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About 4,259 papers for “Iterated function”

Zooming Using Iterated Function Systems

Eric Polidori, Jean‐Luc Dugelay · Fractals · 1997

Iterated Function Systems (I.F.S.) have been recently studied in the field of image coding. In addition to compression, I.F.S. possess some properties of fractals which can be used to sub/oversample an image. In this paper, we review how I.F.S. can be used for image zooming, directly from basic algorithms of still image compression. This study shows that results obtained in such a way do not produce better results than those obtained by using classical spatial interpolators such as the linear one. However, in this paper we also show that modified versions of the basic approach provide a better use of I.F.S.'s coding for zooming.

Cited by 30

Locally Contractive Iterated Function Systems

David Steinsaltz · The Annals of Probability · 1999

An iterated function system on $\mathscr{X}\subset\mathbb{R}^d$ is defined by successively applying an i.i.d.sequence of random Lipschitz functions from to $\mathscr{X}$ to $\mathscr{X}$. This paper shows how $F _n = f_1\circ\ldots\circ f_n$ may converge even in the absence of the strong contraction conditions, for instance, Lipschitz constant smaller than 1 on average,which earlier work has required. Instead, it is posited that there be a region of contraction which compensates for the noncontractive or even expansive part of the functions. Applications to queues, to self-modifying random walks and to random logistic maps are given.

Cited by 60

Attractors of Iterated Function Systems

P. F. Duvall, L. S. Husch · Proceedings of the American Mathematical Society · 1992

In this paper, the question of which compact metric spaces can be attractors of hyperbolic iterated function systems on Euclidean space is studied.It is shown that given any finite-dimensional compact metric X , there is a Cantor set C such that the disjoint union of C and X is an attractor.In the process, it is proved that every such X is the Lipschitz image of a Cantor set.

Cited by 1Open access

Directed graph iterated function systems

Falconer, Kenneth J, Olsen, Lars, Boore, Graeme C · OpenGrey (Institut de l'Information Scientifique et Technique) · 2011

This thesis concerns an active research area within fractal geometry. In the first part, in Chapters 2 and 3, for 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. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing moment of μ[subscript(u)], the self-similar measure at a vertex u, for the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any q ∈ ℝ if the strong separation condition holds, (ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μ[subscript(u)] to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μ[subscript(u)], of the set consisting of all the intersections of the components of F[subscript(u)], is strictly less than the multifractal q Hausdorff dimension with respect to μ[subscript(u)] of F[subscript(u)].

Cited by 0Open access

Foundations of local iterated function systems

Elismar R. Oliveira, Paulo Varandas · arXiv (Cornell University) · 2026

In this paper we present a systematic study of continuous local iterated function systems. We prove local iterated function systems admit compact attractors and, under a contractivity assumption, construct their code space and present an extended shift that describes admissible compositions. In particular, the possible combinatorial structure of a local iterated function system is in bijection with the space of invariant subsets of the full shift. Nevertheless, these objects reveal a degree of unexpectedness relative to the classical framework, as we build examples of local iterated function systems which are not modeled by subshifts of finite type and give rise to non self-similar attractors. We also prove that all attractors of graph-directed IFSs are obtained from local IFSs on an enriched compact metric space. We provide several classes of examples illustrating the scope of our results, emphasizing both their contrasts and connections with the classical theory of iterated function systems.

Cited by 0Open access

Fractal capacities and iterated function systems

Никифорчин, Олег · Mathematical Bulletin of the Scientific Society Shevchenko (Instytut Prykladnykh Problem Mekhaniky I Matematyky Im. Ya. S. Pidstryhacha) · 2013

Cited by 0Open access

Iterated function systems: transitivity and minimality

Hadi Parham, F.H. Ghane, Azam Ehsani · Boletim da Sociedade Paranaense de Matemática · 2019

In this paper, we study the chaotic dynamics of iterated function systems (IFSs) generated by a finite family of maps on a compact metric space. In particular, we restrict ourselves to topological transitivity, fiberwise transitivity, minimality and total minimality of IFSs. First, we pay special attention to the relation between topological transitivity and fiberwise transitivity. Then we generalize the concept of periodic decompositions of continuous maps, introduced by John Banks [1], to iterated function systems. We will focus on the existence of periodic decompositions for topologically transitive IFSs. Finally, we show that each minimal abelian iterated function system generated by a finite family of homeomorphisms on a connected compact metric space X is totally minimal.

Cited by 1Open access

Attractors of topological iterated function system

Dan Dumitru · 2013

The aim of this paper is to prove that the union of a Peano space and a segment with whose intersection is nonempty is the attractor of a topological iterated function system formed by three functions.

Cited by 0

Invariant measures for iterated function systems

Tomasz Szarek · Annales Polonici Mathematici · 2000

A new criterion for the existence of an invariant distribution for Markov operators is presented. Moreover, it is also shown that the unique invariant distribution of an iterated function system is singular with respect to the Hausdorff measure.

Cited by 10Open access

Multifractal Decompositions using Iterated Function Systems

J. L. del Río-Correa, J López García, G Durán Meza · Journal of Physics Conference Series · 2019

Abstract We analyze two types of multifractal decompositions (MD) of fractals F generated by an Iterated Function System (IFS), they are the geometric and the statistical MD, the first is generated by an IFS and the second by an IFS with probability. In the first, F is decomposed in subsets M(φ) of points characterized by the same vector frequency φ, and we evaluate their Hausdorff dimension (HD). In the second, F is decomposed in subsets Jα of points with the same pointwise dimension α; however Jα is composed by an infinite subsets M(φ), therefore Jα is a multifractal, this implies that its HD is the maximum HD of its components M(φ), using a maximizing procedure we find φ∗ such that HD of M(φ∗) is greater than any other M(φ) for a fixed α, this procedure gives in a natural form the auxiliary functions proposed by Cawlin and Mauldin. Thus we present a more simple description of the MD.

Cited by 2Open access