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Journal articles on the topic 'Iterated Function Systems (IFS)'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 September 2023

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1

Iacus, Stefano Maria, and Davide La Torre. "Approximating distribution functions by iterated function systems." Journal of Applied Mathematics and Decision Sciences 2005, no. 1 (January 1, 2005): 33–46. http://dx.doi.org/10.1155/jamds.2005.33.

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An iterated function system (IFS) on the space of distribution functions is built with the aim of proposing a new class of distribution function estimators. One IFS estimator and its asymptotic properties are studied in detail. We also propose a density estimator derived from the IFS distribution function estimator by using Fourier analysis. Relative efficiencies of both estimators, for small and moderate sample sizes, are presented via Monte Carlo analysis.

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2

Mohtashamipour, Maliheh, and Alireza Zamani Bahabadi. "Chaos in Iterated Function Systems." International Journal of Bifurcation and Chaos 30, no. 12 (September 30, 2020): 2050177. http://dx.doi.org/10.1142/s0218127420501771.

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In the present paper, we study chaos in iterated function systems (IFS), namely dynamical systems with several generators. We introduce weak Li–Yorke chaos, chaos in branch, and weak topological chaos to perceive the role of branches to create chaos in an IFS. Moreover, we define another type of chaos, [Formula: see text]-chaos, on an IFS. Further, we find the necessary conditions to create the chaotic iterated function systems.

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Balu, Rinju, and Sunil Mathew. "ON (n, m)-ITERATED FUNCTION SYSTEMS." Asian-European Journal of Mathematics 06, no. 04 (December 2013): 1350055. http://dx.doi.org/10.1142/s1793557113500551.

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One of the most common way to generate a fractal is by using an iterated function system (IFS). In this paper, we introduce an (n, m)-IFS, which is a collection of n IFSs and discuss the attractor of this system. Also we prove the continuity theorem and collage theorem for (n, m)-IFS.

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4

FRAME, MICHAEL, and NIAL NEGER. "FRACTAL VIDEOFEEDBACK AS ANALOG ITERATED FUNCTION SYSTEMS." Fractals 16, no. 03 (September 2008): 275–85. http://dx.doi.org/10.1142/s0218348x08003946.

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We demonstrate that videofeedback augmented with one or two mirrors can produce stationary fractal patterns obtained by an iterated function system (IFS) with two transformations (one with a reflection, one without), and with four transformations (two with reflections, one without, one with a 180° rotation). Camera placement yielding only a partial image in one or both mirrors can be achieved using IFS with memory. The IFS rules are obtained from the images of three non-collinear points in each of the principal pieces of the image.

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5

JONES, HUW. "ITERATED FUNCTION SYSTEMS FOR OBJECT GENERATION AND RENDERING." International Journal of Bifurcation and Chaos 11, no. 02 (February 2001): 259–89. http://dx.doi.org/10.1142/s0218127401002237.

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Iterated function systems (IFS) have been used since the mid 1980s in generating fractal forms and in image compression. They have the property of encoding complex structures as relatively simple and concise algorithms and data sets. The complete information for generation of an IFS structure, known as an "attractor", is held in the definition of a few transformation functions, typically affine transformations described by simple linear equations. These transform 2D or 3D objects by combinations of translation, scaling and rotation, preserving parallel lines. The process of generating an attractor using either of the two standard methods is straightforward, but the inverse problem of identifying the IFS for a particular attractor is more difficult. The paper introduces and reviews the uses of Iterated Function Systems, particularly for 2D image and 3D object generation. As well as a general exposition of the method, results developed by the author for creation of fractal objects are shown. Some are based on regular geometric structures, others mimic a number of naturally occurring forms. These deviate from normal practice in using nonaffine transformations in the IFS definition. A new method for synthesising images of smooth objects using IFS is developed and illustrated as an alternative to standard 3D rendering methods (see the cover picture), which are briefly described.

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Centore, P. M., and E. R. Vrscay. "Continuity of Attractors and Invariant Measures for Iterated Function Systems." Canadian Mathematical Bulletin 37, no. 3 (September 1, 1994): 315–29. http://dx.doi.org/10.4153/cmb-1994-048-6.

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AbstractWe prove the "folklore" results that both the attractor A and invariant measure μ of an N-map Iterated Function System (IFS) vary continuously with variations in the contractive IFS maps as well as the probabilities. This represents a generalization of Barnsley's result showing the continuity of attractors with respect to variations of a parameter appearing in the IFS maps. Some applications are presented, including approximations of attractors and invariant measures of nonlinear IFS, as well as some novel approximations of Julia sets for quadratic complex maps.

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Rajan, Pasupathi, María A. Navascués, and Arya Kumar Bedabrata Chand. "Iterated Functions Systems Composed of Generalized θ-Contractions." Fractal and Fractional 5, no. 3 (July 14, 2021): 69. http://dx.doi.org/10.3390/fractalfract5030069.

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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.

