A. V. Bagaev, D. M. Ganeeva

Nizhny Novgorod State Agrarian and Technological University named after L.Ya. Florentyev (Nizhny Novgorod, Russian Federation)

Abstract. In this paper we consider classical iterated function systems (IFS) consisting of a finite number of contracting mappings for a complete metric space. The main goal is to study the class of IFSs whose attractors are Cantor sets, i.e. perfect totally disconnected sets. Important representatives of this class are totally disconnected IFSs introduced by Barnsley. We have proposed other definitions of a totally disconnected IFS and proved their equivalence to the Barnsley definition. Sufficient conditions for IFS to be totally disconnected are obtained. It is shown that injectivity of mappings from an IFS implies the perfection of the attractor and its uncountability. Also it is proved that if the mappings from an IFS are injective and the sum of their contraction coefficients is less than one, then the attractor is a Cantor set. In general case, these conditions do not guarantee totally disconnectedness of an IFS. Meanwhile, it is shown that if an IFS consists of two injective mappings and the sum of their contraction coefficients is less than one, then the IFS is totally disconnected. Examples of IFS attractors are constructed, demonstrating that conditions of the proven theorems are only sufficient but not necessary.

Key Words: iterated function system, attractor, Cantor set, address space, address function

For citation: A. V. Bagaev, D. M. Ganeeva. Iterated function systems whose attractors are Cantor. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 27:3(2025), 287–301. DOI: https://doi.org/10.15507/2079-6900.27.202503.287-301

Information about the authors:

Andrey V. Bagaev, Associate Professor, Department of Fundamental Mathematics, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603155, Russia), Ph.D. (Phys.-Math.), ORCID: http://orcid.org/0000- 0001-5155-4175, abagaev@hse.ru

Diana M. Ganeeva, Research Assistant of International laboratory of Dynamical Systems and Applications, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603155, Russia), ORCID: http://orcid.org/0009-0001- 4679-9335, dganeeva@hse.ru