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.

Shadowing property and topological ergodicity for iterated function systems

2019 ◽  
Vol 33 (23) ◽  
pp. 1950272
Author(s):  
Yingcui Zhao ◽  
Lidong Wang ◽  
Fengchun Lei
Keyword(s):  
Metric Space ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Skew Product ◽  
Semi Group ◽  
Compact Metric Space ◽  
Shadowing Property ◽  
Iterated Function ◽  
Continuous Maps

Let [Formula: see text] be a compact metric space and [Formula: see text] be two continuous maps on [Formula: see text]. The iterated function system [Formula: see text] is the action of the semi-group generated by [Formula: see text] on [Formula: see text]. In this paper, we introduce the definitions of shadowing property, average shadowing property and topological ergodicity for [Formula: see text] and give some examples. Then we show that (1) if [Formula: see text] has the shadowing property then so do [Formula: see text] and [Formula: see text]; (2) [Formula: see text] has the shadowing property if and only if the step skew product corresponding to [Formula: see text] has the shadowing property. At last, we prove a Lyapunov stable iterated function system having the average shadowing property is topologically ergodic.

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.

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 IFSs consisting of generalized convex contractions

2017 ◽  
Vol 25 (1) ◽  
pp. 77-86 ◽  
Author(s):  
Flavian Georgescu
Keyword(s):  
Metric Space ◽  
Iterated Function System ◽  
Complete Metric Space ◽  
Continuous Functions ◽  
Finite Family ◽  
Function System ◽  
Picard Operator ◽  
Iterated Function ◽  
The Family ◽  
Convex Contractions

Abstract In this paper we introduce the concept of iterated function system consisting of generalized convex contractions. More precisely, given n ∈ ℕ*, an iterated function system consisting of generalized convex contractions on a complete metric space (X; d) is given by a finite family of continuous functions (fi)i ∈I , fi : X → X, having the property that for every ω ∈ λn(I) there exists a family of positive numbers (aω;υ)υ∈Vn(I) such that: x; y ∈ X. Here λn(I) represents the family of words with n letters from I, Vn(I) designates the family of words having at most n - 1 letters from I, while, if ω1 = ω1ω2 ... ωp, by fω we mean fω1 ⃘fω2 ⃘... ⃘ fωp. Denoting such a system by S = ((X; d); n; (fi)i∈I), one can consider the function FS : K(X) → K(X) described by , for all B ∈ K(X), where K(X) means the set of non-empty compact subsets of X. Our main result states that FS is a Picard operator for every iterated function system consisting of generalized convex contractions S.

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.

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.

scholarly journals Dimension and measure for typical random fractals

2013 ◽  
Vol 35 (3) ◽  
pp. 854-882 ◽  
Author(s):  
JONATHAN M. FRASER
Keyword(s):  
Metric Space ◽  
Iterated Function System ◽  
Probabilistic Approach ◽  
Iterated Function Systems ◽  
Function System ◽  
Iterated Function ◽  
Random Attractors ◽  
Finite Set ◽  
Separation Conditions ◽  
Random Iterated Function System

AbstractWe define a random iterated function system (RIFS) to be a finite set of (deterministic) iterated function systems (IFSs) acting on the same metric space. For a given RIFS, there exists a continuum of random attractors corresponding to each sequence of deterministic IFSs. Much work has been done on computing the ‘almost sure’ dimensions of these random attractors. Here we compute the typical dimensions (in the sense of Baire) and observe that our results are in stark contrast to those obtained using the probabilistic approach. Furthermore, we examine the typical Hausdorff and packing measures of the random attractors and give examples to illustrate some of the strange phenomena that can occur. The only restriction we impose on the maps is that they are bi-Lipschitz and we obtain our dimension results without assuming any separation conditions.

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. 

Accessibility on iterated function systems

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Maliheh Mohtashamipour ◽  
Alireza Zamani Bahabadi
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Sufficient Condition ◽  
Skew Product ◽  
Iterated Function ◽  
Topologically Mixing ◽  
Mixing Property

AbstractIn this paper, we define accessibility on an iterated function system (IFS) and show that it provides a sufficient condition for the transitivity of this system and its corresponding skew product. Then, by means of a certain tool, we obtain the topologically mixing property. We also give some results about the ergodicity and stability of accessibility and, further, illustrate accessibility by some examples.

Some Results on Recurrent Fractal Interpolation Function

2020 ◽  
Vol 12 (8) ◽  
pp. 1038-1043
Author(s):  
Wadia Faid Hassan Al-Shameri
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Data Set ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Fractal Functions ◽  
Fractal Approximation

Barnsley (Barnsley, M.F., 1986. Fractal functions and interpolation. Constr. Approx., 2, pp.303–329) introduced fractal interpolation function (FIF) whose graph is the attractor of an iterated function system (IFS) for describing the data that have an irregular or self-similar structure. Barnsley et al. (Barnsley, M.F., et al., 1989. Recurrent iterated function systems in fractal approximation. Constr. Approx., 5, pp.3–31) generalized FIF in the form of recurrent fractal interpolation function (RFIF) whose graph is the attractor of a recurrent iterated function system (RIFS) to fit data set which is piece-wise self-affine. The primary aim of the present research is investigating the RFIF approach and using it for fitting the piece-wise self-affine data set in ℜ2.

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