], where taking [Formula: see text] and then [Formula: see text] will recover the result on the classical Koch curve.

Gromov hyperbolic graphs arising from iterations

For a contractive iterated function system (IFS), it is known that there is a natural hyperbolic graph structure (augmented tree) on the symbolic space of the IFS that reflects the relationship among neighboring cells, and its hyperbolic boundary with the Gromov metric is Hölder equivalent to the attractor K[14,26,28]. This setup was taken up to study the probabilistic potential theory on K[21,20], and the bi-Lipschitz equivalence on K[29]. In this paper, we formulate a broad class of hyperbolic graphs, called expansive hyperbolic graphs, to capture the most essential properties from the augmented trees and the hyperbolic boundaries (e.g., the special geodesics, bounded degree property, metric doubling property, and Hölder equivalence). We also study a new setup of “weighted” IFS and investigate its connection with the self-similar energy form in the analysis of fractals.

Water Extraction Method Based on Multi-Texture Feature Fusion of Synthetic Aperture Radar Images

Lakes play an important role in the water ecosystem on earth, and are vulnerable to climate change and human activities. Thus, the detection of water quality changes is of great significance for ecosystem assessment, disaster warning and water conservancy projects. In this paper, the dynamic changes of the Poyang Lake are monitored by Synthetic Aperture Radar (SAR). In order to extract water from SAR images to monitor water change, a water extraction algorithm composed of texture feature extraction, feature fusion and target segmentation was proposed. Firstly, the fractal dimension and lacunarity were calculated to construct the texture feature set of a water object. Then, an iterated function system (IFS) was constructed to fuse texture features into composite feature vectors. Finally, lake water was segmented by the multifractal spectrum method. Experimental results showed that the proposed algorithm accurately extracted water targets from SAR images of different regions and different imaging modes. Compared with common algorithms such as fuzzy C-means (FCM), the accuracy of the proposed algorithm is significantly improved, with an accuracy of over 98%. Moreover, the proposed algorithm can accurately segment complex coastlines with mountain shadow interference. In addition, the dynamic analysis of the changes of the water area of the Poyang Lake Basin was carried out with the local hydrological data. It showed that the extracted results of the algorithm in this paper are a good match with the hydrological data. This study provides an accurate monitoring method for lake water under complex backgrounds.

A Countable Fractal Interpolation Scheme Involving Rakotch Contractions

The main result of this paper states that for a given countable system of data, there exists a countable iterated function system consisting of Rakotch contractions, such that its attractor is the graph of a fractal interpolation function corresponding to the given countable system of data.

Iterated Functions Systems Composed of Generalized θ-Contractions

The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of Hutchinson IFS, namely generalized θ-contraction IFS, which is a finite collection of generalized θ-contraction functions T1,…,TN from finite Cartesian product space X×⋯×X into X, where (X,d) is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued θ-contraction IFSs are Picard. Finally, when the map θ is continuous, we show the relation between the code space and the attractor of θ-contraction IFS.

Contractive Iterated Function Systems Enriched with Nonexpansive Maps

Motivated by a recent paper of Leśniak and Snigireva [Iterated function systems enriched with symmetry, preprint], we investigate the properties of the semiattractor A_{mathcal {F}cup mathcal {G}}^* of an IFS mathcal {F} enriched by some other IFS mathcal {G}. We show that in natural cases, the semiattractor A_{mathcal {F}cup mathcal {G}}^* is in fact the attractor of certain IFSs related naturally with the IFSs mathcal {F} and mathcal {G}. We also give an example when A_{mathcal {F}cup mathcal {G}}^* is not compact, yet still being the attractor of considered related IFSs. Finally, we use presented machinery to prove that the so called lower transition attractors due to Vince are semiattractors of enriched IFSs.