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The Invariant Measure for a Countable generalized iterated function system

MEDITERRANEAN JOURNAL OF MATHEMATICS 2024年第7期21卷 1-22页

作者： Abraham, Izabella Transilvania Univ Fac Math & Comp Sci Iuliu Maniu 50 Brasov 500091 Romania

The aim of this paper is to answer one of the open questions raised in Strobin [Qual. Theory Dyn. Syst. 19, 85 (2020)] of whether there exists an invariant (Hutchinson) measure for generalized iterated function systems of any order, consisting of a countably infinite number of maps. Our results likewise strengthen those obtained in Secelean [Mediterr. J. Math. 11, 361-372 (2014)], where the existence of the invariant measure is ascertained only for the case of generalized iterated function systems of order 2, consisting of functions which satisfy a particular contractive condition.

关键词： generalized iterated function system countable generalized iterated function system invariant measure Monge-Kantorovich distance

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Fractals in Sb-metric spaces

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COMPLEX & INTELLIGENT systemS 2025年第5期11卷 1-17页

作者： Din, Fahim Ud Nawaz, Sheeza Jhangeer, Adil Tchier, Fairouz Govt Coll Univ Abdus Salam Sch Math Sci Lahore 54600 Pakistan VSB Tech Univ Ostrava IT4 Innovat Ostrava Czech Republic Namal Univ Dept Math 30km Talagang Rd Mianwali 42250 Pakistan King Saud Univ Coll Sci Math Dept POB 22452 Riyadh 11495 Saudi Arabia

This study aims to discover attractors for fractals by using generalized F-contractive iterated function system, which falls within a distinct category of mappings defined on Sb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_b$$\end{document}-metric spaces. In particular, we investigate how these systems, when subjected to specific F-contractive conditions, can lead to the identification of a unique attractor. We achieve a diverse range of outcomes for iterated function systems that adhere to a unique set of generalized F-contractive conditions. Our approach includes a detailed theoretical framework that establishes the existence and uniqueness of attractors in these settings. We provide illustrative examples to bolster the findings established in this work and use the functions given in the example to construct fractals and discuss the convergence of the obtained fractals via iterated function system to an attractor. These examples demonstrate the practical application of our theoretical results, showcasing the convergence behavior of fractals generated by our proposed systems. These outcomes extend beyond the scope of various existing results found in the current body of literature. By expanding the applicability of F-contractive conditions, our findings contribute to the broader understanding of fractal geometry and its applications, offering new insights and potential directions for future research in this area.

关键词： Fractals Attractor generalized iterated function system Fixed point generalized F-contraction S-b-metric space

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On a typical compact set as the attractor of generalized iterated function systems of infinite order

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2020年第2期484卷

作者： Maslanka, Lukasz Polish Acad Sci Inst Math Sniadeckich 8 PL-00656 Warsaw Poland

In 2013 Balka and Mathe showed that in uncountable polish spaces the typical compact set is not a fractal of any IFS. In 2008 Miculescu and Mihail introduced a concept of a generalized iterated function system (GIFS in short), a particular extension of classical IFS, in which they considered families of mappings defined on finite Cartesian product X-m with values in X. Recently, Secelean extended these considerations to mappings defined on the space l(infinity)(X) of all bounded sequences of elements of X endowed with supremum metric. In the paper we show that in Euclidean spaces a typical compact set is an attractor in sense of Secelean and that in general in the polish spaces it can be perceived as selfsimilar in such sense. (C) 2019 The Author(s). Published by Elsevier Inc.

关键词： Fractals iterated function system generalized iterated function system Typical compact sot

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On the geometric ergodicity for a generalized IFS with probabilities

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STOCHASTICS AND DYNAMICS 2022年第1期22卷

作者： Guzik, Grzegorz Kapica, Rafal AGH Univ Sci & Technol Fac Appl Math Al Mickiewicza 30 PL-30059 Krakow Poland

Main goal of this paper is to formulate possibly simple and easy to verify criteria on existence of the unique attracting probability measure for stochastic process induced by generalized iterated function systems with probabilities (GIFSPs). To do this, we study the long-time behavior of trajectories of Markov-type operators acting on product of spaces of Borel measures on arbitrary Polish space. Precisely, we get the desired geometric rate of convergence of sequences of measures under the action of such operator to the unique distribution in the Hutchinson-Wasserstein distance. We apply the obtained results to study limiting behavior of random trajectories of GIFSPs as well as stochastic difference equations with multiple delays.