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8

EROĞLU, KEMAL ILGAR, STEFFEN ROHDE, and BORIS SOLOMYAK. "Quasisymmetric conjugacy between quadratic dynamics and iterated function systems." Ergodic Theory and Dynamical Systems 30, no. 6 (November 24, 2009): 1665–84. http://dx.doi.org/10.1017/s0143385709000789.

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AbstractWe consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the ‘overlap set’ 𝒪 is finite, and which are ‘invertible’ on the attractor A, in the sense that there is a continuous surjection q:A→A whose inverse branches are the contractions of the IFS. The overlap set is the critical set in the sense that q is not a local homeomorphism precisely at 𝒪. We suppose also that there is a rational function p with the Julia set J such that (A,q) and (J,p) are conjugate. We prove that if A has bounded turning and p has no parabolic cycles, then the conjugacy is quasisymmetric. This result is applied to some specific examples including an uncountable family. Our main focus is on the family of IFS {λz,λz+1} where λ is a complex parameter in the unit disk, such that its attractor Aλ is a dendrite, which happens whenever 𝒪 is a singleton. C. Bandt observed that a simple modification of such an IFS (without changing the attractor) is invertible and gives rise to a quadratic-like map qλ on Aλ. If the IFS is post-critically finite, then a result of A. Kameyama shows that there is a quadratic map pc (z)=z2 +c, with the Julia set Jc such that (Aλ,qλ) and (Jc,pc) are conjugate. We prove that this conjugacy is quasisymmetric and obtain partial results in the general (not post-critically finite) case.

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Vasisht, Radhika, Mohammad Salman, and Ruchi Das. "Variants of shadowing properties for iterated function systems on uniform spaces." Filomat 35, no. 8 (2021): 2565–72. http://dx.doi.org/10.2298/fil2108565v.

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In this paper, the notions of topological shadowing, topological ergodic shadowing, topological chain transitivity and topological chain mixing are introduced and studied for an iterated function system (IFS) on uniform spaces. It is proved that if an IFS has topological shadowing property and is topological chain mixing, then it has topological ergodic shadowing and it is topological mixing. Moreover, if an IFS has topological shadowing property and is topological chain transitive, then it is topologically ergodic and hence topologically transitive. Also, these notions are studied for the product IFS on uniform spaces.

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10

DEKKING, F. M., and P. VAN DER WAL. "THE BOUNDARY OF THE ATTRACTOR OF A RECURRENT ITERATED FUNCTION SYSTEM." Fractals 10, no. 01 (March 2002): 77–89. http://dx.doi.org/10.1142/s0218348x0200077x.

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We prove for a subclass of recurrent iterated function systems (also called graph-directed iterated function systems) that the boundary of their attractor is again the attractor of a recurrent IFS. Our method is constructive and permits computation of the Hausdorff dimension of the attractor and its boundary.

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STENFLO, ÖRJAN. "ITERATED FUNCTION SYSTEMS WITH A GIVEN CONTINUOUS STATIONARY DISTRIBUTION." Fractals 20, no. 03n04 (September 2012): 197–202. http://dx.doi.org/10.1142/s0218348x1250017x.

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KAMAE, TETURO, JUN LUO, and BO TAN. "A GLUING LEMMA FOR ITERATED FUNCTION SYSTEMS." Fractals 23, no. 02 (May 28, 2015): 1550019. http://dx.doi.org/10.1142/s0218348x1550019x.

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We obtain a special version of gluing lemma related to the theory of iterated function systems (IFS). As an application, we verify that a family of concrete n-dimensional self-affine tiles {Tn, r : 0 ≤ r < 3, n ≥ 3} are homeomorphic with the unit cube [0,1]n. The tiles Tn, r are "nontrivial" in the sense that each of them is neither a self-affine polytope nor the product of an interval with an (n - 1)-dimensional self-affine tile.

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13

Badger, Matthew, and Vyron Vellis. "Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon." Analysis and Geometry in Metric Spaces 9, no. 1 (January 1, 2021): 90–119. http://dx.doi.org/10.1515/agms-2020-0125.

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Abstract We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/s)-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α > s. At the endpoint, α = s, a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/s)-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s-dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝ n is always at most the ambient dimension n, the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n.

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Dániel Prokaj, R., and Károly Simon. "Piecewise linear iterated function systems on the line of overlapping construction." Nonlinearity 35, no. 1 (November 29, 2021): 245–77. http://dx.doi.org/10.1088/1361-6544/ac355e.

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Abstract In this paper we consider iterated function systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exponent which comes from the most natural system of covers of the attractor.

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15

Lo, Rong-Chin, Tung-Tai Kuo, Ren-Guey Lee, Yuan-Hao Chen, and Chuan Chin Lim. "EVOKED POTENTIAL PRIMITIVES OF RAT MOTOR CORTEX SIGNAL ANALYSIS BASED ON ITERATED FUNCTION SYSTEMS." Biomedical Engineering: Applications, Basis and Communications 32, no. 04 (July 29, 2020): 2050033. http://dx.doi.org/10.4015/s1016237220500337.

