Measure-theoretic Entropy Properties And Calculation Of Nonautonomous Dynamical Systems

The main task of this paper is to introduce the concept of the measure-theoretic entropyfor the nonautonomous dynamical systems and study its properties and calculation method.There are four sections in this paper.In the first section, we give some background of the study of the measure-theoretic entropyand introduce measure-theoretic entropy for the nonautonomous dynamical systems.In the second section, we give some properties for the measure-theoretic entropy of thenonautonomous dynamical systems. For example, if (X, B,m) is a probability space and preserves a measure m on X, then for any k∈N, we have where is induced by . In addition, we show that the measure-theoretic entropy for the nonautonomous dynamical systems has the affine property. That is, if(X,B,m) is a probability space, m,μare f1,∞-invariant measures and p∈[0,1], thenIn the third section, we discuss the relationship between the measure-theoretic entropy andthe topological entropy and obtain the following inequalityIn the forth section, we improve some results of the nonautonomous dynamical systems inthe framework of the random dynamical systems.