Showing papers on "Iterated function system published in 2017"
TL;DR: In this article, it was shown that every self-affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system, and in higher dimensions, under certain assumptions, under the Ledrappier-Young formula, the measures are exact dimensional.
55 citations
01 Mar 2017
TL;DR: In this paper, the authors studied weakly hyperbolic iterated function systems on compact metric spaces and proved the existence of attractors, both in the topological and measure theoretical viewpoint and the ergodicity of invariant measure.
Abstract: We study weakly hyperbolic iterated function systems on compact metric spaces, as defined by Edalat (Inform Comput 124(2):182–197, 1996), but in the more general setting of compact parameter space. We prove the existence of attractors, both in the topological and measure theoretical viewpoint and the ergodicity of invariant measure. We also define weakly hyperbolic iterated function systems for complete metric spaces and compact parameter space, extending the above mentioned definition. Furthermore, we study the question of existence of attractors in this setting. Finally, we prove a version of the results by Barnsley and Vince (Ergodic Theory Dyn Syst 31(4):1073–1079, 2011), about drawing the attractor (the so-called the chaos game), for compact parameter space.
19 citations
12 Sep 2017
TL;DR: In this paper, a new model of random graph directed fractals is proposed, and the Hausdorff and upper box counting dimension coincide almost surely, irrespective of overlap, under the assumption of the uniform strong separation condition.
Abstract: In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We study its dimensional properties for similarities with and without overlaps. In particular we show that for the two classes of $1$-variable and $\infty$-variable random graph directed attractors we introduce, the Hausdorff and upper box counting dimension coincide almost surely, irrespective of overlap. Under the additional assumption of the uniform strong separation condition we give an expression for the almost sure Hausdorff and Assouad dimension.
14 citations
TL;DR: In this paper, the authors point out an error in proving some results on finite iterated function systems consisting of Kannan maps, Reich maps and Chatterjea type maps.
Abstract: In this paper we will point out an error in proving some results on finite iterated function systems consisting of Kannan maps, Reich maps and Chatterjea type maps. In this respect, some counter-examples are given. We also answer some open questions on iterated function systems consisting of contractive maps of Reich type and we present revisions of some theorems on iterated function systems consisting of Kannan, Reich and Chatterjea type maps, by adding a commutativity assumption on the maps.
14 citations
TL;DR: In this article, an infinite graph-directed iterated function system associated with finite iterated functions is constructed, and a new approach for proving the differentiability of the L q -spectrum and establishing the multifractal formalism of certain self-similar measures with overlaps is developed.
13 citations
24 Mar 2017
11 citations
TL;DR: The kinetic equations of DNA replication are shown to be exactly solved in terms of iterated function systems, running along the template sequence and giving the statistical properties of the copy sequences, as well as the kinetic and thermodynamic properties ofThe replication process.
Abstract: The kinetic equations of DNA replication are shown to be exactly solved in terms of iterated function systems, running along the template sequence and giving the statistical properties of the copy sequences, as well as the kinetic and thermodynamic properties of the replication process. With this method, different effects due to sequence heterogeneity can be studied, in particular, a transition between linear and sublinear growths in time of the copies, and a transition between continuous and fractal distributions of the local velocities of the DNA polymerase along the template. The method is applied to the human mitochondrial DNA polymerase $\ensuremath{\gamma}$ without and with exonuclease proofreading.
10 citations
TL;DR: In this paper, the authors present iterated function systems on a compact metric space consisting of $$\varphi $$ -contractions and prove that such a system necessarily has an associated fractal set and an associated measure.
Abstract: In present times, there has been a considerable effort to generalize the classical notion of iterated function system. We’ll present in this paper iterated function systems on a compact metric space consisting of $$\varphi $$
-contractions and prove that such an iterated function system necessarily has an associated fractal set and an associated fractal measure.
9 citations
TL;DR: In this article, it was shown that the set of all the unique codings is a sub-shift of finite type if and only if it is a sofic shift.