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The function of the brain has been the focus of neuroscience studies for nearly half a century. The studies found that the evoked potential signals of the motor cortex are the main source to command action. An action signal is composed of several signal primitives that are mainly generated by the motor cortex. It was found that signal primitives of the motor cortex can be produced by several fixed rules and a group of codes called iterated function systems (IFS) code. The goal of our research is to find the relationships between the signal primitives of the motor cortex and actions. We recorded the action signals of the rat motor cortex using 8-channel micro-electrodes and used independent component analysis (ICA) to find the independent source signals called signal primitives. Then, the IFS algorithm was used to find the signal primitive codes, which is the IFS code. The experimental results showed that the source signals of actions produce the IFS rules and a set of codes by the IFS algorithm and conversely, using the IFS rules and the set of codes can reconstruct the source signals. Every 20-character length of action signals will generate unique 6-character IFS codes, meaning that the action signals can be replaced with IFS codes to achieve the compression. We found that the IFS rules and codes can be used to represent different cortex commands which have distinct IFS codes that can be used to classify the movements of rat. The classification result reached 78.75% for rough movement and nearly 50% for subtle movement, where the rough movement is that the rat performs two motions and the subtle movement is three motions. This result shows that the motor cortex command can consist of distinct signal primitives and the huge file size of the motor cortex command is reduced three times by the IFS algorithm.

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Leśniak, Krzysztof, Nina Snigireva, Filip Strobin, and Andrew Vince. "Transition phenomena for the attractor of an iterated function system*." Nonlinearity 35, no. 10 (September 23, 2022): 5396–426. http://dx.doi.org/10.1088/1361-6544/ac8af1.

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Abstract 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.

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Demers, Matthew, Herb Kunze, and Davide La Torre. "ON RANDOM ITERATED FUNCTION SYSTEMS WITH GREYSCALE MAPS." Image Analysis & Stereology 31, no. 2 (May 17, 2012): 109. http://dx.doi.org/10.5566/ias.v31.p109-120.

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In the theory of Iterated Function Systems (IFSs) it is known that one can find an IFS with greyscale maps (IFSM) to approximate any target signal or image with arbitrary precision, and a systematic approach for doing so was described. In this paper, we extend these ideas to the framework of random IFSM operators. We consider the situation where one has many noisy observations of a particular target signal and show that the greyscale map parameters for each individual observation inherit the noise distribution of the observation. We provide illustrative examples.

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BOUBOULIS, P. "PSEUDO RANDOM NUMBER GENERATION WITH THE AID OF ITERATED FUNCTION SYSTEMS ON ℝ2." International Journal of Modern Physics C 18, no. 05 (May 2007): 861–82. http://dx.doi.org/10.1142/s0129183107010590.

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Two new pseudorandom number generators, based on iterated function systems (IFS), are introduced. An IFS is created based on an arbitrary seed and a set is constructed using the deterministic iteration algorithm (DIA). From this set pseudo random numbers have been constructed. The generators have big periods and pass all major statistical tests, indicating that they can be used in any application requiring random numbers, such as cryptography.

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YE, YUAN-LING. "Ruelle operator with weakly contractive iterated function systems." Ergodic Theory and Dynamical Systems 33, no. 4 (August 31, 2012): 1265–90. http://dx.doi.org/10.1017/s0143385712000211.

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AbstractThe Ruelle operator has been studied extensively both in dynamical systems and iterated function systems (IFSs). Given a weakly contractive IFS $(X, \{w_j\}_{j=1}^m)$ and an associated family of positive continuous potential functions $\{p_j\}_{j=1}^m$, a triple system $(X, \{w_j\}_{j=1}^m, \{p_j\}_{j=1}^m)$is set up. In this paper we study Ruelle operators associated with the triple systems. The paper presents an easily verified condition. Under this condition, the Ruelle operator theorem holds provided that the potential functions are Dini continuous. Under the same condition, the Ruelle operator is quasi-compact, and the iterations sequence of the Ruelle operator converges with a specific geometric rate, if the potential functions are Lipschitz continuous.

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FORTE, BRUNO, and EDWARD R. VRSCAY. "SOLVING THE INVERSE PROBLEM FOR FUNCTION/IMAGE APPROXIMATION USING ITERATED FUNCTION SYSTEMS I: THEORETICAL BASIS." Fractals 02, no. 03 (September 1994): 325–34. http://dx.doi.org/10.1142/s0218348x94000429.

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We are concerned with function approximation and image representation using Iterated Function Systems (IFS) over ℒp (X, µ): An N-map IFS with grey level maps (IFSM), to be denoted as (w, Φ), is a set w of N contraction maps wi: X → X over a compact metric space (X, d) (the "base space") with an associated set Φ of maps ϕi: R → R. Associated with each IFSM is a contractive operator T with fixed point [Formula: see text]. We provide a rigorous solution to the following inverse problem: Given a target υ ∈ ℒp(X, µ) and an ∊ > 0, find an IFSM whose attractor satisfies [Formula: see text].

