Showing papers on "Iterated function system published in 2016"

BOX DIMENSIONS OF α-FRACTAL FUNCTIONS

Abstract: The box dimension of the graph of non-affine, continuous, nowhere differentiable function fα which is a fractal analogue of a continuous function f corresponding to a certain iterated function system (IFS), is investigated in the present paper. The estimates for box dimension of the graph of α-fractal function fα for equally spaced as well as arbitrary data sets are found.

Fractional Calculus on Fractal Interpolation for a Sequence of Data with Countable Iterated Function System

Abstract: In recent years, the concept of fractal analysis is the best nonlinear tool towards understanding the complexities in nature. Especially, fractal interpolation has flexibility for approximation of nonlinear data obtained from the engineering and scientific experiments. Random fractals and attractors of some iterated function systems are more appropriate examples of the continuous everywhere and nowhere differentiable (highly irregular) functions, hence fractional calculus is a mathematical operator which best suits for analyzing such a function. The present study deals the existence of fractal interpolation function (FIF) for a sequence of data $${\{(x_n,y_n):n\geq 2\}}$$ with countable iterated function system, where $${x_n}$$ is a monotone and bounded sequence, $${y_n}$$ is a bounded sequence. The integer order integral of FIF for sequence of data is revealed if the value of the integral is known at the initial endpoint or final endpoint. Besides, Riemann–Liouville fractional calculus of fractal interpolation function had been investigated with numerical examples for analyzing the results.

Reich-type iterated function systems

Abstract: In this paper, we introduce the concept of Reich-type iterated function system and prove the existence and uniqueness of the attractor of such a system. Moreover, we study the properties of the canonical projection from the code space onto the attractor of such a system. We also present an iterated function system consisting of continuous Reich contractions having more than one attractor.

Algorithms generating images of attractors of generalized iterated function systems

Abstract: The paper is devoted to searching algorithms which will allow to generate images of attractors of generalized iterated function systems (GIFS in short), which are certain generalization of classical iterated function systems, defined by Mihail and Miculescu in 2008, and then intensively investigated in the last years (the idea is that instead of selfmaps of a metric space X, we consider mappings form the Cartesian product X×...×X to X). Two presented algorithms are counterparts of classical deterministic algorithm and so-called chaos game. The third and fourth one is fitted to special kind of GIFSs - to affine GIFS, which are, in turn, also investigated.

Dimension of self-affine sets for fixed translation vectors.

Abstract: An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension of the self-affine set is the affinity dimension for Lebesgue almost every translation vectors. Similar statement was proven by Jordan, Pollicott, and Simon in 2007 for the dimension of self-affine measures. In this article, we have an orthogonal approach. We introduce a class of self-affine systems in which, given translation vectors, we get the same results for Lebesgue almost all matrices. The proofs rely on Ledrappier-Young theory that was recently verified for affine iterated function systems by Barany and Kaenmaki, and a new transversality condition, and in particular they do not depend on properties of the Furstenberg measure. This allows our results to hold for self-affine sets and measures in any Euclidean space.

Two-dimensional self-affine sets with interior points, and the set of uniqueness

Abstract: Let M be a real matrix with both eigenvalues less than 1 in modulus. Consider two self-affine contraction maps from , where . We are interested in the properties of the attractor of the iterated function system (IFS) generated by T m and T p , i.e. the unique non-empty compact set A such that . Our two main results are as follows: If both eigenvalues of M are between and 1 in absolute value, and the IFS is non-degenerate, then A has non-empty interior. For almost all non-degenerate IFS, the set of points which have a unique address is of positive Hausdorff dimension—with the exceptional cases fully described as well. This paper continues our work begun in [11].

Rigorous approximation of stationary measures and convergence to equilibrium for iterated function systems

Abstract: We study the problem of the rigorous computation of the stationary measure and of the rate of convergence to equilibrium of an iterated function system described by a stochastic mixture of two or more dynamical systems that are either all uniformly expanding on the interval, either all contracting. In the expanding case, the associated transfer operators satisfy a Lasota–Yorke inequality, we show how to compute a rigorous approximations of the stationary measure in the L 1 norm and an estimate for the rate of convergence. The rigorous computation requires a computer-aided proof of the contraction of the transfer operators for the maps, and we show that this property propagates to the transfer operators of the IFS. In the contracting case we perform a rigorous approximation of the stationary measure in the Wasserstein–Kantorovich distance and rate of convergence, using the same functional analytic approach. We show that a finite computation can produce a realistic computation of all contraction rates for the whole parameter space. We conclude with a description of the implementation and numerical experiments.