Abstract: Let be a self-similar set generated by some iterated function system. In this paper we prove, under some assumptions, that K can be identified with a subshift of finite type. With this identification, we can calculate the Hausdorff dimension of K as well as the set of elements in K with unique codings using the machinery of Mauldin and Williams (1988 Trans. Am. Math. Soc. 309 811–29). We give three different applications of our main result. Firstly, we calculate the Hausdorff dimension of the set of points of K with multiple codings. Secondly, in the setting of β-expansions, when the set of all the unique codings is not a subshift of finite type, we can calculate in some cases the Hausdorff dimension of the univoque set. Motivated by this application, we prove that the set of all the unique codings is a subshift of finite type if and only if it is a sofic shift. This equivalent condition was not mentioned by de Vries and Komornik (2009 Adv. Math. 221 390–427, theorem 1.8). Thirdly, for the doubling map with asymmetrical holes, we give a sufficient condition such that the survivor set can be identified with a subshift of finite type. The third application partially answers a problem posed by Alcaraz Barrera (2014 PhD Thesis University of Manchester).
8 citations
TL;DR: In this paper, Secelean extended these considerations to mappings defined on the space ∑ ∞ ( X ) of all bounded sequences of elements of X and obtained versions of the Hutchinson-Barnsley theorem for appropriate families of such functions.
8 citations
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TL;DR: In this paper, it was shown that any iterated function system of circle homeomorphisms with at least one of them having dense orbit is asymptotically stable, and the corresponding Perron-Frobenius operator is shown to satisfy the e-property.
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.
04 Oct 2017
TL;DR: A general class of unital associative complex algebras, defined by commutation relations associated with group or semigroup actions of dynamical systems and iterated function systems, is considered in this article.
Abstract: A general class of multi-parametric families of unital associative complex algebras, defined by commutation relations associated with group or semigroup actions of dynamical systems and iterated function systems, is considered. A generalization of these commutation relations in three generators is also considered, modifying Lie algebra type commutation relations, typical for usual differential or difference operators, to relations satisfied by more general twisted difference operators associated with general twisting maps. General reordering and nested commutator formulas for arbitrary elements in these algebras are presented, and some special cases are considered, generalizing some well-known results in mathematics and physics.
TL;DR: In this article, the authors consider a class of iterated function systems (IFSs) of contracting similarities of Rn, introduced by Hutchinson, for which the invariant set possesses a natural Holder continuous parameterization by the unit interval.
Abstract: We consider a class of iterated function systems (IFSs) of contracting similarities of Rn, introduced by Hutchinson, for which the invariant set possesses a natural Holder continuous parameterization by the unit interval. When such an invariant set is homeomorphic to an interval, we give necessary conditions in terms of the similarities alone for it to possess a quasisymmetric (and as a corollary, bi-Holder) parameterization. We also give a related necessary condition for the invariant set of such an IFS to be homeomorphic to an interval.
TL;DR: In this paper, the authors consider two dynamical systems associated to the nearest integer continued fraction and show that both of them have full Hausdorff dimension spectrum, which is the same as the one we consider in this paper.
05 Oct 2017
TL;DR: In this paper, random independent and identically distributed iterations of functions from an iterated function system of homeomorphisms on the circle which is minimal are analyzed in terms of iterated functions with probabilities which are nonexpansive on average.
Abstract: We study random independent and identically distributed iterations of functions from an iterated function system of homeomorphisms on the circle which is minimal. We show how such systems can be analyzed in terms of iterated function systems with probabilities which are non-expansive on average.
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TL;DR: In this paper, the authors investigated the pressure function and affinity dimension for iterated function systems associated to the "box-like" selfaffine fractals investigated by D.-J. Feng, Y. Wang and J.M. Fraser.
Abstract: In this article we investigate the pressure function and affinity dimension for iterated function systems associated to the "box-like" self-affine fractals investigated by D.-J. Feng, Y. Wang and J.M. Fraser. Combining previous results of V. Yu. Protasov, A. K\"aenm\"aki and the author we obtain an explicit formula for the pressure function which makes it straightforward to compute the affinity dimension of box-like self-affine sets. We also prove a variant of this formula which allows the computation of a modified singular value pressure function defined by J.M. Fraser. We give some explicit examples where the Hausdorff and packing dimensions of a box-like self-affine fractal may be easily computed.
TL;DR: In this article, a fractal theory of the Julia sets of infinitely generated semigroups of rational maps was established, and a new class of semigroup which is called nicely expanding rational semiigroups was introduced, and Bowen's formula for the Hausdorff dimension of the pre-Julia sets was proved.