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Al-Shameri, Wadia Faid Hassan. "Some Results on Recurrent Fractal Interpolation Function." Nanoscience and Nanotechnology Letters 12, no. 8 (August 1, 2020): 1038–43. http://dx.doi.org/10.1166/nnl.2020.3198.

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Barnsley (Barnsley, M.F., 1986. Fractal functions and interpolation. Constr. Approx., 2, pp.303–329) introduced fractal interpolation function (FIF) whose graph is the attractor of an iterated function system (IFS) for describing the data that have an irregular or self-similar structure. Barnsley et al. (Barnsley, M.F., et al., 1989. Recurrent iterated function systems in fractal approximation. Constr. Approx., 5, pp.3–31) generalized FIF in the form of recurrent fractal interpolation function (RFIF) whose graph is the attractor of a recurrent iterated function system (RIFS) to fit data set which is piece-wise self-affine. The primary aim of the present research is investigating the RFIF approach and using it for fitting the piece-wise self-affine data set in ℜ2.

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LASOTA, A., and J. MYJAK. "FRACTALS, SEMIFRACTALS AND MARKOV OPERATORS." International Journal of Bifurcation and Chaos 09, no. 02 (February 1999): 307–25. http://dx.doi.org/10.1142/s0218127499000195.

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The paper contains a review of results concerning the theory of iterated function systems (IFS) acting on an arbitrary metric space (without any assumption of compactness). First we discuss IFS acting on sets and we define fractals and semifractals using topological limits. Then we study IFS with probabilities acting on measures and we show a relationship with the theory of Markov operators and Markov processes.

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BAKER, SIMON. "An analogue of Khintchine's theorem for self-conformal sets." Mathematical Proceedings of the Cambridge Philosophical Society 167, no. 3 (August 3, 2018): 567–97. http://dx.doi.org/10.1017/s030500411800049x.

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AbstractKhintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFS's). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and satisfies the open set condition. The divergence condition we introduce incorporates the inhomogeneity present within the IFS. We demonstrate via an example that such an approach is essential. We also formulate an analogue of the Duffin–Schaeffer conjecture and show that it holds for a set of full Hausdorff dimension.Combining our results with the mass transference principle of Beresnevich and Velani [4], we prove a general result that implies the existence of exceptional points within the attractor of our IFS. These points are exceptional in the sense that they are “very well approximated”. As a corollary of this result, we obtain a general solution to a problem of Mahler, and prove that there are badly approximable numbers that are very well approximated by quadratic irrationals.The ideas put forward in this paper are introduced in the general setting of iterated function systems that may contain overlaps. We believe that by viewing iterated function systems from the perspective of metric number theory, one can gain a greater insight into the extent to which they overlap. The results of this paper should be interpreted as a first step in this investigation.

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DUTKAY, DORIN ERVIN, PALLE E. T. JORGENSEN, and GABRIEL PICIOROAGA. "Unitary representations of wavelet groups and encoding of iterated function systems in solenoids." Ergodic Theory and Dynamical Systems 29, no. 6 (March 12, 2009): 1815–52. http://dx.doi.org/10.1017/s0143385708000904.

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AbstractFor points indreal dimensions, we introduce a geometry for general digit sets. We introduce a positional number system where the basis for our representation is a fixedd by dmatrix over ℤ. Our starting point is a given pair (A,𝒟) with the matrixAassumed expansive, and 𝒟 a chosen complete digit set, i.e., in bijective correspondence with the points in ℤd/ATℤd. We give an explicit geometric representation and encoding with infinite words in letters from 𝒟. We show that the attractorX(AT,𝒟) for an affine Iterated Function System (IFS) based on (A,𝒟) is a set of fractions for our digital representation of points in ℝd. Moreover our positional ‘number representation’ is spelled out in the form of an explicit IFS-encoding of a compact solenoid 𝒮Aassociated with the pair (A,𝒟). The intricate part (Theorem 6.15) is played by the cycles in ℤdfor the initial (A,𝒟)-IFS. Using these cycles we are able to write down formulas for the two maps which do the encoding as well as the decoding in our positional 𝒟-representation. We show how some wavelet representations can be realized on the solenoid, and on symbolic spaces.

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BARNSLEY, MICHAEL F., and ANDREW VINCE. "THE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM." Bulletin of the Australian Mathematical Society 88, no. 2 (June 11, 2013): 267–79. http://dx.doi.org/10.1017/s0004972713000348.

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AbstractWe investigate the topological and metric properties of attractors of an iterated function system (IFS) whose functions may not be contractive. We focus, in particular, on invertible IFSs of finitely many maps on a compact metric space. We rely on ideas of Kieninger [Iterated Function Systems on Compact Hausdorff Spaces (Shaker, Aachen, 2002)] and McGehee and Wiandt [‘Conley decomposition for closed relations’, Differ. Equ. Appl. 12 (2006), 1–47] restricted to what is, in many ways, a simpler setting, but focused on a special type of attractor, namely point-fibred invariant sets. This allows us to give short proofs of some of the key ideas.

