Research On Complexity Of Several Dynamical Systems

In the investigation of topological dynamical systems,we can generate corresponding dynamical system according to different ways,such as iterative system,(generalized)inverse limit systems,dynamic system under group action,multiple mappings on hyperspace system and g-fuzzification.All of Iterative system,dynamical system under group action and multiple mappings are based on finite self-continuous maps.There is a close relationship between set-valued mapping on hyperspace and fuzzy system.Therefore,chaotic theories can be extended to these dynamical systems,and the complexity of these dynamical systems can be investigated.It is improtant to research the shadowing property,sensitivity and chaotic property for dynamical systems.In this paper,we mainly research the complexity of the above five dynamical systems.1.we introduce the definitions of shadowing property,average shadowing property and topological ergodicity for IFS(f0,f1)and give some examples.Then we show that(1)if IFS(f0,f1)has the shadowing property then so does f0 and f1;(2)IFS(f0,f1)has the shadowing property if and only if the step skew product corresponding to IFS(f0,f1)has the shadowing property.At last,we prove a Lyapunov stable iterated function system having the average shadowing property is topologically ergodic.2.We study the properties of shadowing,transitivity,weakly mixing,mixing,chain transitivity and chain mixing with respect to set-valued maps.For each of these problems we show that if the shift map on the inverse limit of a set-valued function has the property,then so does the set-valued function.We also provide a counter example in each case to show that the converse does not hold.Also,we prove shadowing has the iterative invariant property for set-valued maps.We show that for a set-valued map with shaodowing,the following statements are equivalent:(1)it is weakly mixing,(2)it is mixing,(3)it is chain mixing.3.we define and study Li-Yorke chaos and distributional chaos along a sequence for finitely generated semigroup actions.Let X be a compact space with metric d and G be a semigroup generated by f1,f2,…fm which are finitely many continuous mappings from X to itself.Then we show if(X,G)is transitive and there exists a common fixed point for all the above mappings,then(X,G)is chaotic in the sense of Li-Yorke.And we give a sufficient condition for(X,G)to be uniformly distributionally chaotic along a sequence and chaotic in the strong sense of LiYorke.At the end of this paper,an example on the one-sided symbolic dynamical system for(X,G)to be chaotic in the strong sense of Li-Yorke and uniformly distributionally chaotic along a sequence is given.4.We investigate Hausdorff metric Li-Yorke chaos,distributional chaos and distributional chaos in a sequence from a set-valued view.On the basis of this research,we draw main conclusions as follows:(i)If F has a distributionally chaotic pair,especially F is distributionally chaotic,the strongly non-wandering set S ?(F)contains at least two points.(?)We give a sufficient condition for F to be distributionally chaotic in a sequence and chaotic in the strong sense of Li-Yorke.Finally,an example to verify(?)is given.5.Firstly,we give a simpler proof of the necessity for a theorem in[1]which states that(K(X),f)is sensitive if and only if(F1(X),fg)is sensitive.Then we study the relations among the various forms of sensitivity of the systems(X,f),(K(X),f)and(F1(X),fg).We show that(1)(K(X),f)is thickly sensitive(resp.,thickly syndetically sensitive,thickly periodically sensitive,confinitely sensitive,ergodically sensitive,pointwise sensitive,colletively sensitive,asymptotically sensitive,Li-Yorke sensitive)if and only if(F1(X),fg)is thickly sensitive(resp.,thickly syndetically sensitive,thickly periodically sensitive,confinitely sensitive,ergodically sensitive,pointwise sensitive,colletively sensitive,asymptotically sensitive,Li-Yorke sensitive)for any g ? Dm(I)with g(-1)(1)={1}.(2)If(K(X),f)is thickly sensitive(resp.,thickly syndetically sensitive,thickly periodically sensitive),then(X,f)is thickly sensitive(resp.,thickly syndetically sensitive,thickly periodically sensitive).(3)(X,f)is cofinitely sensitive(resp.,strongly sensitive)(?)(K(X),f)is cofinitely sensitive(resp.,strongly sensitive)(?)(F1(X),fg)is cofinitely sensitive(resp.,strongly sensitive)for any g ?Dm(I)with g(-1)(1)={1}.(4)Construct a dynamical system(X,f)such that(X,f)is syndetically sensitive(resp.,ergodically sensitive,pointwise sensitive,collectively sensitive),but(K(X),f)is not syndetically sensitive(resp.,ergodically sensitive,pointwise sensitive,collectively sensitive).