Dynamical system of translations in the space of multi-valued functions with closed images

In the work there is considered the dynamical system of translations in the space $\mathfrak R$ of continuous multi-valued functions with images in complete metric space $(\mathrm{clos}(\mathbb R^n),\rho_\mathrm{cl})$ of nonempty closed subsets of $\mathbb R^n$. The distance between such functions is measured by means of the metric analogous to the Bebutov metric constructed for the space of continuous real-valued functions defined on the whole real line. It is shown that for compactness of the trajectory's closure in $\mathfrak R$ it is sufficient to have initial function bounded and uniformly continuous in the $\rho_\mathrm{cl}$ metric. As consequence, it is also proved that the trajectory's closure of a recurrent or an almost periodic motion is compact in $\mathfrak R$.