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MELNIKOV, ALEXANDER V. "APPROXIMATION OF FRACTAL SETS BY JULIA SET ATTRACTORS OF POLYNOMIAL ITERATED FUNCTION SYSTEMS." Fractals 07, no. 01 (March 1999): 41–49. http://dx.doi.org/10.1142/s0218348x99000062.

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A novel computational solution of the fractal inverse problem is suggested. The method presented is based on the idea of approximating fractal sets by Julia set attractors of polynomial Iterated Function Systems (IFS). An implementation of this method has shown good results and stable convergence both for polynomial Julia sets and other sets with similar features.

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Nikitenko, Oleg V. "Nonregular Dynamics. Barnsley Iterated Function Systems (Barnsley's IFS) and Fractal Filling." Journal of Automation and Information Sciences 30, no. 6 (1998): 90–102. http://dx.doi.org/10.1615/jautomatinfscien.v30.i6.100.

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BAHAR, SONYA. "TIME-DELAY EMBEDDINGS OF IFS ATTRACTORS." Fractals 07, no. 02 (June 1999): 133–38. http://dx.doi.org/10.1142/s0218348x99000153.

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A modified type of iterated function system (IFS) has recently been shown to generate images qualitatively similar to "classical" chaotic attractors. Here, we use time-delay embedding reconstructions of time-series from this system to generate three-dimentional projections of IFS attractors. These reconstructions may be used to access the topological structure of the periodic orbits embedded within the attractor. This topological characterization suggests an approach by which a rigorous comparison of IFS attractors and classical chaotic systems may be attained.

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JORGENSEN, PALLE E. T. "ITERATED FUNCTION SYSTEMS, REPRESENTATIONS, AND HILBERT SPACE." International Journal of Mathematics 15, no. 08 (October 2004): 813–32. http://dx.doi.org/10.1142/s0129167x04002569.

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In this paper, we are concerned with spectral-theoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τi, i=1,…,N, in some Hausdorff space Y. There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X⊂Y. Typically, some kind of contractivity property for the maps τi is assumed, but our present considerations relax this restriction. This means that there is then not a natural equilibrium measure μ available which allows us to pass the point-maps τi to operators on the Hilbert space L2(μ). Instead, we show that it is possible to realize the maps τi quite generally in Hilbert spaces ℋ(X) of square-densities on X. The elements in ℋ(X) are equivalence classes of pairs (φ,μ), where φ is a Borel function on X, μ is a positive Borel measure on X, and ∫X|φ|2 dμ<∞. We say that (φ,μ)~(ψ,ν) if there is a positive Borel measure λ such that μ≪λ, ν≪λ, and [Formula: see text] We prove that, under general conditions on the system (X,τi), there are isometries [Formula: see text] in ℋ(X) satisfying [Formula: see text] the identity operator in ℋ(X). For the construction we assume that some mapping σ:X→X satisfies the conditions σ◦τi= id X, i=1,…,N. We further prove that this representation in the Hilbert space ℋ(X) has several universal properties.

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Xu, You. "Fractal n-hedral tilings of ℝd." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 66, no. 3 (June 1999): 403–17. http://dx.doi.org/10.1017/s1446788700036697.

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AbstractAn n-hedral tiling of ℝd is a tiling with each tile congruent to one of the n distinct sets. In this paper, we use the iterated function systems (IFS) to generate n-hedral tilings of ℝd. Each tile in the tiling is similar to the attractor of the IFS. These tiles are fractals and their boundaries have the Hausdorff dimension less than d. Our results generalize a result of Bandt.

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RI, SONGIL. "A NEW CONSTRUCTION OF THE FRACTAL INTERPOLATION SURFACE." Fractals 23, no. 04 (December 2015): 1550043. http://dx.doi.org/10.1142/s0218348x15500437.

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In this paper, we introduce a new construction of the fractal interpolation surface (FIS) using an even more general iterated function systems (IFS) which can generate self-affine and non self-affine fractal surfaces. Here we present the general types of fractal surfaces that are based on nonlinear IFSs.

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ZHANG, TONG, and ZHUO ZHUANG. "MULTI-DIMENSIONAL SELF-AFFINE FRACTAL INTERPOLATION MODEL." Advances in Complex Systems 09, no. 01n02 (March 2006): 133–46. http://dx.doi.org/10.1142/s0219525906000641.

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Iterated function system (IFS) models have been explored to represent discrete sequences where the attractor of an IFS is self-affine either in R2 or R3 (R is the set of real numbers). In this paper, the self-affine IFS model is extended from R3 to Rn (n is an integer and greater than 3), which is called the multi-dimensional self-affine fractal interpolation model. This new model is presented by introducing the defined parameter "mapping partial derivative." A constrained inverse algorithm is given for the identification of the model parameters. The values of new model depend continuously on all of the variables. That is, the function is determined by the coefficients of the possibly multi-dimensional affine maps. So the new model is presented as much more general and significant.