Abstract: We investigate the dynamics of semigroups of rational maps on the Riemann sphere. To establish a fractal theory of the Julia sets of infinitely generated semigroups of rational maps, we introduce a new class of semigroups which we call nicely expanding rational semigroups. More precisely, we prove Bowen’s formula for the Hausdorff dimension of the pre-Julia sets, which we also introduce in this paper. We apply our results to the study of the Julia sets of non-hyperbolic rational semigroups. For these results, we do not assume the cone condition, which has been assumed in the study of infinite contracting iterated function systems. Similarly, we show that Bowen’s formula holds for the limit set of a contracting conformal iterated function system without the cone condition.
TL;DR: In this article, the authors studied the pressure function of a conformal iterated function system and proved that it is Gateaux differentiable and the multifractal formalism holds, if the IFS has nonoverlapping.
Abstract: Let X = [0,1]. Given a non-uniformly contracting conformal iterated function system (IFS) and a family of positive Dini continuous potential functions , the triple system , under some conditions, determines uniquely a probability invariant measure, denoted by μ. In this paper, we study the pressure function of the system and multifractal structure of μ. We prove that the pressure function is Gateaux differentiable and the multifractal formalism holds, if the IFS has non-overlapping.
TL;DR: A new type of C 1 Fractal Interpolation Function (FIF) is developed using the Iterated Function System (IFS) which contains the rational spline.
TL;DR: In this paper, it was shown that the family of cylinders of generalized Luroth expansions are not faithful for the Hausdorff dimension calculation, and sufficient conditions for the non-faithfulness of such families of cylinders are also found.
Abstract: We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. We show that the systems of cylinders of generalized Luroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' approach, we prove sufficient conditions for the non-faithfulness of such families of cylinders. On the other hand, rather general sufficient conditions for the faithfulness of such covering systems are also found. As a corollary, we obtain the non-faithfullness of the family of cylinders generated by the classical Luroth expansion.
We also develop new approach to the study of subsets of Q∞-essentially non-normal numbers and prove that this set has full Hausdorff dimension. This result answers the open problem mentioned in [2] and completes the metric, dimensional and topological classification of real numbers via the asymptotic behaviour of frequencies their digits in the generalized Luroth expansion.
17 Jan 2017
TL;DR: The constrained bivariate interpolation discussed herein includes a method to construct fractal interpolation surfaces that preserve positivity inherent in a prescribed data set that are constrained suitably such that these fractal boundary curves areabove the plane whenever the given interpolation data along the grid lines are above the plane.
Abstract: This paper addresses a method to obtain rational cubic fractal functions, which generate surfaces that lie above a plane via blending functions. In particular, the constrained bivariate interpolation discussed herein includes a method to construct fractal interpolation surfaces that preserve positivity inherent in a prescribed data set. The scaling factors and shape parameters involved in fractal boundary curves are constrained suitably such that these fractal boundary curves are above the plane whenever the given interpolation data along the grid lines are above the plane. Our rational cubic spline FIS is above the plane whenever the corresponding fractal boundary curves are above the plane. We illustrate our interpolation scheme with some numerical examples.
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TL;DR: It is shown that the algorithm reconstructs well-known fractals from data, with the model converging to high precision parameters, and the utility of the model as an approximation for datasources outside the IFS model class is shown.
Abstract: We present an Expectation-Maximization algorithm for the fractal inverse problem: the problem of fitting a fractal model to data. In our setting the fractals are Iterated Function Systems (IFS), with similitudes as the family of transformations. The data is a point cloud in ${\mathbb R}^H$ with arbitrary dimension $H$. Each IFS defines a probability distribution on ${\mathbb R}^H$, so that the fractal inverse problem can be cast as a problem of parameter estimation. We show that the algorithm reconstructs well-known fractals from data, with the model converging to high precision parameters. We also show the utility of the model as an approximation for datasources outside the IFS model class.
03 Jul 2017
TL;DR: This paper uses IFS to model Speech and Electroencephalographic signals and compares the results, showing that the estimation algorithm has a very good convergence to the global minimum and the best tradeoff between performance and computational complexity.
Abstract: In this paper, we describe an algorithm to estimate the parameters of Iterated Function System (IFS) fractal models. We use IFS to model Speech and Electroencephalographic signals and compare the results. The IFS parameters estimation is performed by means of a genetic optimization approach. We show that the estimation algorithm has a very good convergence to the global minimum. This can be successfully exploited by pattern recognition tools. However, the set-up of the genetic algorithm should be properly tuned. In this paper, besides the optimal set-up description, we describe also the best tradeoff between performance and computational complexity. To simplify the optimization problem some constraints are introduced. A comparison with suboptimal algorithms is reported. The performance of IFS modeling of the considered signals are in accordance with known measures of the fractal dimension.
