Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems

In this paper, we introduced the concept of pseudo-convex open covering of topological spaces and entropy for topological spaces with such covering. Entropy was previously defined only for open coverings of compact topological spaces. It is shown by examples that the classes of topological spaces for which the concept of entropy is defined is quite wide. A discrete random process describing the evolution of the phase space of closed dynamical systems is built. Two random processes are constructed, one in which the elements of the transition matrices depend on the first indices, and the second Markov`s process, one in which the elements of the transition matrices do not depend from the first indices. The construction of transition matrices is based on the fact that the probability of a change in the phase space of a system to another space is proportional to the entropy of this other space. Based on the concept of entropy of topological spaces and on the well-known construction of an infinite product of probability measures which is also probabilistic introduced the concept entropy of trajectory of evolution of phase space of system. Previously, the entropy of the trajectory was defined only for the motion of a structureless material point, in this article is defined entropy of trajectory of evolution of structured objects represented in the form of topological spaces. On the basis of the concept entropy of trajectory, a method is determined for finding the most probable trajectory of evolution of the phase space of a closed system.