Topological Complexities Of Group Actions

This paper mainly introduces the various complexities of different group action systems and expounds the relationship between them.Without special explanation,the groups in this paper are countable discrete groups,and all group actions are continuous.For a given group,it can act continuously on a compact metric space.This group and this group action are regarded as a whole,which is called group action system.In particular,for a dynamical system,it can be regarded as an integral group action.For group action systems,the parameters of qualitative analysis describing the complexity of the system include sensitivity,chaos,mixing,etc.For these parameters,the common ones are Li-Yorke chaos,distributed chaos,multi-sensitivity,mean sensitivity,weak mixing,etc.The parameters of quantitative analysis are: topological dimension,topological entropy,topological sequence entropy,topological mean dimension,etc,which are topological conjugate invariants.Precisely because there are many parameters describing the complexity of group action systems and not all variables have a fixed strength relationship,so the study of the relationship between these parameters describing the complexity of group action systems is very meaningful.For amenable group actions,this paper mainly studies the relationship between topological entropy and Banach mean sensitivity and the calculation of the mean dimension of full shifts on amenable group action,some results are obtained.For the action of free semi-groups,the definition of mean dimension on it is given and some good properties of mean dimension under this definition are proved.For the action of general countable groups,the relationship between topological sequence entropy and multi-sensitivity is given.The specific conclusions are as follows:1.The concept of Banach mean equicontinuity for amenable group actions is defined,and it is proved that Banach-,Weyl-,and Besicovitch mean equicontinuity are equivalent for abelian group systems.Moreover,it is proved a transitive almost Banach mean equicontinuous abelian group action system has topological entropy 0.Because Banach mean sensitivity and almost Banach equicontinuity are inversely defined under transitive action,this conclusion is equivalent to that for transitive abelian group action,if it has positive topological entropy,then it is Banach mean sensitive2.It is proved that the mean dimension of full shifts on finite dimensional compact metric space over amenable group actions still equal to the stable dimension of base space.This generalizes the version of Tsukamoto under the action of integral group.3.The definition of mean dimension under the action of free semi-groups is given.Although under this definition,the mean dimension of Bernoulli shift under the free semigroup action is equal to infinity rather than 1,other important properties of subgroup formula,addition formula,finite entropy implies mean dimension 0 and SBP implies mean dimension 0 continue.4.For general countable group action,it is proved that if it is multi-sensitive,then there is a sequence such that the topological sequence entropy respect to the sequence is greater than 0.If a locally connected condition is added,the supremum of the topological sequence entropy will be equal to infinity.In addition to these main results,the author also obtains some results for the mean dimension of the hyperspace induced by the dynamic system: for the full shifts on the finite dimensional compact metric space,the value that the mean dimension of the hyperspace system induced by it can only take 0 or infinity,and the mean dimension of the hyperspace is equal to infinity if and only if the topological dimension of the base space is greater than 0.