On intrinsic and extrinsic rational approximation to Cantor sets

In this paper we consider classical iterated function systems (IFS) consisting of a finite number of contracting mappings for a complete metric space. The main goal is to study the class of IFSs whose attractors are Cantor sets, i.e. perfect totally disconnected sets. Important representatives of this class are totally disconnected IFSs introduced by Barnsley. We have proposed other definitions of a totally disconnected IFS and proved their equivalence to the Barnsley definition. Sufficient conditions for IFS to be totally disconnected are obtained. It is shown that injectivity of mappings from an IFS implies the perfection of the attractor and its uncountability. Also it is proved that if the mappings from an IFS are injective and the sum of their contraction coefficients is less than one, then the attractor is a Cantor set. In general case, these conditions do not guarantee totally disconnectedness of an IFS. Meanwhile, it is shown that if an IFS consists of two injective mappings and the sum of their contraction coefficients is less than one, then the IFS is totally disconnected. Examples of IFS attractors are constructed, demonstrating that conditions of the proven theorems are only sufficient but not necessary.

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.

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

In nature, objects are not single fractal sets but are a collection of complex multiple fractals that characterise the multi-fractal space, a generalisation of fractal space. While fractal space includes a fractal set, a multi-fractal space includes the union of fractals. A fuzzy fractal space is a fuzzy metric space and is an approach for the construction, analysis, and approximation of sets and images that exhibit fractal characteristics. The finite Cartesian product of fuzzy fractal spaces is called the multi-fuzzy fractal space. We propose in this paper, a theoretical proof to define the multi-fractal dimensions FD of a multi-fuzzy fractal attractor of n objects for the self-similar fractals sets A=∏i=1nAi=(A1,A2,…An) of the contraction mapping W**:∏i=1nH(F(Xi)→∏i=1nH(F(Xi)) with contractivity factor r = max{ri, i = 1, 2,…n} where H(F(Xi) is a fuzzy fractal space for each i = 1, 2,=, n; over a complete metric space (∏i=1nH(F(Xi)),D*) then for all Bi that belong toH(F(Xi)), there exists B* belonging to (∏i=1nH(F(Xi))) such that W**(B*=∏i=1nBi)=∏i=1n(∪j=1n∪k=1k(i,j)ωij*k(Bj)=∏i=1nWi(B*)). By supposing that M(t)=(∑k(rij*k)FD)n×n is the matrix associated with the the contraction mapping ωij*k with contraction factor rij*k, ∀i, j=1, 2,…, n, ∀k=1, 2, …, k(i, j), for all t≥0, and h (t)=det(M (t)-I). Then, we prove that if there exists a FD such that; h(FD)=0, then FD is the multi fractal dimension for the multi fuzzy-fractal sets of IFS; and M(FD) has a fixed point in Rn.In nature, objects are not single fractal sets but are a collection of complex multiple fractals that characterise the multi-fractal space, a generalisation of fractal space. While fractal space includes a fractal set, a multi-fractal space includes the union of fractals. A fuzzy fractal space is a fuzzy metric space and is an approach for the construction, analysis, and approximation of sets and images that exhibit fractal characteristics. The finite Cartesian product of fuzzy fractal spaces is called the multi-fuzzy fractal space. We propose in this paper, a theoretical proof to define the multi-fractal dimensions FD of a multi-fuzzy fractal attractor of n objects for the self-similar fractals sets A=∏i=1nAi=(A1,A2,…An) of the contraction mapping W**:∏i=1nH(F(Xi)→∏i=1nH(F(Xi)) with contractivity factor r = max{ri, i = 1, 2,…n} where H(F(Xi) is a fuzzy fractal space for each i = 1, 2,=, n; over a complete metric space (∏i=1nH(F(Xi)),D*) then for all Bi that belong toH(F(Xi)), there exists B* belonging to...

Arbitrary affine transformation and their composition effects for two-dimensional fractal sets