30 May 2017
TL;DR: In this article, the concept of weakly hyperbolic iterated function systems on compact metric spaces has been considered and the existence and uniqueness of the invariant measure of a $$\mathbb {P}$$� -weakly hyper-bolic IFS with compact parameter space has been proved.
Abstract: In this paper we will consider the concept of $$\mathbb {P}$$
-weakly hyperbolic iterated function systems on compact metric spaces that generalizes the concept of weakly hyperbolic iterated function systems, as defined by Edalat (Inf Comput 124(2):182–197, 1996) and by Arbieto, Santiago and Junqueira (Bull Braz Math Soc New Ser 2016) for a more general setting where the parameter space is a compact metric space. We prove the existence and uniqueness of the invariant measure of a $$\mathbb {P}$$
-weakly hyperbolic IFS. Furthermore, we prove an ergodic theorem for $$\mathbb {P}$$
-weakly hyperbolic IFS with compact parameter space.
TL;DR: In this article, a new notion named thick chaotic IFSs is introduced, and the relationship between thick chaos and another properties of some notions in dynamical systems is studied, including the relation between chaotic properties and the properties of iterated function systems.
Abstract: This paper is devoted to study some chaotic properties of iterated function systems (IFSs). Specially, a new notion named thick chaotic IFSs is introduced. The relationship between thick chaos and another properties of some notions in dynamical systems are studied.
TL;DR: In this article, the authors established a strong invariance principle for unbounded observables on the limit set of a sequential iterated function system and showed that the measure in LIL is continuous to the Hausdorff measure.
Abstract: We establish a law of the iterated logarithm (LIL) for the set of real numbers whose $n$-th partial quotient is bigger than $\alpha_n$, where $(\alpha_n)$ is a sequence such that $\sum 1/\alpha_n$ is finite. This set is shown to have Hausdorff dimension $1/2$ in many cases and the measure in LIL is absolutely continuous to the Hausdorff measure. The result is obtained as an application of a strong invariance principle for unbounded observables on the limit set of a sequential iterated function system.
01 Jan 2017
Journal Article•
TL;DR: In this article, the authors construct a fractal set of Iterated function systems, which in their case are a collection of mappings defined in an extended b-metric space, of compact subsets of the space.
Abstract: Iterated function systems are method of constructing fractals, which are based on the mathematical foundations laid by Hutchinson[1] and Barnsley[2]. Formally an Iterated function systems is a finite set of ‘contraction mappings’, on a complete metric space X. In this paper we construct a fractal set of Iterated function systems, which in our case are a collection of mappings defined in an extended b-metric space, of compact subsets of the space. We will prove that the Hutchinson operator defined with the help of a finite family of ‘generalized F-contraction mappings’ on a complete extended b- metric space is itself a generalized F- contraction mapping on a family of compact subsets of X. Then by successive application of a generalized F-Hutchinson operator we obtain a final fractal in an extended b- metric space
26 Jul 2017
TL;DR: Fractal interpolation surfaces were constructed over triangular as well as over polygonal lattices with triangular subdomains by using affine transformations as mentioned in this paper, and the resulting self-affine or self-similar graphs were generated through a family of discrete dynamic systems, the iterated function system.
Abstract: This chapter provides a brief and coarse discussion on the theory of fractal interpolation functions and their recent developments including some of the research made by the authors. It focuses on fractal interpolation as well as on recurrent fractal interpolation in one and two dimensions. The resulting self-affine or self-similar graphs, which usually have non-integral dimension, were generated through a family of (discrete) dynamic systems, the iterated function system, by using affine transformations. Specifically, the fractal interpolation surfaces presented here were constructed over triangular as well as over polygonal lattices with triangular subdomains. A further purpose of this chapter is the exploration of the existent breakthroughs and their application to a flexible and integrated software that constructs and visualises the above-mentioned models. We intent to supply both a panoramic view of interpolating functions and a useful source of links to assist a novice as well as an expert in fractals. The ideas or findings contained in this paper are not claimed to be exhaustive, but are intended to be read before, or in parallel with, technical papers available in the literature on this subject.