Geodesic
Attractors of semigroups generated by a finite family of contraction transformations of a complete metric space
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 26 (2024) no. 4, pp. 359-375.

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

The present paper is devoted to the properties of semigroup dynamical systems $(G,X)$, where the semigroup $G$ is generated by a finite family of contracting transformations of the complete metric space $X$. It is proved that such dynamical systems $(G,X)$ always have a unique global attractor $\mathcal{A}$, which is a non-empty compact subset in $X$, with $\mathcal{A}$ being unique minimal set of the dynamical system $(G,X)$. It is shown that the dynamical system $(G,X)$ and the dynamical system $(G_{\mathcal{A}},\mathcal{A})$ obtained by restricting the action of $G$ to $\mathcal{A}$ both are not sensitive to the initial conditions. The global attractor $\mathcal{A}$ can have either a simple or a complex structure. The connectivity of the global attractor $\mathcal{A}$ is also studied. A condition is found under which $\mathcal{A}$ is not a totally disconnected set. In particular, for semigroups $G$ generated by two one-to-one contraction mappings, a connectivity condition for the global attractor $\mathcal{A}$ is indicated. Also, sufficient conditions are obtained under which $\mathcal{A}$ is a Cantor set. Examples of global attractors of dynamical systems from the considered class are presented.
Keywords: semigroup dynamical system, global attractor, minimal set, sensitivity to initial conditions, system of iterated functions, Cantor set 
@article{SVMO_2024_26_4_a0,
     author = {A. V. Bagaev},
     title = {Attractors of semigroups generated by a finite family of contraction transformations of a complete metric space},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {359--375},
     publisher = {mathdoc},
     volume = {26},
     number = {4},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2024_26_4_a0/}
}
TY  - JOUR
AU  - A. V. Bagaev
TI  - Attractors of semigroups generated by a finite family of contraction transformations of a complete metric space
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2024
SP  - 359
EP  - 375
VL  - 26
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2024_26_4_a0/
LA  - ru
ID  - SVMO_2024_26_4_a0
ER  - 
%0 Journal Article
%A A. V. Bagaev
%T Attractors of semigroups generated by a finite family of contraction transformations of a complete metric space
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2024
%P 359-375
%V 26
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2024_26_4_a0/
%G ru
%F SVMO_2024_26_4_a0
A. V. Bagaev. Attractors of semigroups generated by a finite family of contraction transformations of a complete metric space. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 26 (2024) no. 4, pp. 359-375. http://geodesic.mathdoc.fr/item/SVMO_2024_26_4_a0/

[1] E. Kontorovich, M. Megrelishvili, “A note on sensitivity of semigroup actions”, Semigroup Forum, 76:1 (2008), 133–141 | DOI | MR | Zbl

[2] F.M. Schneider, S. Kerkhoff, M. Behrisch, S. Siegmund, “Chaotic actions of topological semigroups”, Semigroup Forum, 87 (2013), 590–598 | DOI | MR | Zbl

[3] J. Iglesias, A. Portela, “Almost open semigroup actions”, Semigroup Forum, 98 (2019), 261–270 | DOI | MR | Zbl

[4] A. Nagar, M. Singh, Topological dynamics of enveloping semigroups, Springer, Singapore, 2023, 87 pp. | MR

[5] N.I. Zhukova, “Sensitivity and chaoticity of some classes of semigroup actions”, Regular and Chaotic Dynamics, 29:1 (2024), 174–189 | DOI | MR | Zbl

[6] R. M. Kronover, Fraktaly i khaos v dinamicheskikh sistemakh. Osnovy teorii, Postmarket, M., 2000, 352 pp.

[7] M. F. Barnsley, Fractals everywhere, Academic Press, Boston, 1988, 394 pp. | MR | Zbl

[8] J. E. Hutchinson, “Fractals and self-similarity”, Indiana University Mathematics Journal, 30 (1981), 713–-747 | DOI | MR | Zbl

[9] K.J. Falconer, Fractal geometry: mathematical foundations and applications, John Wiley and Sons, New York, 2014, 400 pp. | MR | Zbl

[10] M. Yamaguti, M. Hata, J. Kigami, Mathematics of Fractals, American Mathematical Society, Providence, 1997, 90 pp. | DOI | MR | Zbl

[11] N.I. Zhukova, “Minimal sets of Cartan foliations”, Proceedings of the Steklov Institute of Mathematics, 256 (2007), 105–135 | DOI | MR | Zbl

[12] A.V. Bagaev, A.V. Kiseleva, “O mnogomernykh analogakh treugolnika Serpinskogo”, XXVI Mezhdunarodnaya nauchno-tekhnicheskaya konferentsiya «Informatsionnye sistemy i tekhnologii-2020» (N.Novgorod, 27-28 aprelya 2020 goda), Nizhegorodskii gosudarstvennyi tekhnicheskii universitet im. R.E.Alekseeva, N.Novgorod, 2020, 1148–1152