arXiv subjects

Krzysztof Caban

Publications and source records attributed to Krzysztof Caban.

2 recordsLinked to original sources

arXiv2026

A non-logarithmic approach to the rate of convergence of the deterministic chaos game

The aim of this paper is to provide a different perspective in the study of the rate of convergence of the chaos game algorithm to the attractor of an iterated function system. We prove that for any function $ψ$ with $\lim\limits_{\ve\to 0}ψ(\ve)=\infty$, a typical (in the sense of the Baire category) driver yields a rate of recovery comparable to $ψ$. This result extends the main theorem from Leśniak et al. (Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 157, 2024). Moreover, thanks to the change of perspective, we are able to prove that a typical driver gives arbitrarily slow rate of recovery.

math.DS↗

arXiv2025

A topological approach to fuzzy iterated function systems

In the paper we unify two extensions of the classical Hutchinson--Barnsley theory - the topological and the fuzzy-set approaches. We show that a fuzzy iterated function system (fuzzy IFS) on a Tychonoff space $X$ which is contracting w.r.t. some admissible multimetric, generates a natural fuzzy attractor in the hyperspace $\mathcal{K}_{\mathcal{F}}(X)$ of all compact fuzzy sets. As a consequence, we prove that a fuzzy IFS on a Hausdorff topological space which is topologically contracting admits a fuzzy attractor in a bit weaker sense. Our discussion involves investigations on topologies on the hyperspace $\mathcal{K}_{\mathcal{F}}(X)$ which are suitable for establishing convergence of sequences of iterations of a fuzzy Hutchinson operator.

math.DS↗