关键词： generalized iterated function system asymptotic stability Markov operator Hutchinson-Wasserstein norm stochastic difference equation with delays

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Cyclic generalized iterated function systems

COMPUTATIONAL AND MATHEMATICAL METHODS

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COMPUTATIONAL AND MATHEMATICAL METHODS 2021年第6期3卷

作者： Pasupathi, Rajan Chand, Arya Kumar Bedabrata Navascues, Maria Antonia Sebastian, Maria Victoria Indian Inst Technol Madras Dept Math Chennai Tamil Nadu India Univ Zaragoza Escuela Ingn & Arquitectura Dept Matemat Aplicada Zaragoza Spain Ctr Univ Def Acad Gen Mil Ctra Huesca S-N Zaragoza 50090 Spain

In this article, we introduce the notion of cyclic generalized iterated function system (GIFS), which is a family of functions f(1), f(2), ..., f(M): X-k -> X, where each f(i) is a cyclic generalized phi-contraction (contractive) map on a collection of subsets {B-j}(j=1)(p) of a complete metric space (X,d) respectively, and k, M, p are natural numbers. When B-j, j = 1,2, ..., p are closed subsets of X, we show the existence of attractor of this cyclic GIFS, and investigate its properties. Further, we extend our ideas to cyclic countable GIFS.

关键词： cyclic phi-contractions fixed point theory fractals generalized iterated function system iterated function systems

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The Ergodic Theorem for a new kind of attractor of a GIFS

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CHAOS SOLITONS & FRACTALS 2017年 98卷 63-71页

作者： Oliveira, Elismar R. Univ Fed Rio Grande do Sul Inst Matemat & Estat Av Bento Goncalves 9500 BR-91500 Porto Alegre RS Brazil

In 1987, Elton [11], has proved the first fundamental result on the convergence of IFS, the Elton's Ergodic Theorem. In this work we prove the natural extension of this theorem to the projected Hutchinson measure mu(alpha) associated to a GIFSpdp S = (X, (phi(j) : X-m -> X)(j=0,1...,n-1,) (p(j))(j=0,1,...,n-1)) in a compact metric space (X, d). More precisely, the average along of the trajectories chi(n)(a) of the GIFS, starting in any initial points X-0 ,..., Xm-1 is an element of X satisfies, for any f is an element of C(X, R), lim(N ->+infinity)1/N Sigma(N-1)(n=0) f(X-n(a)) = integral(X)f(t)d mu alpha(t), for almost all a is an element of Omega = {0, 1,..., n-1}(N), the symbolic space. Additionally, we give some examples and applications to Chaos Games and Nonautonomous Dynamical systems defined by finite difference equations. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Fractals generalized iterated function system Markov operator Hutchinson measure Ergodic Theorem iterated function systems Dynamical systems Chaos Games

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Algorithms generating images of attractors of generalized iterated function systems

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NUMERICAL ALGORITHMS 2016年第2期73卷 477-499页

作者： Jaros, Patrycja Maslanka, Lukasz Strobin, Filip Lodz Univ Technol Div Dynam Stefanowskiego 1-15 PL-90924 Lodz Poland Lodz Univ Technol Inst Math Wolczanska 215 PL-93005 Lodz Poland

The paper is devoted to searching algorithms which will allow to generate images of attractors of generalized iterated function systems (GIFS in short), which are certain generalization of classical iterated function systems, defined by Mihail and Miculescu in 2008, and then intensively investigated in the last years (the idea is that instead of selfmaps of a metric space X, we consider mappings form the Cartesian product Xx...xX to X). Two presented algorithms are counterparts of classical deterministic algorithm and so-called chaos game. The third and fourth one is fitted to special kind of GIFSs - to affine GIFS, which are, in turn, also investigated.

关键词： Fractal iterated function system generalized iterated function system Chaos game Deterministic algorithm Affine mappings

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generalized F-iterated function systems on product of metric spaces

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JOURNAL OF FIXED POINT THEORY AND APPLICATIONS 2015年第3期17卷 575-595页

作者： Secelean, Nicolae-Adrian Lucian Blaga Univ Sibiu Dept Math & Informat Fac Sci Sibiu Romania

In this paper, we start from an F-contraction defined on a metric space X into itself, introduced by Wardowski [Fixed Point Theory Appl. 2012 (2012), doi:10.1186/1687-1812-2012-94], and extend it to the case of mappings defined on the space X (I) into X endowed with the supremum metric, where I is a set of positive integers. Next, we consider the generalized iterated function systems composed of F-contractions on X (I) improving some fixed point results from the classical Hutchinson-Barnsley theory of iterated function systems. Some illustrative examples are given.

关键词： F-contraction generalized F-contraction generalized iterated function system attractor Hausdorff-Pompeiu metric fixed point

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