Аттракторы полугрупп, порожденных конечным семейством сжимающих преобразований полного метрического пространства

Журнал Средневолжского математического общества. 2024. Т. 26. № 4. С. 359–375.

Bagaev A.

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 \scrA , which is a non-empty compact subset in X, with \scrA being unique minimal set of the dynamical system (G, X). It is shown that the dynamical system (G, X) and the dynamical system (G\scrA , \scrA ) obtained by restricting the action of G to \scrA both are not sensitive to the initial conditions. The global attractor \scr A can have either a simple or a complex structure. The connectivity of the global attractor \scr A is also studied. A condition is found under which \scr 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 \scr A is indicated. Also, sufficient conditions are obtained under which \scr A is a Cantor set. Examples of global attractors of dynamical systems from the considered class are presented.

Keywords: global attractor, глобальный аттрактор, minimal set, минимальное множество, sensitivity to initial conditions, Cantor set, semigroup dynamical system, system of iterated functions, канторово множество, система итерированных функций, полугрупповая динамическая система, чувствительность к начальным условиям