scholarly journals Binary Operations in Metric Spaces Satisfying Side Inequalities

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 11
Author(s):  
María A. Navascués ◽  
Pasupathi Rajan ◽  
Arya Kumar Bedabrata Chand
Keyword(s):  
Metric Spaces ◽  
Initial Conditions ◽  
Fractal Theory ◽  
Iterated Function Systems ◽  
Riesz Bases ◽  
Real Interval ◽  
Compact Sets ◽  
Binary Operations ◽  
Iterated Function ◽  
Bessel Sequences

The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space E. We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space.

scholarly journals Duality results for iterated function systems with a general family of branches

2017 ◽  
Vol 17 (03) ◽  
pp. 1750021 ◽  
Author(s):  
Jairo K. Mengue ◽  
Elismar R. Oliveira
Keyword(s):  
Cost Function ◽  
Relative Entropy ◽  
Metric Spaces ◽  
Iterated Function Systems ◽  
Compact Spaces ◽  
Compact Metric Spaces ◽  
Duality Results ◽  
General Family ◽  
Iterated Function ◽  
Kantorovich Duality

Given [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] compact metric spaces, we consider two iterated function systems [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are contractions. Let [Formula: see text] be the set of probabilities [Formula: see text] with [Formula: see text]-marginal being holonomic with respect to [Formula: see text] and [Formula: see text]-marginal being holonomic with respect to [Formula: see text]. Given [Formula: see text] and [Formula: see text], let [Formula: see text] be the set of probabilities in [Formula: see text] having [Formula: see text]-marginal [Formula: see text] and [Formula: see text]-marginal [Formula: see text]. Let [Formula: see text] be the relative entropy of [Formula: see text] with respect to [Formula: see text] and [Formula: see text] be the relative entropy of [Formula: see text] with respect to [Formula: see text]. Given a cost function [Formula: see text], let [Formula: see text]. We will prove the duality equation: [Formula: see text] In particular, if [Formula: see text] and [Formula: see text] are single points and we drop the entropy, the equation above can be rewritten as the Kantorovich duality for the compact spaces [Formula: see text] and a continuous cost function [Formula: see text].

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.

Iterated function systems and attractors in the KM fuzzy metric spaces

2015 ◽  
Vol 267 ◽  
pp. 100-116 ◽  
Author(s):  
Jian-Zhong Xiao ◽  
Xing-Hua Zhu ◽  
Pan-Pan Jin
Keyword(s):  
Metric Spaces ◽  
Iterated Function Systems ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Iterated Function

A new class of markov processes for image encoding

1988 ◽  
Vol 20 (01) ◽  
pp. 14-32 ◽  
Author(s):  
Michael F. Barnsley ◽  
John H. Elton
Keyword(s):  
Invariant Measures ◽  
Metric Spaces ◽  
Iterated Function Systems ◽  
Interesting Case ◽  
Image Encoding ◽  
New Class ◽  
Affine Maps ◽  
Iterated Maps ◽  
Iterated Function ◽  
Infinite Extent

A new class of iterated function systems is introduced, which allows for the computation of non-compactly supported invariant measures, which may represent, for example, greytone images of infinite extent. Conditions for the existence and attractiveness of invariant measures for this new class of randomly iterated maps, which are not necessarily contractions, in metric spaces such as , are established. Estimates for moments of these measures are obtained. Special conditions are given for existence of the invariant measure in the interesting case of affine maps on . For non-singular affine maps on , the support of the measure is shown to be an infinite interval, but Fourier transform analysis shows that the measure can be purely singular even though its distribution function is strictly increasing.

scholarly journals Hölder Parameterization of Iterated Function Systems and a Self-Aflne Phenomenon

2021 ◽  
Vol 9 (1) ◽  
pp. 90-119
Author(s):  
Matthew Badger ◽  
Vyron Vellis
Keyword(s):  
Metric Spaces ◽  
Optimal Parameter ◽  
Complete Metric Space ◽  
Iterated Function Systems ◽  
Open Set Condition ◽  
Interesting Phenomenon ◽  
Open Set ◽  
Iterated Function ◽  
Self Similar ◽  
Path Connected

Abstract We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/s)-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α > s. At the endpoint, α = s, a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/s)-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s-dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝ n is always at most the ambient dimension n, the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n.

A new class of markov processes for image encoding

1988 ◽  
Vol 20 (1) ◽  
pp. 14-32 ◽  
Author(s):  
Michael F. Barnsley ◽  
John H. Elton
Keyword(s):  
Invariant Measures ◽  
Metric Spaces ◽  
Iterated Function Systems ◽  
Interesting Case ◽  
Image Encoding ◽  
New Class ◽  
Affine Maps ◽  
Iterated Maps ◽  
Iterated Function ◽  
Infinite Extent

A new class of iterated function systems is introduced, which allows for the computation of non-compactly supported invariant measures, which may represent, for example, greytone images of infinite extent. Conditions for the existence and attractiveness of invariant measures for this new class of randomly iterated maps, which are not necessarily contractions, in metric spaces such as , are established. Estimates for moments of these measures are obtained.Special conditions are given for existence of the invariant measure in the interesting case of affine maps on . For non-singular affine maps on , the support of the measure is shown to be an infinite interval, but Fourier transform analysis shows that the measure can be purely singular even though its distribution function is strictly increasing.

scholarly journals Asymptotics of Divide-And-Conquer Recurrences Via Iterated Function Systems

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
John Kieffer
Keyword(s):  
Initial Conditions ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Continuous Functions ◽  
Bounded Sequence ◽  
Divide And Conquer ◽  
Real Numbers ◽  
Fixed Integer ◽  
Iterated Function ◽  
International Audience

International audience Let $k≥2$ be a fixed integer. Given a bounded sequence of real numbers $(a_n:n≥k)$, then for any sequence $(f_n:n≥1)$ of real numbers satisfying the divide-and-conquer recurrence $f_n = (k-mod(n,k))f_⌊n/k⌋+mod(n,k)f_⌈n/k⌉ + a_n, n ≥k$, there is a unique continuous periodic function $f^*:\mathbb{R}→\mathbb{R}$ with period 1 such that $f_n = nf^*(\log _kn)+o(n)$. If $(a_n)$ is periodic with period $k, a_k=0$, and the initial conditions $(f_i:1 ≤i ≤k-1)$ are all zero, we obtain a specific iterated function system $S$, consisting of $k$ continuous functions from $[0,1]×\mathbb{R}$ into itself, such that the attractor of $S$ is $\{(x,f^*(x)): 0 ≤x ≤1\}$. Using the system $S$, an accurate plot of $f^*$ can be rapidly obtained.

scholarly journals Iterated Function Systems Consisting of Generalized Convex Contractions in the Framework of Complete Strong b-metric Spaces

2017 ◽  
Vol 55 (2) ◽  
pp. 119-142
Author(s):  
Flavian Georgescu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Iterated Function Systems ◽  
Iterated Function ◽  
Convex Contractions

AbstractThe concept of generalized convex contraction was introduced and studied by V. Istrăţescu and the notion ofb-metric space was introduced by I. A. Bakhtin and S. Czerwik. In this paper we combine these two elements by studying iterated function systems consisting of generalized convex contractions on the framework ofb-metric spaces. More precisely we prove the existence and uniqueness of the attractor of such a system providing in this way a generalization of Istrăţescu’s convex contractions fixed point theorem in the setting of complete strongb-metric spaces.

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