# Structure of Genetic Regulatory Networks: Evidence for Scale Free Networks (L S Liebovitch et al.) # Modelling Fractal Dynamics (B J West) # Fractional Relaxation of Distributed Order (R Gorenflo & F Mainardi) # Fractional Time: Dishomogeneous Poisson Processes vs. Homogeneous Non-Poisson Processes (P Allegrini et al.) # Markov Memory in Multifractal Natural Processes (N Papasimakis & F Pallikari) # Description of Complex Systems in Terms of Self-Organization Processes of Prime Integer Relations (V Korotkikh & G Korotkikh) # Dynamical Decomposition of Multifractal Time Series as Fractal Evolution and Long-Term Cycles: Applications to Foreign Currency Exchange Market (A Turiel & C Perez-Vicente) # Fractal Sets from Noninvertible Maps (Ch Mira) # A Generative Construction and Visualization of 3D Fractal Measures (T Martyn) # Complexity in Nature and Society: Complexity Management in the Age of Globalization (K Mainzer) # Fractals, Complexity and Chaos in Supply Chain Networks (M Pearson) # The Effects of Different Competition Rules on the Power-Law Exponent of High Income Distribution (K Yamamoto et al.) # Synergetics on Its Way to the Life Sciences (H Haken) # Complexity, Fractals, Nature and Industrial Design: Some Connections (N Sala) # Modelling Pattern Formation Upon Laser-Induced Etching (A Mora et al.) # Fractal Properties of Some Machined Surfaces (T R Thomas & B-G Rosen) # Fractals, Morphological Spectrum and Complexity of Interfacial Patterns in Non-Equilibrium Solidification (P K Galenko & D M Herlach) # Competition of Doublon Structure in the Phase-Field Model (S Tokunaga & H Sakaguchi) # Study of Thermal Field in Composite Materials (P Stefkova et al.) # Simulation of Geochemical Banding in Acidization-Precipitation Experiments In Situ (M Msharrafieh et al.) # Bulk Mediated Surface Diffusion: Non Markovian and Biased Behavior (J A Revelli et al) # Multifractal Formalism for Remote Sensing: A Contribution to the Description and the Understanding of Meteorological Phenomena in Satellite Images (J Grazzini et al.) # The Distance Ratio Fractal Image (X-Z Zhang et al.) # Iterated Function Systems in Mixed Euclidean and p-Adic Spaces (B Sing) # Multifractality of the Multiplicative Autogressive Point Processes (B Kaulakys et al.) # Analysis of Geographical Distribution Patterns in Plants Using Fractals (A Bari et al.) # Hierarchy of Cellular Automata in Relation to Control of Chaos or Anticontrol (M Markus et al.) # Fractal Analysis of the Images Using Wavelet Transformation (P Jerabkova et al.) # Clustering Phenomena in the Time Distribution of Lightning (L Telesca et al.) # A Cornucopia of Connections: Finding Four Familiar Fractals in the Tower of Hanoi (D R Camp) # Monitoring the Depth of Anaesthesia Using Fractal Complexity Method (W Klonowski et al.) # The Complex Couplings and Gompertzian Dynamics (P Waliszewski & J Konarski)

Modern complex network theory reveals that real-world systems possess not only small-world and scale-free topological features, but may also exhibit statistical self-similarity and fractal scaling. Within the framework of fractal geometry and iterated function systems, this paper investigates the geometric dimension of a class of networks generated from binary structures. Inspired by the Frostman measure theory, we adopt the definition of the Frostman dimension for a family of networks and systematically compute the dimension for networks derived from the full binary tree and from a restricted subtree where consecutive left-child nodes are forbidden. Through precise combinatorial analysis and asymptotic estimation, we prove that the Frostman dimension of the binary tree is [Formula: see text], while the dimension of Fibonacci network — whose node growth follows the Fibonacci sequence — is [Formula: see text], where [Formula: see text] is the golden ratio. The results show that the geometric dimension of such binary networks coincides with the fractal dimension of classical sets such as the Cantor set, and in the Fibonacci case, is closely related to the Fibonacci entropy. This establishes a deeper connection between network science and fractal geometry. The study provides new theoretical insights and computational methods for understanding the fractal properties of networks under regular and constrained growth mechanisms.

A discussion is presented of software which runs on a personal computer and is designed to decode and partially encode IFS (iterated function system) codes in real time. When used with a random-iteration procedure, an IFS is a mathematical model that can be used to generate a fractal set. This fractal set is an attractor of the IFS used in its generation. Two basis concepts of IFS are the collage theorem, which suggests an interactive geometric modeling algorithm for finding IFS codes, which is known as encoding, and the random iteration algorithm for computing the geometry of and rendering images starting from IFS codes, which is known as decoding. It is shown that the program works successfully on an IBM PS 2 model 80 and was used to generate beautiful fractals. Although this program does not generate an IFS code for the given image, it can be used to create different patterns and is limited only by the imagination of the user. Since this software displays both attractor as well as building blocks, it becomes very easy to learn the effects of IFS code on the attractor. >

Fractal sets manipulation and modeling is a difficult task due to their complexity and unpredictability. One of the basic problems is to determine bounds of a fractal set given by some recursive definitions, for example by an Iterated Function System (IFS). Here we propose a method of bounding an IFS-generated fractal set by a minimal simplex that is affinely identical to the standard simplex. First, it will be proved that for a given IFS attractor, such simplex exists and it is unique. Such simplex is then used for definition of an Affine invariant Iterated Function System (AIFS) that then can be used for affine transformation of a given fractal set and for its modeling.KeywordsConvex HullIterate Function SystemNonempty Compact SubsetStandard SimplexStandard Orthonormal BasisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In computer graphics, geometric modeling of complex objects is a difficult process. An important class of complex objects arise from natural phenomena: trees, plants, clouds, mountains, etc. Researchers are at present investigating a variety of techniques for extending modeling capabilities to include these as well as other classes. One mathematical concept that appears to have significant potential for this is fractals. Much interest currently exists in the general scientific community in using fractals as a model of complex natural phenomena. However, only a few methods for generating fractal sets are known. We have been involved in the development of a new approach to computing fractals. Any set of linear maps (affine transformations) and an associated set of probabilities determines an Iterated Function System (IFS). Each IFS has a unique "attractor" which is typically a fractal set (object). Specification of only a few maps can produce very complicated objects. Design of fractal objects is made relatively simple and intuitive by the discovery of an important mathematical property relating the fractal sets to the IFS. The method also provides the possibility of solving the inverse problem. given the geometry of an object, determine an IFS that will (approximately) generate that geometry. This paper presents the application of the theory of IFS to geometric modeling.

