Filip Strobin
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View article: Topological prevalence of variable speed of convergence in the deterministic chaos game
Topological prevalence of variable speed of convergence in the deterministic chaos game Open
Let A be the attractor of a Banach contractive iterated function system (IFS) on a complete space. We prove that the orbit generated by a typical (in the sense of Baire category) driver recovers A with every possible speed. Our result exte…
View article: Highly Non-contractive Iterated Function Systems on Euclidean Space Can Have an Attractor
Highly Non-contractive Iterated Function Systems on Euclidean Space Can Have an Attractor Open
Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. Moreover, contractivity of the functions in the IFS has been central to the theory of iterated functions s…
View article: A Controlled Hahn-Mazurkiewicz Theorem and its Applications
A Controlled Hahn-Mazurkiewicz Theorem and its Applications Open
For a metric Peano continuum $X$, let $S_X$ be a Sierpiński function assigning to each $\varepsilon>0$ the smallest cardinality of a cover of $X$ by connected subsets of diameter $\le \varepsilon$. We prove that for any increasing function…
View article: Transition phenomena for the attractor of an iterated function system*
Transition phenomena for the attractor of an iterated function system* Open
Iterated function systems (IFSs) and their attractors have been central in fractal geometry. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Two natural questions concerning contractivit…
View article: Strongly-Fibred Iterated Function Systems and the Barnsley--Vince triangle
Strongly-Fibred Iterated Function Systems and the Barnsley--Vince triangle Open
We review the theory of semiattractors associated with non-contractive Iterated Function Systems (IFSs) and demonstrate its applications on a concrete example. In particular, we present criteria for the existence of semiattractors due to L…
View article: Transition Phenomena for the Attractor of an Iterated Function System
Transition Phenomena for the Attractor of an Iterated Function System Open
Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. And contractivity of the functions in the IFS has been central to the theory of iterated functions systems…
View article: A FRACTAL TRIANGLE ARISING IN THE AIMD DYNAMICS
A FRACTAL TRIANGLE ARISING IN THE AIMD DYNAMICS Open
The organizers of the conference Contemporary Mathematics in Kielce 2020 warmly welcome all the mathematicians interested in that field to participate in our meeting. The conference is hosted by the Jan Kochanowski University in Kielce, Po…
View article: Existence of invariant idempotent measures by contractivity of idempotent Markov operators
Existence of invariant idempotent measures by contractivity of idempotent Markov operators Open
We prove that the idempotent Markov operator generated by contractive max plus normalized iterated function system (IFS) is also a contractive map w.r.t. natural metrics on the space of idempotent measures. This gives alternative proofs of…
View article: Fuzzy-set approach to invariant idempotent measures
Fuzzy-set approach to invariant idempotent measures Open
We provide a new approach to the Hutchinson-Barnsley theory for idempotent measures first presented in N. Mazurenko, M. Zarichnyi, Invariant idempotent measures, Carpathian Math. Publ., 10 (2018), 1, 172--178. The main feature developed he…
View article: Contractive Iterated Function Systems Enriched with Nonexpansive Maps
Contractive Iterated Function Systems Enriched with Nonexpansive Maps Open
Motivated by a recent paper of Leśniak and Snigireva [ Iterated function systems enriched with symmetry , preprint], we investigate the properties of the semiattractor $$A_{\mathcal {F}\cup \mathcal {G}}^*$$ of an IFS $$\mathcal {F…
View article: Embedding fractals in Banach, Hilbert or Euclidean spaces
Embedding fractals in Banach, Hilbert or Euclidean spaces Open
By a metric fractal we understand a compact metric space K endowed with a finite family \mathcal F of contracting self-maps of K such that K=\bigcup_{f\in\mathcal F}f(K) . If K is a subset of a metric space X and each f\in\mathcal F extend…
View article: On the existence of the Hutchinson measure for generalized iterated function systems
On the existence of the Hutchinson measure for generalized iterated function systems Open
We prove that each generalized (in the sense of Miculescu and Mihail) IFS consisting of contractive maps generates the unique generalized Hutchinson measure. This result extends the earlier result due to Miculescu in which the assertion is…
View article: Connectedness of attractors of a certain family of IFSs
Connectedness of attractors of a certain family of IFSs Open
Let X be a Banach space and f,g\colon X\rightarrow X be contractions. We investigate the set C_{f,g}:=\{w\in X\colon\mathrm {the\: attractor\: of\: IFS} \:\mathcal F_w=\{f,g+w\}\: \mathrm {is\: connected}. The motivation for our research c…