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ISLAM, MD SHAFIQUL, and STEPHEN CHANDLER. "APPROXIMATION BY ABSOLUTELY CONTINUOUS INVARIANT MEASURES OF ITERATED FUNCTION SYSTEMS WITH PLACE-DEPENDENT PROBABILITIES." Fractals 23, no. 04 (December 2015): 1550038. http://dx.doi.org/10.1142/s0218348x15500383.

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Let [Formula: see text] be the attractor (fractal) of a contractive iterated function system (IFS) with place-dependent probabilities. An IFS with place-dependent probabilities is a random map [Formula: see text] where the probabilities [Formula: see text] of switching from one transformation to another are functions of positions, that is, at each step, the random map [Formula: see text] moves the point [Formula: see text] to [Formula: see text] with probability [Formula: see text]. If the random map [Formula: see text] has a unique invariant measure [Formula: see text], then the support of [Formula: see text] is the attractor [Formula: see text]. For a bounded region [Formula: see text], we prove the existence of a sequence [Formula: see text] of IFSs with place-dependent probabilities whose invariant measures [Formula: see text] are absolutely continuous with respect to Lebesgue measure. Moreover, if [Formula: see text] is a compact metric space, we prove that [Formula: see text] converges weakly to [Formula: see text] as [Formula: see text] We present examples with computations.

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YUAN, ZHIHUI. "SHRINKING TARGET PROBLEM FOR RANDOM IFS." Fractals 26, no. 06 (December 2018): 1850085. http://dx.doi.org/10.1142/s0218348x18500858.

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We describe the shrinking target problem for random iterated function systems which are semi-conjugate to random subshifts. We get the Hausdorff dimension of the set based on shrinking target problems with given targets. The main idea is an extension of ubiquity theorem which plays an important role to get the lower bound of the dimension. Our method can be used to deal with the sets with respect to more general targets and the sets based on the quantitative Poincaré recurrence properties.

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35

Forte, B., M. Lo Schiavo, and E. R. Vrscay. "Continuity properties of attractors for iterated fuzzy set systems." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 36, no. 2 (October 1994): 175–93. http://dx.doi.org/10.1017/s0334270000010341.

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AbstractAn N-map Iterated Fuzzy Set System (IFZS), introduced in [4] and to be denoted as (w, Φ), is a system of N contraction maps wi: X → X over a compact metric space (X, d), with associated “grey level” maps øi: [0, 1] → [0, 1]. Associated with an IFZS (w, Φ) is a fixed point u ∈ f*(X), the class of normalized fuzzy sets on X, u: X → [0, 1]. We are concerned with the continuity properties of u with respect to changes in the wi, and the φi. Establishing continuity for the fixed points of IFZS is more complicated than for traditional Iterated Function Systems (IFS) with probabilities since a composition of functions is involved. Continuity at each specific attractor u may be established over a suitably restricted domain of φi maps. Two applications are (i) animation of images and (ii) the inverse problem of fractal construction.

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Bagheri, N., L. R. Knudsen, M. Henricksen, B. Sadeghyian, and M. Naderi. "Cryptanalysis of an Iterated Halving-based hash function: CRUSH." IET Information Security 3, no. 4 (December 1, 2009): 129–38. http://dx.doi.org/10.1049/iet-ifs.2009.0055.

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Nevart A. Minas and Faten H. Al-Qadhee. "Digital Video Compression Using DCT-Based Iterated Function System (IFS)." Tikrit Journal of Pure Science 22, no. 6 (January 30, 2023): 125–30. http://dx.doi.org/10.25130/tjps.v22i6.800.

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Large video files processing involves a huge volume of data. The codec, storage systems and network needs resource utilization, so it becomes important to minimize the used memory space and time to distribute these videos over the Internet using compression techniques. Fractal image and video compression falls under the category of lossy compression. It gives best results when used for natural images. This paper presents an efficient method to compress an AVI (Audio Video Interleaved) file with fractal video compression(FVC). The video first is separated into a sequence of frames that are a color bitmap images, then images are transformed from RGB color space to Luminance/Chrominance components (YIQ) color space; each of these components is compressed alone with Enhanced Partition Iterated Function System (EPIFS), then fractal codes are saved. The classical IFS suffers from a very long encoding time that needed to find the best matching for each range block when compared with the domain image blocks. In this work, the (FVC) is enhanced by enhancing the IFS of the fractal image compression using a classification scheme based on the Discrete Cosine Transform (DCT). Experimentation is performed by considering different block sizes and jump steps to reduce number of the tested domain blocks. Results shows a significant reduction in the encoding time with good quality and high compression ratio for different video files.

