Geodesic
On the attractors of weakly hyperbolic IFS's with condensation
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2024), pp. 100-108 
Vasile Glavan; Valeriu Guţu. On the attractors of weakly hyperbolic IFS's with condensation. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2024), pp. 100-108. http://geodesic.mathdoc.fr/item/BASM_2024_1_a6/
@article{BASM_2024_1_a6,
     author = {Vasile Glavan and Valeriu Gu\c{t}u},
     title = {On the attractors of weakly hyperbolic {IFS's} with condensation},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {100--108},
     year = {2024},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BASM_2024_1_a6/}
}
TY  - JOUR
AU  - Vasile Glavan
AU  - Valeriu Guţu
TI  - On the attractors of weakly hyperbolic IFS's with condensation
JO  - Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
PY  - 2024
SP  - 100
EP  - 108
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/BASM_2024_1_a6/
LA  - en
ID  - BASM_2024_1_a6
ER  - 
%0 Journal Article
%A Vasile Glavan
%A Valeriu Guţu
%T On the attractors of weakly hyperbolic IFS's with condensation
%J Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
%D 2024
%P 100-108
%N 1
%U http://geodesic.mathdoc.fr/item/BASM_2024_1_a6/
%G en
%F BASM_2024_1_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

We show that for any weakly hyperbolic IFS with condensation in $\mathbb{R}^n$ whose condensation set is a union of a finite collection of convex compact sets, there exists a standard weakly hyperbolic IFS with the same attractor.

[1] Banakh T., Nowak M., “A 1-dimensional Peano continuum which is not an IFS attractor”, Proc. Amer. Math. Soc., 141:3 (2013), 931–935 | DOI | MR | Zbl

[2] Barnsley M. F., Fractals Everywhere, Acad. Press Profess., Boston, 1993 | MR | Zbl

[3] Barnsley M. F., Leśniak K., “The chaos game on a general iterated function system from a topological point of view”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24:11 (2014), 1450139, 10 pp. | DOI | MR | Zbl

[4] Barnsley M. F., Vince A., “The chaos game on a general iterated function system”, Ergod. Th. Dynam. Sys., 31 (2011), 1073–1079 | DOI | MR | Zbl

[5] Bosman Ir. A. E., Het Wondere Onderzoekingsveld der Vlakke Meetkunde, N. V. Uitgeversmaatschappij Parcival, Breda, 1957

[6] Crovisier S., Rams M., “IFS attractors and Cantor sets”, Topology and its Applications, 153 (2006), 1849–1859 | DOI | MR | Zbl

[7] D'Aniello E., “Non-self-similar sets in $[0,1]^N$ of arbitrary dimension”, J. Math. Anal. Appl., 456:2 (2017), 1123–1128 | DOI | MR | Zbl

[8] D'Aniello E., Steele T. H., “Attractors for iterated function systems”, J. Fractal Geom., 3:2 (2016), 95–117 | DOI | MR | Zbl

[9] D'Aniello E., Steele T. H., “Attractors for classes of iterated function systems”, Eur. J. Math., 5:1 (2019), 116–137 | DOI | MR | Zbl

[10] Duvall P. F., Husch L. S., “Attractors of iterated function systems”, Proc. Amer. Math. Soc., 116 (1992), 279–284 | DOI | MR | Zbl

[11] Edalat A., “Power domains and iterated functions systems”, Inform. and Comput., 124:2 (1996), 182–197 | DOI | MR | Zbl

[12] Gadomski L., Glavan V., Guţu V., “Fractals by weak contractions and CAS Mathematica”, Computer Algebra Systems in Teaching and Research, 4th Intern. Workshop, CASTR 2007, Proceedings (Siedlce, Poland, Jan 31 - Feb 3, 2007), Wydawnictwo Akademii Podlaskiej, Siedlce, 2007, 129–134

[13] Gadomski L., Glavan V., Guţu V., “Playing Chaos game on the Pythagoras Tree with CAS Mathematica”, Computer Algebra Systems in Teaching and Research. Differential Equations, Dynamical Systems and Celestial Mechanics, Wydawnictwo Collegium Mazovia, Siedlce, 2011, 38–45

[14] Glavan V., Guţu V., “Attractors and fixed points of weakly contracting relations”, Fixed Point Theory, 5:2 (2004), 265–284 | MR | Zbl

[15] Glavan V., Guţu V., “How to construct the Pythagoras tree using CAS Mathematica”, Computer Algebra Systems in Teaching and Research, IX, Siedlce University of Natural Sciences and Humanities, Siedlce, 2020, 29–41

[16] Guţu V., “How to Construct Compact Convex Spots on the Plane Using CAS Mathematica”, Computer Algebra Systems in Teaching and Research, IV, no. 1, Wydawnictwo Collegium Mazovia, Siedlce, 2013, 16–23

[17] Guţu V., “Sums of convex compacta as attractors of hyperbolic IFS's”, Topol. Methods Nonlinear Anal., 54:2B (2019), 967–978 | MR | Zbl

[18] Hata M., “On the structure of self-similar sets”, Japan J. Appl. Math., 2 (1985), 381–414 | DOI | MR | Zbl

[19] Hutchinson J. E., “Fractals and self similarity”, Indiana Univ. Math. J., 30:5 (1981), 713–747 | DOI | MR | Zbl

[20] Kulczycki M., Nowak M., “A class of continua that are not attractors of any IFS”, Cent. Eur. J. Math., 10:6 (2012), 2073–2076 | MR | Zbl

[21] Kwieciński M., “A locally connected continuum which is not an IFS attractor”, Bull. Pol. Acad. Sci. Math., 47:2 (1999), 127–132 | MR | Zbl

[22] Leśniak K., Snigireva N., Strobin F., “Weakly contractive iterated function systems and beyond: a manual”, J. Difference Equ. Appl., 26:8 (2020), 1114–1173 | DOI | MR

[23] Leśniak K., Snigireva N., Strobin F., Vince A., “Transition phenomena for the attractor of an iterated function system”, Nonlinearity, 35:10 (2022), 5396-5426 | DOI | MR

[24] Matias E., Diaz L. J., Non-hyperbolic Iterated Function Systems: attractors and stationary measures | DOI

[25] Riddle L., (seen 19.09.2024) https://larryriddle.agnesscott.org/ifs/pythagorean/pythTree.htm

[26] Rus I. A., Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001 | MR | Zbl

[27] Sanders M. J., “An $n$-cell in $\mathbb{R}^{n+1}$ that is not the attractor of any IFS on $\mathbb{R}^{n+1}$”, Missouri J. Math. Sci., 21:1 (2009), 13–20 | DOI | MR | Zbl