The characteristics of attainability set connected with invariancy of control systems on the finite time interval

We study the statistical characteristics of the attainability set $A(t,\sigma,X)$ of the control system which is parametrized by means of a topological dynamical system $(\Sigma,h^t).$ We obtain the lower estimates for characteristics connected with invariance of given set on a finite time interval. We also consider the following problem arising in many applications. Let numbers $\lambda_0\in (0,1]$ and $\vartheta>0$ are given. It is necessary to find the conditions which the control system and set $X$ should satisfy providing that for given $\sigma\in\Sigma$ relative frequency of containing of the attainability set $A(t,\sigma,X)$ in the given set $M$ on any interval of time length $\vartheta$ would be not less then $\lambda_0$. Let's notice, that the characteristic $\vartheta$ is assumed given depending on an applying problems. In particular, if control process is periodic, then $\vartheta$ is the period of the process. Results are illustrated by examples of the control systems which describe different models of population growth.