Entropy Dimension And Preimage Entropy Dimension Of Semigroup Actions

Entropy is an important invariant to describe the complexity of dynamical systems.Entropy and preimage entropy of free semigroup actions describe the complexity of non-zero entropy systems. In order to study the complexity of zero entropy and zero preimageentropy of the free semigroup actions, we introduce the entropy dimension and preimageentropy dimension of them. The main aim of this paper is to introduce entropy dimensionand preimage entropy dimension for the free semigroup actions, and investigate theproperties of them. The main results are as follows:Firstly, we introduce the concept of topological entropy dimension of free semigroupactions by open covers and the separated sets, spanning sets, and study its basic proper-ties. We prove that it is a topological equi-conjugacy invariant.Secondly, we introduce the defnition of the measure-theoretic entropy dimension offree semigroup actions and study its basic properties. We show that it has the afneproperty, and obtain the relation between topological dimension.Thirdly, we introduce the concepts of several kinds of preimage entropy dimension offree semigroup actions by open covers and spanning sets, separated sets. We prove thatthey are all invariant under topological equi-conjugate,and obtian the relations betweenthem.