Dam structural behavior identification and prediction by using variable dimension fractal model and iterated function system

Abstract: Display OmittedThe better forecast effect can be obtained by the combination of IFS and variable dimension fractal model. The method has the certain adaptive ability with high forecasting speed and without convergence problem. Fractal characteristic of dam structural behavior is identified with MF-DFA method.IFS is introduced to build the model fitting the measured dam structural behavior.The variable dimension fractal model and IFS are combined to forecast the dam structural behavior. According to the observations of dam structural health monitoring, iterated function system is adopted to implement the analysis and forecast for dam structural behavior. Firstly, the multifractal detrended fluctuation analysis (MF-DFA) method is employed to identify the fractal characteristics in the measured data series of dam structural behavior. Secondly, the iterated function system algorithm is studied to build the fitting model. The ways to determine the interpolating points (position and number) and vertical scaling factors are given in detail. Thirdly, the variable dimension fractal model and iterated function system are combined to forecast the dam structural behavior. Lastly, the displacement behavior of one concrete gravity dam is analyzed and predicted by the proposed approach. It is shown that the whole trend and detail characteristics of dam structural behavior observed can be described well, and the prediction precision can be improved.

On the chaos game of iterated function systems

Abstract: Every quasi-attractor of an iterated function system (\rom{IFS}) of continuous functions on a first-countable Hausdorff topological space is renderable by the probabilistic chaos game. By contrast, we prove that the backward minimality is a necessary condition to get the deterministic chaos game. As a consequence, we obtain that an \rom{IFS} of homeomorphisms of the circle is renderable by the deterministic chaos game if and only if it is forward and backward minimal. This result provides examples of attractors (a forward but no backward minimal \rom{IFS} on the circle) that are not renderable by the deterministic chaos game. We also prove that every well-fibred quasi-attractor is renderable by the deterministic chaos game as well as quasi-attractors of both, symmetric and non-expansive \rom{IFS}s.

A Fibonacci control system with application to hyper-redundant manipulators

Abstract: We study a model for snake-like robots based on the Fibonacci sequence. The present paper includes an investigation of the reachable workspace, a more general analysis of the control system underlying the model, its reachability and local controllability properties. In addition, we establish some fractal properties of the reachable workspace by means the theory of iterated function systems.

Old wine in fractal bottles I: Orthogonal expansions on self-referential spaces via fractal transformations

Abstract: Our results and examples show how transformations between self-similar sets may be continuous almost everywhere with respect to measures on the sets and may be used to carry well known notions from analysis and functional analysis, for example flows and spectral analysis, from familiar settings to new ones. The focus of this paper is on a number of surprising applications including what we call fractal Fourier analysis, in which the graphs of the basis functions are Cantor sets, discontinuous at a countable dense set of points, yet have good approximation properties. In a sequel, the focus will be on Lebesgue measure-preserving flows whose wave-fronts are fractals. The key idea is to use fractal transformations to provide unitary transformations between Hilbert spaces defined on attractors of iterated function systems.

Algorithms generating images of attractors of generalized iterated function systems

Abstract: The paper is devoted to searching algorithms which will allow to generate images of attractors of \emph{generalized iterated function systems} (GIFS in short), which are certain generalization of classical iterated function systems, defined by Mihail and Miculescu in 2008, and then intensively investigated in the last years (the idea is that instead of selfmaps of a metric space $X$, we consider mappings form the Cartesian product $X\times...\times X$ to $X$). Two presented algorithms are counterparts of classical \emph{deterministic algorithm} and so-called \emph{chaos game}. The third and fourth one one is fitted to special kind of GIFSs - to \emph{affine} GIFS, which are, in turn, also investigated.

Stability of iterated function systems on the circle

Abstract: We prove that any Iterated Function System of circle homeomorphisms with at least one of them having dense orbit, is asymptotically stable. The corresponding Perron-Frobenius operator is shown to satisfy the e-property, that is, for any continuous function its iterates are equicontinuous. The Strong Law of Large Numbers for trajectories starting from an arbitrary point for such function systems is also proved.

Counterexamples for IFS-attractors

Abstract: In this paper, we deal with the part of Fractal Theory related to finite families of (weak) contractions, called iterated function systems (IFS, herein). An attractor is a compact set which remains invariant for such a family. Thus, we consider spaces homeomorphic to attractors of either IFS or weak IFS, as well, which we will refer to as Banach and topological fractals, respectively. We present a collection of counterexamples in order to show that all the presented definitions are essential, though they are not equivalent in general.

Space-filling curves of self-similar sets (I): iterated function systems with order structures

Abstract: This paper is the first part of a series which provides a systematic treatment of the space-filling curves of self-similar sets. In the present paper, we introduce a notion of linear graph-directed IFS (linear GIFS in short). We show that to construct a space-filling curve of a self-similar set, it amounts to exploring its linear GIFS structures. Compared to the previous methods, such as the L-system or recurrent set method, the linear GIFS approach is simpler, more rigorous and leads to further studies on this topic. We also propose a new algorithm for the beautiful visualization of space-filling curves. In a series of papers Dai et al (2015 arXiv:1511.05411 [math.GN]), Rao and Zhang (2015) and Rao and Zhang (2015), we investigate for a given self-similar set how to get 'substitution rules' for constructing space-filling curves, which was obscure in the literature. We solve the problem for self-similar sets of finite type, which covers most of the known results on constructions of space-filling curves.

