(untitled) (European Journal of Pure and Applied Mathematics, 2026)
This research has two parts, in the first part: we study the statistical characteristics of the reachability set R(t,σ,X) of the control system x=f(h^tσ,x,u), uU(h^tσ,x) (t,σ,x)R×Σ× R^n, which is parameterized using a metric dynamic system (Σ,P,μ, h^t). We examine such characteristics as the relative absorption frequency and the upper and lower relative frequencies of absorption of the reachability set. For the above system, we also provide sufficient conditions under which a given set M is statistically invariant with respect to the controlled system. In addition, we compare the distribution function defined on the control linear space with the corresponding relative frequency. In the second part, we investigate the role of the distribution function and its relationship to the stability of the dynamical system under different types of distributions.