scholarly journals Multi Fractals of Generalized Multivalued Iterated Function Systems in b-Metric Spaces with Applications

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 967 ◽  
Author(s):  
Sudesh Kumari ◽  
Renu Chugh ◽  
Jinde Cao ◽  
Chuangxia Huang
Keyword(s):  
Inverse Problem ◽  
Fixed Points ◽  
Natural Number ◽  
Metric Space ◽  
Metric Spaces ◽  
Iterated Function System ◽  
Hausdorff Metric ◽  
Iterated Function Systems ◽  
Iterated Function ◽  
Collage Theorem

In this paper, we obtain multifractals (attractors) in the framework of Hausdorff b-metric spaces. Fractals and multifractals are defined to be the fixed points of associated fractal operators, which are known as attractors in the literature of fractals. We extend the results obtained by Chifu et al. (2014) and N.A. Secelean (2015) and generalize the results of Nazir et al. (2016) by using the assumptions imposed by Dung et al. (2017) to the case of ciric type generalized multi-iterated function system (CGMIFS) composed of ciric type generalized multivalued G-contractions defined on multifractal space C ( U ) in the framework of a Hausdorff b-metric space, where U = U 1 × U 2 × ⋯ × U N , N being a finite natural number. As an application of our study, we derive collage theorem which can be used to construct general fractals and to solve inverse problem in Hausdorff b-metric spaces which are more general spaces than Hausdorff metric spaces.

scholarly journals Applications of Multivalued Contractions on Graphs to Graph-Directed Iterated Function Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-16
Author(s):  
T. Dinevari ◽  
M. Frigon
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Metric Space ◽  
Metric Spaces ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Complete Metric Spaces ◽  
Fixed Point Result ◽  
Iterated Function ◽  
Multivalued Contractions

We apply a fixed point result for multivalued contractions on complete metric spaces endowed with a graph to graph-directed iterated function systems. More precisely, we construct a suitable metric space endowed with a graphGand a suitableG-contraction such that its fixed points permit us to obtain more information on the attractor of a graph-directed iterated function system.

Attractors of Infinite Iterated Function Systems Containing Con

2013 ◽  
Vol 59 (2) ◽  
pp. 281-298
Author(s):  
Dan Dumitru
Keyword(s):  
Metric Space ◽  
Existence And Uniqueness ◽  
Iterated Function System ◽  
Sufficient Conditions ◽  
Infinite Family ◽  
Iterated Function Systems ◽  
Function System ◽  
Iterated Function

Abstract We consider a complete ε-chainable metric space (X, d) and an infinite iterated function system (IIFS) formed by an infinite family of (ε, φ)-functions on X. The aim of this paper is to prove the existence and uniqueness of the attractors of such infinite iterated systems (IIFS) and to give some sufficient conditions for these attractors to be connected. Similar results are obtained in the case when the IIFS is formed by an infinite family of uniformly ε-locally strong Meir-Keeler functions.

scholarly journals The Tarski–Kantorovitch prinicple and the theory of iterated function systems

2000 ◽  
Vol 61 (2) ◽  
pp. 247-261 ◽  
Author(s):  
Jacek Jachymski ◽  
Leslaw Gajek ◽  
Piotr Pokarowski
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Iterated Function Systems ◽  
Ordered Sets ◽  
Iterated Function ◽  
Fixed Point Principle ◽  
Countably Compact ◽  
Sequential Spaces

We show how some results of the theory of iterated function systems can be derived from the Tarski–Kantorovitch fixed–point principle for maps on partialy ordered sets. In particular, this principle yields, without using the Hausdorff metric, the Hutchinson–Barnsley theorem with the only restriction that a metric space considered has the Heine–Borel property. As a by–product, we also obtain some new characterisations of continuity of maps on countably compact and sequential spaces.

