Opuscula Math. 45, no. 5 (2025), 601-621
https://doi.org/10.7494/OpMath.2025.45.5.601
Opuscula Mathematica
From set-valued dynamical processes to fractals
Grzegorz Guzik
Grzegorz Kleszcz
Abstract. We present a general theory of topological semiattractors and attractors for set-valued semigroups. Our results extend and unify those previously obtained by Lasota and Myjak. In particular, we naturally generalize the concept of semifractals for the systems acting on Hausdorff topological spaces. The main tool in our analysis is the notion of topological (Kuratowski) limits. We especially focus on the forward asymptotic behavior of discrete set-valued processes generated by sequences of iterated function systems. In this context, we establish sufficient conditions for the existence of fractal-type limit sets.
Keywords: topological limit, lower semicontinuous multifunction, iterated function system, set-valued process, attractor.
Mathematics Subject Classification: 28A80, 54H20, 26E25, 47H04, 93E03.
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- Grzegorz Guzik
https://orcid.org/0000-0002-6099-9499
- AGH University of Krakow, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Kraków, Poland
- Grzegorz Kleszcz (corresponding author)
https://orcid.org/0000-0003-3044-5097
- AGH University of Krakow, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Kraków, Poland
- Communicated by Marek Galewski.
- Received: 2025-05-27.
- Revised: 2025-07-18.
- Accepted: 2025-07-24.
- Published online: 2025-09-13.