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.


Introduction
The evolution of real dynamical systems is reflected in the change intheir entropy. This fact must be taken into account in the mathematical modeling of such systems, so it is important to define the entropy concepts of the mathematical objects in which the dynamic systems are represented in the models. Often modeling of a dynamical systems (it`s phase spaces) is considered to be a continuous or discrete sequence of topological spaces that describe the continuous or discrete time variability of the system. This sequence we called trajectory of changes of phase space. The notion of entropy of topological spaces has not been defined until recently. Until now, has been defined the concept of the entropy of covering of compact topological spaces [8,9]. In this work we have tried to define entropy directly for topological spaces and define entropy of trajectory.
This notion of entropy of a trajectory differs from other similar definitions. In other definitions, the trajectory is considered as trace of the movement of an unstructured point in the phase space [8][9][10]15] and here as the trajectory of the changes of structured object, phase space of the system.

Entropy for Topological Space
Definition1. We called a pseudo-convex open covering of space X the covering containing only contractible [5] elements, such sets whose intersections with a finite number are also contractible.  [5] to X . Every convex set is contractible. The space X is deformation retract [5] in is the basis consisting bay convex sets in , since r deformation retraction [5]. The set which is not contractible. So we got a contradiction non contractible set homotopy equivalent to contractible. It follows that our assumption is wrong. Because for open convex sets from has a pseudo-convex covering, similarly to previous cases the set X has a pseudoconvex covering.
So, we see that the above class of spaces for which entropy is defined is quite wide.
It is clear that if { } O α is finite pseudo convex-covering of compact topological space X , then geometric realization of the nerve ({ }) N O α homotopy equivalent to X Let   1  2  3 , , ,..., ,... i X X X X the discrete sequence of topological spaces that describe the evolution of phase space of a some dynamical system in discrete time and let , 1, 2,... i X i = compact topological spaces which admits pseudo-convex coverings. If a system is closed, then with the evolution of such a system, the entropy grows. Therefore  i with equality value of entropy. We construct a random process with discrete time [2,13] in this way Let sequence This matrix is the transition matrix for transition random objects (classes) set i ℜ to random objects set 1 i+ ℜ .

Evolution of Phase Space of Closed Dynamical Systems
In this way we have built the random process [2,13] with discrete time which describe evolution phase space of closed dynamic system.
If we do not admit that the entropy of phase space of a dynamical system does not always increases, then spaces which minimal pseudo convex covering contains, i n element, which, as we known, has homotopy type of geometric realization of nerve of this covering. The lasses of every geometric simplicial complex [5] with i n vertex also is included in set i ℜ .
If we choose element from each class we will get sequence The mathematical expectation of this random quantity will be

The Infinity Product of Probability Measures and Most Probabilistic Trajectories
In this section we consider the question in another aspect.
Consider the direct product of sets A ≠ ℜ , these sets are called cylindrical sets, they make up a semi algebra of sets [11,12], we denote this semi algebra by ( ) Β ℜ and smallest σ -algebra generated by ( ) Β ℜ so ( ( )) σ Β ℜ .