Infinite iterated function systems with overlaps

Abstract: We formulate two natural but different extensions of the weak separation condition to infinite iterated function systems of conformal contractions with overlaps, and study the associated topological pressure functions. We obtain a formula for the Hausdorff dimension of the limit sets under these weak separation conditions.

Fractals of generalized F-Hutchinson operator

Abstract: The aim of this paper is to construct a fractal with the help of a finite family of F−contraction mappings, a class of mappings more general than contraction mappings, defined on a complete metric ...

Asymptotic perturbation of graph iterated function systems

Abstract: We study an asymptotic perturbation of the limit set generated from a nitely family of conformal contraction maps endowed with a directed graph. We show that if those maps have asymptotic expansions under weak conditions, then the Hausdor dimension of the limit set behaves asymptotically by the same order. We also prove that the Gibbs measure of a suitable potential and the measure theoretic entropy of this measure have asymptotic expansions under an additional condition. In nal section, we demonstrate degeneration of graph iterated function systems. Mathematics Subject Classi cation (2010). Primary: 37B10; Secondary: 37C45, 37D35.

9 – Fractal surfaces

Abstract: This chapter deals with RmRm-valued multivariate continuous fractal functions f:X⊆Rn→Rm and some of their properties. The existence and construction of these multivariate functions is already implicitly contained in Theorems 71–73 Theorem 71 Theorem 72 Theorem 73; simply choose X to be a nonempty compact subset of RnRn and Y:=Rm. Here, however, the issues have to be readdressed. There are two reasons for this: firstly, the more complex geometry is hidden within the construction and needs to be investigated more closely; secondly, the graphs of these RmRm-valued multivariate continuous fractal functions, the so-called fractal surfaces , can be used to construct wavelet bases in RnRn. This construction is based on results from the theory of Coxeter and affine reflection groups, and certain issues involving the geometry of fractal surfaces need to be clarified. Chapter 10 will deal exclusively with this last question. The fractal surfaces introduced there required planar boundary conditions but used different vertical scaling factors. This construction used arbitrary boundary values but only one vertical scaling factor. The latter two constructions use recurrent iterated function systems. Of course, it is always possible to construct fractal surfaces as tensor products of univariate continuous fractal functions. However, these tensor product fractal surfaces lack most of the exciting features of the aforementioned fractal surfaces. Dubuc and his coworkers have also constructed fractal surfaces using a multidimensional iterative interpolation process. The first section in this chapter introduces tensor product fractal surfaces. The construction of nontensor product fractal surfaces is presented next. This construction is based on recurrent iterated function systems and emphasizes the interpolatory nature of fractal surfaces in Rn+mRn+m. The special case m = 1 is then considered as an illustrative example. A few comments are made about Dubuc’s fractal surfaces. In the third section, properties of fractal surfaces such as Holder continuity, oscillation, box dimension, and regularity are discussed. The final section deals with fractal surfaces of higher smoothness; that is, fractal surfaces of class Ck, k∈Nk∈N.

Law of the iterated logarithm for some Markov operators

Abstract: The law of the iterated logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell cycle model examined by Lasota and Mackey (J. Math. Biol. 38 (1999), 241-261).

Crowding Optimization Method to Improve Fractal Image Compressions Based Iterated Function

Abstract: Fractals are geometric patterns generated by Iterated Function System theory. A popular technique known as fractal image compression is based on this theory, which assumes that redundancy in an image can be exploited by block-wise self-similarity and that the original image can be approximated by a finite iteration of fractal codes. This technique offers high compression ratio among other image compression techniques. However, it presents several drawbacks, such as the inverse proportionality between image quality and computational cost. Numerous approaches have been proposed to find a compromise between quality and cost. As an efficient optimization approach, genetic algorithm is used for this purpose. In this paper, a crowding method, an improved genetic algorithm, is used to optimize the search space in the target image by good approximation to the global optimum in a single run. The experimental results for the proposed method show good efficiency by decreasing the encoding time while retaining a high quality image compared with the classical method of fractal image compression.

Fractal dual substitution tilings

Abstract: Starting with a substitution tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles have fractal boundary. We show that each of the new tilings is mutually locally derivable to the original tiling. Thus, at the tiling space level, the new substitution rules are expressing geometric and combinatorial, rather than topological, features of the original. Our method is easy to apply to particular substitution tilings, permits experimentation, and can be used to construct border-forcing substitution rules. For a large class of examples we show that the combinatorial dual tiling has a realization as a substitution tiling. Since the boundaries of our new tilings are fractal we are led to compute their fractal dimension. As an application of our techniques we show how to compute the \v{C}ech cohomology of a (not necessarily border-forcing) tiling using a graph iterated function system of a fractal tiling.