In computer graphics, geometric modeling of complex objects is a difficult process. An important class of complex objects arise from natural phenomena: trees, plants, clouds, mountains, etc. Researchers are at present investigating a variety of techniques for extending modeling capabilities to include these as well as other classes. One mathematical concept that appears to have significant potential for this is fractals. Much interest currently exists in the general scientific community in using fractals as a model of complex natural phenomena. However, only a few methods for generating fractal sets are known. We have been involved in the development of a new approach to computing fractals. Any set of linear maps (affine transformations) and an associated set of probabilities determines an Iterated Function System (IFS). Each IFS has a unique "attractor" which is typically a fractal set (object). Specification of only a few maps can produce very complicated objects. Design of fractal objects is made relatively simple and intuitive by the discovery of an important mathematical property relating the fractal sets to the IFS. The method also provides the possibility of solving the inverse problem. given the geometry of an object, determine an IFS that will (approximately) generate that geometry. This paper presents the application of the theory of IFS to geometric modeling.

The self-similarity feature of fractal compression makes it work based on iterated function system. Discrete Wavelet transform of an image provides a set of wavelet coefficients which represent the image at multiresolution. There exists a connection between wavelet transform and fractal compression because the self-similarity existed in wavelet transform could be exploited by fractal theory. The purpose of this paper is to propose a hybrid image coder based on fractal coding and wavelet coding schemes. The similarity among different subbands of the same orientation in a wavelet decomposition of the image is exploited for block prediction in fractal coding scheme. The optimal sizes of range blocks for each subband have been evaluated by experiments. The evaluated data is used in the hybrid image coding scheme. The experimental results show that PSNR and CR are 29.2, 19.8 respectively in our method. the performance of our hybrid image coder is superior to that obtained from baseline fractal image coder for both visual quality and mean square error.

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.

Let $K$ be the attractor of a linear iterated function system (IFS) $S_{j}(x)={\it\rho}_{j}x+b_{j},j=1,\ldots ,m$, on the real line $\mathbb{R}$ satisfying the generalized finite type condition (whose invariant open set ${\mathcal{O}}$ is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [A generalized finite type condition for iterated function systems. Adv. Math.208 (2007), 647–671] as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of $K$ coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let ${\it\alpha}$ be the dimension of $K$. In this paper, we state that $$\begin{eqnarray}{\mathcal{H}}^{{\it\alpha}}(K\cap J)\leq |J|^{{\it\alpha}}\end{eqnarray}$$ for all intervals $J\subset \overline{{\mathcal{O}}}$, and $$\begin{eqnarray}{\mathcal{P}}^{{\it\alpha}}(K\cap J)\geq |J|^{{\it\alpha}}\end{eqnarray}$$ for all intervals $J\subset \overline{{\mathcal{O}}}$ centered in $K$, where ${\mathcal{H}}^{{\it\alpha}}$ denotes the ${\it\alpha}$-dimensional Hausdorff measure and ${\mathcal{P}}^{{\it\alpha}}$ denotes the ${\it\alpha}$-dimensional packing measure. This result extends a recent work of Olsen [Density theorems for Hausdorff and packing measures of self-similar sets. Aequationes Math.75 (2008), 208–225] where the open set condition is required. We use these inequalities to obtain some precise density theorems for the Hausdorff and packing measures of $K$. Moreover, using these density theorems, we describe a scheme for computing ${\mathcal{H}}^{{\it\alpha}}(K)$ exactly as the minimum of a finite set of elementary functions of the parameters of the IFS. We also obtain an exact algorithm for computing ${\mathcal{P}}^{{\it\alpha}}(K)$ as the maximum of another finite set of elementary functions of the parameters of the IFS. These results extend previous ones by Ayer and Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets. Trans. Amer. Math. Soc.351 (1999), 3725–3741] and by Feng [Exact packing measure of Cantor sets. Math. Natchr.248–249 (2003), 102–109], respectively, and apply to some new classes allowing us to include Cantor sets in $\mathbb{R}$ with overlaps.

In this paper, we study -calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the -calculus on the generalized Cantor sets known as middle- Cantor sets. We have suggested a calculus on the middle- Cantor sets for different values of with . Differential equations on the middle- Cantor sets have been solved, and we have presented the results using illustrative examples. The conditions for super-, normal, and sub-diffusion on fractal sets are given.