Some classical theorems, properties and examples of ergodic theory of dynamical systems

Long periods of observation of particles with the same erosion in a certain area of the phase space of microstates require the use and justification of the hypothesis that all available microstates are equiprobable. This hypothesis is equivalent to provisions on ergodicity in Hamiltonian systems; the ergodicity is that consecutive measurements of the states of an individual particle give the same result as measurements of the state of a whole system. The article discusses a number of successive theorems and properties of measure-preserving transformations of dynamic systems. We have generalized some important properties of dynamical systems such as ergodicity, confusion, isomorphism, identified their relationships, improved proof of some classical theorems.