ITERATED FUNCTION SYSTEMS ON FUNCTIONS OF BOUNDED VARIATION

Fractals ◽  
2016 ◽  
Vol 24 (02) ◽  
pp. 1650019 ◽  
Author(s):  
DAVIDE LA TORRE ◽  
FRANKLIN MENDIVIL ◽  
EDWARD R. VRSCAY
Keyword(s):  
Inverse Problems ◽  
Bounded Variation ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Functions Of Bounded Variation ◽  
Complete Space ◽  
Iterated Function ◽  
Collage Theorem ◽  
Contraction Maps

We show that under certain hypotheses, an iterated function system on mappings (IFSM) is a contraction on the complete space of functions of bounded variation (BV). It then possesses a unique attractor of BV. Some BV-based inverse problems based on the Collage Theorem for contraction maps are considered.

scholarly journals Fractals of Generalized F-Hutchinson Operator in b-Metric Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Talat Nazir ◽  
Sergei Silvestrov ◽  
Xiaomin Qi
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Iterated Function System ◽  
Finite Family ◽  
Function System ◽  
Contraction Mappings ◽  
Iterated Function

The aim of this paper is to construct a fractal with the help of a finite family of generalized F-contraction mappings, a class of mappings more general than contraction mappings, defined in the setup of b-metric space. Consequently, we obtain a variety of results for iterated function system satisfying a different set of contractive conditions. Our results unify, generalize, and extend various results in the existing literature.

scholarly journals GPU-Accelerated Rendering of Unbounded Nonlinear Iterated Function System Fixed Points

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Orion Sky Lawlor
Keyword(s):  
Fixed Points ◽  
Iterated Function System ◽  
Nonlinear Function ◽  
Texture Mapping ◽  
Iterated Function Systems ◽  
Function System ◽  
Graphics Hardware ◽  
Iteration Algorithm ◽  
High Definition ◽  
Iterated Function

Nonlinear functions, including nonlinear iterated function systems, have interesting fixed points. We present a non-Lipschitz theoretical approach to nonlinear function system fixed points which generalizes to noncontractive functions, compare several methods for evaluating such fixed points on modern graphics hardware, and present a nonlinear generalization of Barnsley’s Deterministic Iteration Algorithm. Unlike the many existing randomized rendering algorithms, this deterministic method avoids noncoherent branching and memory access and takes advantage of programmable texture mapping hardware. Together with the performance potential of modern graphics hardware, this allows us to animate high-quality and high-definition fixed points in real time.

ON (n, m)-ITERATED FUNCTION SYSTEMS

2013 ◽  
Vol 06 (04) ◽  
pp. 1350055 ◽  
Author(s):  
Rinju Balu ◽  
Sunil Mathew
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Continuity Theorem ◽  
Iterated Function ◽  
Collage Theorem

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.

scholarly journals THE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM

2013 ◽  
Vol 88 (2) ◽  
pp. 267-279 ◽  
Author(s):  
MICHAEL F. BARNSLEY ◽  
ANDREW VINCE
Keyword(s):  
Metric Space ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Invariant Sets ◽  
Hausdorff Spaces ◽  
Compact Metric Space ◽  
Metric Properties ◽  
Iterated Function

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.

scholarly journals Iterated function systems: transitivity and minimality

2019 ◽  
Vol 38 (3) ◽  
pp. 97-109 ◽  
Author(s):  
Hadi Parham ◽  
F. H. Ghane ◽  
A. Ehsani
Keyword(s):  
Metric Space ◽  
Chaotic Dynamics ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Finite Family ◽  
Function System ◽  
Topological Transitivity ◽  
Compact Metric Space ◽  
Iterated Function ◽  
Continuous Maps

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.

The Hutchinson-Barnsley theory for infinite iterated function systems

2005 ◽  
Vol 72 (3) ◽  
pp. 441-454 ◽  
Author(s):  
Gertruda Gwóźdź-Lukawska ◽  
Jacek Jachymski
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Iterated Function System ◽  
Complete Metric Space ◽  
Iterated Function Systems ◽  
Function System ◽  
Iterated Function ◽  
Infinite Case

We show that some results of the Hutchinson-Barnsley theory for finite iterated function systems can be carried over to the infinite case. Namely, if {Fi:i∈ ℕ} is a family of Matkowski's contractions on a complete metric space (X, d) such that (Fix0)i∈Nis bounded for somex0∈X, then there exists a non-empty bounded and separable setKwhich is invariant with respect to this family, that is,. Moreover, given σ ∈ ℕℕandx∈X, the limit exists and does not depend onx. We also study separately the case in which (X, d) is Menger convex or compact. Finally, we answer a question posed by Máté concerning a finite iterated function system {F1,…,FN} with the property that each ofFihas a contractive fixed point.

Sign in / Sign up

Export Citation Format

Share Document