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ZAMMOURI, IKBAL, ERIC TOSAN, and BÉCHIR AYEB. "A THEORETICAL FRAMEWORK MAPPING GRAMMAR BASED SYSTEMS AND FRACTAL DESCRIPTION." Fractals 16, no. 04 (December 2008): 389–401. http://dx.doi.org/10.1142/s0218348x08004058.

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In this work, we propose a new method to describe fractal shapes using parametric L-systems. This method consists of introducing scaling factors in the production rules of the parametric L-system grammars. We present a turtle monoid on which we base our calculations to show the exact mathematical relation between L-systems and iterated function systems (IFS); we then establish the conditions for the scaling factors to produce plants' and curves' fractal shapes from parametric L-systems. We demonstrate that with specific values of the scaling factors, we find the exact relationship established by Prusinkiewicz and Hammel between L-systems and IFS. Finally, we present some examples of fractal plant forms and curves created using parametric L-systems with scaling factors.

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Deng, Fei-Fei, and Yuan-Ling Ye. "Vector-valued Ruelle operator for weakly contractive IFS and Dini matrix potentials." Nonlinearity 36, no. 7 (May 31, 2023): 3661–83. http://dx.doi.org/10.1088/1361-6544/acd21e.

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Abstract The (scalar) Ruelle operator theory is well-known both in fractal geometry and dynamical systems. In this paper we consider vector-valued Ruelle operators for weakly contractive iterative function systems associated with Dini matrix potentials. We generalize the result in the paper (J. Math. Anal. Appl. 299 341–56) to weakly contractive iterated function system. More exactly, our main theorem gives a sufficient condition for the vector-valued Ruelle operator with the Perron–Frobenius property.

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STANKEWITZ, RICH. "UNIFORMLY PERFECT ANALYTIC AND CONFORMAL ATTRACTOR SETS." Bulletin of the London Mathematical Society 33, no. 3 (May 2001): 320–30. http://dx.doi.org/10.1017/s0024609301007950.

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Conditions are given which imply that analytic iterated function systems (IFSs) in the complex plane [Copf ] have uniformly perfect attractor sets. In particular, it is shown that the attractor set of a finitely generated conformal IFS is uniformly perfect when it contains two or more points. Also, an example is given of a finitely generated analytic attractor set which is not uniformly perfect.

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ZHANG, TONG, JIANLIN LIU, and ZHUO ZHUANG. "MULTI-DIMENSIONAL PIECE-WISE SELF-AFFINE FRACTAL INTERPOLATION MODEL IN TENSOR FORM." Advances in Complex Systems 09, no. 03 (September 2006): 287–93. http://dx.doi.org/10.1142/s0219525906000756.

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Iterated Function System (IFS) models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R2 or R3 (R is the set of real numbers). In this paper, the piece-wise self-affine IFS model is extended from R3 to Rn (n is an integer greater than 3), which is called the multi-dimensional piece-wise self-affine fractal interpolation model. This model uses a "mapping partial derivative" and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the model parameters, and represent most data which are not multi-dimensional self-affine in Rn. Therefore, the result is very general. Moreover, the multi-dimensional piece-wise self-affine fractal interpolation model in tensor form is more terse than in the usual matrix form.

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Druoton, L., L. Garnier, and R. Langevin. "Iterative construction of Dupin cyclide characteristic circles using non-stationary Iterated Function Systems (IFS)." Computer-Aided Design 45, no. 2 (February 2013): 568–73. http://dx.doi.org/10.1016/j.cad.2012.10.042.

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FORTE, BRUNO, and EDWARD R. VRSCAY. "SOLVING THE INVERSE PROBLEM FOR FUNCTION/IMAGE APPROXIMATION USING ITERATED FUNCTION SYSTEMS II: ALGORITHM AND COMPUTATIONS." Fractals 02, no. 03 (September 1994): 335–46. http://dx.doi.org/10.1142/s0218348x94000430.

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In this paper, we provide an algorithm for the construction of IFSM approximations to a target set [Formula: see text], where X ⊂ RD and µ = m(D) (Lebesgue measure). The algorithm minimizes the squared "collage distance" [Formula: see text]. We work with an infinite set of fixed affine IFS maps wi: X → X satisfying a certain density and nonoverlapping condition. As such, only an optimization over the grey level maps ϕi: R+ → R+ is required. If affine maps are assumed, i.e. ϕi = αit + βi, then the algorithm becomes a quadratic programming (QP) problem in the αi and βi. We can also define a "local IFSM" (LIFSM) which considers the actions of contractive maps wi on subsets of X to produce smaller subsets. Again, affine ϕi maps are used, resulting in a QP problem. Some approximations of functions on [0,1] and images in [0, 1]2 are presented.

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Wu, Zhongke, Mingquan Zhou, and Xingce Wang. "Interactive Modeling of 3D Tree with Ball B-Spline Curves." International Journal of Virtual Reality 8, no. 2 (January 1, 2009): 101–7. http://dx.doi.org/10.20870/ijvr.2009.8.2.2731.

