Attractors of semigroups generated by a finite family of contraction transformations of a complete metric space

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.