Some considerations on the bifurcation of the fixed point generated by iterated function systems

In this paper, we present a construction of new nonlinear recurrent hidden variable fractal interpolation curves. In order to get new fractal curves, we use Rakotch’s fixed point theorem. We construct recurrent hidden variable iterated function systems with function vertical scaling factors to generate more flexible fractal interpolation curves. We also give an illustrative example to demonstrate the effectiveness of our results.

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Generalized Iterated Function Systems Containing Functions of Integral Type

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i3.31.18280 ◽

2018 ◽

Vol 7 (3.31) ◽

pp. 126

Author(s):

Minirani S ◽

. .

Keyword(s):

Fixed Point ◽

Existence And Uniqueness ◽

Product Space ◽

Iterated Function System ◽

Complete Metric Space ◽

Iterated Function Systems ◽

Function System ◽

Finite Collection ◽

Integral Type ◽

Iterated Function

A finite collection of mappings which are contractions on a complete metric space constitutes an iterated function system. In this paper we study the generalized iterated function system which contain generalized contractions of integral type from the product space . We prove the existence and uniqueness of the fixed point of such an iterated function system which is also known as its attractor.

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

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700022243 ◽

2000 ◽

Vol 61 (2) ◽

pp. 247-261 ◽

Cited By ~ 13

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.

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

Abstract and Applied Analysis ◽

10.1155/2015/345856 ◽

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.

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NEW NONLINEAR RECURRENT HIDDEN VARIABLE FRACTAL INTERPOLATION SURFACES

Fractals ◽

10.1142/s0218348x20500383 ◽

2020 ◽

Vol 28 (02) ◽

pp. 2050038

Author(s):

HYONJIN KIM ◽

JINMYONG KIM ◽

HAKMYONG MUN

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Iterated Function Systems ◽

Vertical Scaling ◽

Fractal Interpolation ◽

Scaling Factors ◽

Fractal Surfaces ◽

Hidden Variable ◽

Iterated Function

In this paper, we present the construction of new nonlinear recurrent hidden variable fractal interpolation surfaces (RHVFISs) with function vertical scaling factors. We use Rakotch’s fixed point theorem which is a generalization of Banach’s fixed point theorem to get new nonlinear fractal surfaces. We construct recurrent vector-valued iterated function systems (IFSs) with function vertical scaling factors on rectangular grids and generate flexible and diverse RHVFISs which are attractors of the IFSs. We also give an explicit example to show the effectiveness of obtained results.

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Invariant sets and Knaster-Tarski principle

Open Mathematics ◽

10.2478/s11533-012-0109-4 ◽

2012 ◽

Vol 10 (6) ◽

Cited By ~ 1

Author(s):

Krzysztof Leśniak

Keyword(s):

Fixed Point ◽

Dynamical Systems ◽

Discrete Time ◽

Iterated Function Systems ◽

Invariant Sets ◽

Iterated Function ◽

Fixed Point Principle

AbstractOur aim is to point out the applicability of the Knaster-Tarski fixed point principle to the problem of existence of invariant sets in discrete-time (multivalued) semi-dynamical systems, especially iterated function systems.

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Another characterization of hyperbolic diameter diminishing to zero IFSs

Carpathian Journal of Mathematics ◽

10.37193/cjm.2021.02.08 ◽

2021 ◽

Vol 37 (2) ◽

pp. 217-226

Author(s):

RADU MICULESCU ◽

ALEXANDRU MIHAIL ◽

CRISTINA-MARIA PĂCURAR

Keyword(s):

Fixed Point ◽

Fixed Point Theory ◽

Iterated Function System ◽

Iterated Function Systems ◽

Point Theory ◽

Function System ◽

Iterated Function ◽

By Products

"In this paper we provide another characterization of hyperbolic diameter diminishing to zero iterated function systems that were studied in [R. Miculescu, A. Mihail, Diameter diminishing to zero IFSs, arXiv:2101.12705]. The primary tool that we use is an operator H_{\mathcal{S}}, associated to the iterated function system \mathcal{S}, which is inspired by the similar one utilized in Mihail (Fixed Point Theory Appl., 2015:75, 2015). Some fixed point results are also obtained as by products of our main result."

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Inventing the Formula of the Trees: A Solution of the Representation of Self Similar Objects

Fractals ◽

10.1142/s0218348x97000632 ◽

1997 ◽

Vol 05 (supp01) ◽

pp. 51-64

Author(s):

Erwin Hocevar ◽

Walter G. Kropatsch

Keyword(s):

Fixed Point ◽

Fixed Points ◽

Efficient Algorithm ◽

Iterated Function Systems ◽

Binary Image ◽

The Self ◽

Compact Representation ◽

Iterated Function ◽

Similar Images ◽

Self Similar

Iterated Function Systems (IFS) seem to be used best to represent objects in the nature, because many of them are self similar. An IFS is a set of affine and contractive transformations. The union (so-called collage) of the subimages generated by transforming the whole image produces the image again - the self similar attractor of these transformations, which can be described by a binary image. For a fast and compact representation of those images, it would be desirable to calculate the transformations (the IFS-Codes) directly from the image that means to solve the inverse IFS-Problem. The solution presented in this paper will directly use the features of the self similar image. Subsets of the entire image and the subimage to be calculated are identified by the computation of the set difference between the pixels of the original and a rotated copy. The rotation and the scale factor of the transformation can be computed by the mapping of this two subsets onto each other, if the translation part - the fixed point - is predefined. The calculation of the transformation has to be repeated for each subimage. It will be proved, that with this method the IFS-Codes can be calculated for not convex, undistorted, and self similar images as long as the fixed point is known. An efficient algorithm for the identification of these fixed points within the image is introduced. Different ways to achieve this solutions are presented. In the conclusion the class of images, which can be coded by this method is defined, the results are pointed out, the advantages resp. the disadvantages of the method are evaluated, and possible ways to extend the method are discussed.

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The Hutchinson-Barnsley theory for infinite iterated function systems

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700035267 ◽

2005 ◽

Vol 72 (3) ◽

pp. 441-454 ◽

Cited By ~ 28

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.

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