Entropy Dimension Of Non-autonomous Dynamical Systems

Entropy dimension is a significant invariant for distinguishing the complexity ofzero entropy dynamical systems. There are many good results of studying autonomousdynamical systems. The main aim of this paper is to introduc entropy dimension for thenonautonomous dynamical systems and to investigate its properties.The main results areas follows:Firstly, we introduce the concept of topological entropy dimension by the separatedsets, spanning sets and open covers for the nonautonomous topological dynamical systemsand study its basic properties. We prove that it is a topological equi-conjugacy invariant.Secondly, we introduce the concept of the measure-theoretic entropy dimension forthe nonautonomous dynamical systems and study its basic properties.We show that ithas the affine property, and moreover, there is a inequality which relates it and thetopological entropy dimension.Thirdly, we take some examples to calculate entropy dimension.For the arbitrarypositive real number s, we construct a sequence of monotone maps{f_}_i~∞=1on circles.Such that the topological entropy dimension and the measure-theoretic entropy dimensionare all equal to s. Hence we notice that unlike that for the autonomous systems, theentropy dimension for nonautonomous systems not only distinguishes the complexity ofzero entropy systems but also measures the complexity of the systems with positive, eveninfinite, entropies.