View article: Weakly contractive iterated function systems and beyond: a manual
Weakly contractive iterated function systems and beyond: a manual Open
We give a systematic account of iterated function systems (IFS) of weak contractions of different types (Browder, Rakotch, topological). We show that the existence of attractors and asymptotically stable invariant measures, and the validit…
View article: Zero-dimensional compact metrizable spaces as attractors of generalized iterated function systems
Zero-dimensional compact metrizable spaces as attractors of generalized iterated function systems Open
R. Miculescu and A. Mihail in 2008 introduced the concept of a generalized iterated function system (GIFS in short), a particular extension of the classical IFS. The idea is that, instead of families of selfmaps of a metric space $X$, GIFS…
View article: Zero-dimensional compact metrizable spaces as attractors of generalized iterated function systems
Zero-dimensional compact metrizable spaces as attractors of generalized iterated function systems Open
Miculescu and Mihail in 2008 introduced the concept of a \emph{generalized iterated function system} (GIFS in~short), a particular extension of the classical IFS. The idea is that, instead of families of selfmaps of a metric space~$X$, GIF…
View article: Dense free subgroups of automorphism groups of homogeneous partially ordered sets
Dense free subgroups of automorphism groups of homogeneous partially ordered sets Open
A countable poset is ultrahomogeneous if every isomorphism between its finite subposets can be extended to an automorphism. If A is such a poset, then the group Aut ( A ) {\operatorname{Aut}(A)} has a natural topology in which Au…
View article: On generalized iterated function systems defined on $\ell_\infty$-sum of a metric space
On generalized iterated function systems defined on $\ell_\infty$-sum of a metric space Open
Miculescu and Mihail in 2008 introduced a concept of a generalized iterated function system (GIFS in short), a particular extension of classical IFS. Instead of families of selfmaps of a metric space $X$, they considered families of mappin…
View article: A fixed point theorem for mappings on the $\ell_\infty$-sum of a metric space and its application
A fixed point theorem for mappings on the $\ell_\infty$-sum of a metric space and its application Open
The aim of this paper is to prove a counterpart of the Banach fixed point principle for mappings $f: \ell_\infty(X) \to X$, where $X$ is a metric space and $\ell_\infty(X)$ is the space of all bounded sequences of elements from~$X$. Our re…
View article: A fixed point theorem for mappings on the l∞-sum of a metric space and its application
A fixed point theorem for mappings on the l∞-sum of a metric space and its application Open
The aim of this paper is to prove a counterpart of the Banach fixed point principle for mappings f: l?(X)? X, where X is a metric space and l?(X) is the space of all bounded sequences of elements from X. Our result generalizes the theorem …
View article: Fuzzy Attractors Appearing from GIFZS
Fuzzy Attractors Appearing from GIFZS Open
Cabrelli, Forte, Molter and Vrscay in 1992 considered a {fuzzy} version of the theory of iterated function systems (IFSs in short) and their fractals%The idea was to extend the classical Hutchinson-Barnsley operator to selfmaps of a metric…
View article: Algorithms generating images of attractors of generalized iterated function systems
Algorithms generating images of attractors of generalized iterated function systems Open
The paper is devoted to searching algorithms which will allow to generate images of attractors of \emph{generalized iterated function systems} (GIFS in short), which are certain generalization of classical iterated function systems, define…
View article: Algorithms generating images of attractors of generalized iterated\n function systems
Algorithms generating images of attractors of generalized iterated\n function systems Open
The paper is devoted to searching algorithms which will allow to generate\nimages of attractors of \\emph{generalized iterated function systems} (GIFS in\nshort), which are certain generalization of classical iterated function\nsystems, de…
View article: Contractive function systems, their attractors and metrization
Contractive function systems, their attractors and metrization Open
In this paper we study the Hutchinson-Barnsley theory of fractals in the setting of multimetric spaces (which are sets endowed with point separating families of pseudometrics) and in the setting of topological spaces. We find natural conne…
View article: Embedding topological fractals in universal spaces
Embedding topological fractals in universal spaces Open
A compact metric space X is called a Rakotch (Banach) fractal if\linebreak X=\bigcup_{f\in\mathcal F}f(X) for some finite system \mathcal F of Rakotch (Banach) contracting self-maps of X . A Hausdorff topological space X is called a topolo…