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A novel approach to modeling realistic tree easily through interactive methods based on ball B-Spline Curves (BBSCs) and an efficient graph based data structure of tree model is proposed in the paper. As BBSCs are flexible for modifying, deforming and editing, these methods provide intuitive interaction and more freedom for users to model trees. If conjuncted with other methods like generating tree models through L-systems or iterated function systems (IFS), the models are more realistic and natural through modifying and editing. The method can be applied to the design of bonsai tree models.

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BRINKS, RALPH. "A HYBRID ALGORITHM FOR THE SOLUTION OF THE INVERSE PROBLEM IN FRACTAL INTERPOLATION." Fractals 13, no. 03 (September 2005): 215–26. http://dx.doi.org/10.1142/s0218348x05002866.

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A new algorithm for the solution of the inverse problem concerning interpolatory iterated functions systems (IFS) is presented. It combines the advantages of the greedy least squares approach of Mazel and Hayes, and the self-affinity in the wavelet scalogram. Thus, the algorithm is computationally cheap and yields results with high accuracy.

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Devi, T. T., and K. B. Mangang. "On Topological Shadowing and Chain Properties of IFS on Uniform Spaces." Malaysian Journal of Mathematical Sciences 16, no. 3 (September 26, 2022): 501–15. http://dx.doi.org/10.47836/mjms.16.3.07.

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We define notions such as pseudo-orbit, topological shadowing, and topological chain transitivity of iterated function systems on compact uniform spaces. We prove that these notions are invariant under topological conjugacy on a compact uniform space. For an IFS on a compact uniform space with topological shadowing property, we show that the topological chain transitivity implies topological transitivity. We also show that in a connected compact uniform space, notions such as topological chain mixing, totally topological chain transitive, topological chain transitive, and topological chain recurrent are equivalent.

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Li, Yu Hao, Jing Chun Feng, Y. Li, and Yu Han Wang. "Stochastic Fractal Modeling and Process Planning for Machining Using Two-Dimensional Iterated Function System." Key Engineering Materials 392-394 (October 2008): 575–79. http://dx.doi.org/10.4028/www.scientific.net/kem.392-394.575.

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Self-affine and stochastic affine transforms of R2 Iterated Function System (IFS) are investigated in this paper for manufacturing non-continuous objects in nature that exhibit fractal nature. A method for modeling and fabricating fractal bio-shapes using machining is presented. Tool path planning algorithm for numerical control machining is presented for the geometries generated by our fractal generation function. The tool path planning algorithm is implemented on a CNC machine, through executing limited number of iteration. This paper describes part of our ongoing research that attempts to break through the limitation of current CAD/CAM and CNC systems that are oriented to Euclidean geometry objects.

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VASS, JÓZSEF. "ON INTERSECTING IFS FRACTALS WITH LINES." Fractals 22, no. 04 (November 12, 2014): 1450014. http://dx.doi.org/10.1142/s0218348x14500145.

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IFS fractals — the attractors of Iterated Function Systems — have motivated plenty of research to date, partly due to their simplicity and applicability in various fields, such as the modeling of plants in computer graphics, and the design of fractal antennas. The statement and resolution of the Fractal-Line Intersection Problem is imperative for a more efficient treatment of certain applications. This paper intends to take further steps towards this resolution, building on the literature. For the broad class of hyperdense fractals, a verifiable condition guaranteeing intersection with any line passing through the convex hull of a planar IFS fractal is shown, in general ℝd for hyperplanes. The condition also implies a constructive algorithm for finding the points of intersection. Under certain conditions, an infinite number of approximate intersections are guaranteed, if there is at least one. Quantification of the intersection is done via an explicit formula for the invariant measure of IFS.

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BOORE, G. C., and K. J. FALCONER. "Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure." Mathematical Proceedings of the Cambridge Philosophical Society 154, no. 2 (October 31, 2012): 325–49. http://dx.doi.org/10.1017/s0305004112000576.

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AbstractFor directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known.

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Li, Jian, Chun Liang Zhang, and Xia Yue. "A Compression Algorithm for Vibration Signal of Rotating Machinery Based on Partitioned Iterated Function Systems." Advanced Materials Research 139-141 (October 2010): 1718–22. http://dx.doi.org/10.4028/www.scientific.net/amr.139-141.1718.

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The partitioned iterated function systems (PIFS) were introduced into the compression of vibration signal. The actual vibration data of ZLH600-2 pump were adopted to verify the performance of PIFS. Also the compression ratios and the computation time were good, but the spectrum and amplitude, as important performances, were deformed after compression. If the concerned frequency of users was significantly lower than the sampling frequency and the required compression ratios was not more than 2, the compression using PIFS in vibration signal of rotating machinery was comfortable. Otherwise, the information loss in the compression could not be ignored. The decoded signals with the different compression parameters were listed in the paper and it was a meaningful exploration of the IFS on diagnosis and laid the foundation for further research.

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