Fractal Top for Contractive and Non-Contractive Transformations

2012 ◽  
Vol 3 (4) ◽  
pp. 49-65
Author(s):  
Sarika Jain ◽  
S. L. Singh ◽  
S. N. Mishra
Keyword(s):  
Fixed Point ◽  
Iterated Function System ◽  
Function System ◽  
Feedback Process ◽  
Iterated Function ◽  
Spatial Part ◽  
One Step ◽  
The One ◽  
Definition Of

Barnsley (2006) introduced the notion of a fractal top, which is an addressing function for the set attractor of an Iterated Function System (IFS). A fractal top is analogous to a set attractor as it is the fixed point of a contractive transformation. However, the definition of IFS is extended so that it works on the colour component as well as the spatial part of a picture. They can be used to colour-render pictures produced by fractal top and stealing colours from a natural picture. Barnsley has used the one-step feed- back process to compute the fractal top. In this paper, the authors introduce a two-step feedback process to compute fractal top for contractive and non-contractive transformations.

scholarly journals Generalized Iterated Function Systems Containing Functions of Integral Type

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. 

Some properties of the fractal convolution of functions

2021 ◽  
Vol 24 (6) ◽  
pp. 1735-1757
Author(s):  
María Navascués ◽  
Ram N. Mohapatra ◽  
Arya K.B. Chand
Keyword(s):  
Binary Operation ◽  
Iterated Function System ◽  
Function System ◽  
Real Interval ◽  
Topological Properties ◽  
Convolution Operators ◽  
Iterated Function ◽  
Left And Right ◽  
Fréchet Differentiable ◽  
The One

Abstract We consider the fractal convolution of two maps f and g defined on a real interval as a way of generating a new function by means of a suitable iterated function system linked to a partition of the interval. Based on this binary operation, we consider the left and right partial convolutions, and study their properties. Though the operation is not commutative, the one-sided convolutions have similar (but not equal) characteristics. The operators defined by the lateral convolutions are both nonlinear, bi-Lipschitz and homeomorphic. Along with their self-compositions, they are Fréchet differentiable. They are also quasi-isometries under certain conditions of the scale factors of the iterated function system. We also prove some topological properties of the convolution of two sets of functions. In the last part of the paper, we study stability conditions of the dynamical systems associated with the one-sided convolution operators.

scholarly journals Generalized modular fractal spaces and fixed point theorems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Alireza Alihajimohammad ◽  
Reza Saadati
Keyword(s):  
Fixed Point ◽  
Hausdorff Distance ◽  
Fixed Point Theorems ◽  
Iterated Function System ◽  
Function System ◽  
Topological Properties ◽  
Iterated Function ◽  
Fractal Space ◽  
Nonempty Compact ◽  
Contraction Theory

AbstractIn this article, we introduce a new concept of Hausdorff distance through generalized modular metric on nonempty compact subsets and study some topological properties of it. This concept with contraction theory and the iterated function system (IFS) helps us to define a generalized modular fractal space.

scholarly journals Another characterization of hyperbolic diameter diminishing to zero IFSs

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."

K-ITERATED FUNCTION SYSTEM

Fractals ◽  
2010 ◽  
Vol 18 (01) ◽  
pp. 139-144 ◽  
Author(s):  
D. R. SAHU ◽  
ANINDITA CHAKRABORTY ◽  
R. P. DUBEY
Keyword(s):  
Iterated Function System ◽  
Function System ◽  
Contraction Condition ◽  
Kannan Mapping ◽  
Iterated Function ◽  
Definition Of ◽  
Collage Theorem

In 1969, Kannan1 gave the definition of a new mapping which had presented a condition which is more lenient than contraction condition. The purpose of this note is to introduce K-Iterated Function System using Kannan mapping which will cover a larger range of mappings. We also prove the Collage theorem for the K-Iterated Function System.

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.

The Programming of Snow Flower Fractal Graphics Based on Iterated Function System

2011 ◽  
Vol 143-144 ◽  
pp. 765-769
Author(s):  
Fu Cheng You ◽  
Yu Jie Chen
Keyword(s):  
Fixed Point ◽  
Learning Environment ◽  
Traditional Method ◽  
Iterated Function System ◽  
Function System ◽  
Iterated Function ◽  
New Learning ◽  
E Learning ◽  
Black Color ◽  
Generation Procedure

The traditional method to introduce the design of snow flower fractal graphics is to give fixed graphics in black color. In this paper, a new learning environment of triangle fractal graphics based on E-Learning is proposed, development of which is also introduced in detail. In this new learning environment, students can repeat drawing the colorful snow flower fractal graphics with single color or random color, which may arouse the students' interests and attract their attentions. By the learning environment, the fractal limitation or fixed point can be seen, and two different kinds of snow flower can be got when the IFS code is changed, which will do benefit to students' learning concepts of fractal graphics. By this new learning environment, it is very easy for students to grasp the programming procedure of fractal graphics, and understand the generation procedure and structure of triangle fractal graphics.

Parrondo's paradox for homoeomorphisms

2021 ◽  
pp. 1-9
Author(s):  
A. Gasull ◽  
L. Hernández-Corbato ◽  
F. R. Ruiz del Portal
Keyword(s):  
Fixed Point ◽  
Iterated Function System ◽  
Function System ◽  
Asymptotically Stable ◽  
Stable Fixed Point ◽  
Globally Asymptotically Stable ◽  
Iterated Function ◽  
Parrondo’S Paradox ◽  
First Time ◽  
Odd Dimensions

We construct two planar homoeomorphisms $f$ and $g$ for which the origin is a globally asymptotically stable fixed point whereas for $f \circ g$ and $g \circ f$ the origin is a global repeller. Furthermore, the origin remains a global repeller for the iterated function system generated by $f$ and $g$ where each of the maps appears with a certain probability. This planar construction is also extended to any dimension $>$ 2 and proves for first time the appearance of the dynamical Parrondo's paradox in odd dimensions.

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