scholarly journals Variants of shadowing properties for iterated function systems on uniform spaces

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2565-2572
Author(s):  
Radhika Vasisht ◽  
Mohammad Salman ◽  
Ruchi Das
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Uniform Spaces ◽  
Shadowing Property ◽  
Topological Mixing ◽  
Topologically Transitive ◽  
Iterated Function ◽  
Chain Transitivity

In this paper, the notions of topological shadowing, topological ergodic shadowing, topological chain transitivity and topological chain mixing are introduced and studied for an iterated function system (IFS) on uniform spaces. It is proved that if an IFS has topological shadowing property and is topological chain mixing, then it has topological ergodic shadowing and it is topological mixing. Moreover, if an IFS has topological shadowing property and is topological chain transitive, then it is topologically ergodic and hence topologically transitive. Also, these notions are studied for the product IFS on uniform spaces.

Shadowing and average shadowing properties for iterated function systems

2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Alireza Zamani Bahabadi
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Skew Product ◽  
Shadowing Property ◽  
Iterated Function ◽  
Chain Transitivity

AbstractIn this paper, we introduce the definitions of shadowing and average shadowing properties for iterated function systems and give some examples characterizing these definitions. We prove that an iterated function system has the shadowing property if and only if the step skew product corresponding to the iterated function system has the shadowing property. Also, we study some notions such as transitivity, chain transitivity, chain mixing and mixing for iterated function systems.

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 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.

ON POINTS WITH POSITIVE DENSITY OF THE DIGIT SEQUENCE IN INFINITE ITERATED FUNCTION SYSTEMS

2016 ◽  
Vol 102 (3) ◽  
pp. 435-443
Author(s):  
ZHEN-LIANG ZHANG ◽  
CHUN-YUN CAO
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Positive Density ◽  
Digit Sequence ◽  
Iterated Function ◽  
Banach Density ◽  
Image Position

Let $\{f_{n}\}_{n\geq 1}$ be an infinite iterated function system on $[0,1]$ and let $\unicode[STIX]{x1D6EC}$ be its attractor. Then, for any $x\in \unicode[STIX]{x1D6EC}$, it corresponds to a sequence of integers $\{a_{n}(x)\}_{n\geq 1}$, called the digit sequence of $x$, in the sense that $$\begin{eqnarray}x=\lim _{n\rightarrow \infty }f_{a_{1}(x)}\circ \cdots \circ f_{a_{n}(x)}(1).\end{eqnarray}$$ In this note, we investigate the size of the points whose digit sequences are strictly increasing and of upper Banach density one, which improves the work of Tong and Wang and Zhang and Cao.

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.

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.

THE BOUNDARY OF THE ATTRACTOR OF A RECURRENT ITERATED FUNCTION SYSTEM

Fractals ◽  
2002 ◽  
Vol 10 (01) ◽  
pp. 77-89 ◽  
Author(s):  
F. M. DEKKING ◽  
P. VAN DER WAL
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Iterated Function

We prove for a subclass of recurrent iterated function systems (also called graph-directed iterated function systems) that the boundary of their attractor is again the attractor of a recurrent IFS. Our method is constructive and permits computation of the Hausdorff dimension of the attractor and its boundary.

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.

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