Cyclic Cohomology And Minimal Unique Ergodic Dynamic Systems

We focus our thesis on classifying diffeomorphism in the sense of s-mooth flip conjugate by smooth crossed product algebras. Let M be a compact finite dimensional smooth manifold, α be a minimal unique er-godic diffeomorphism of it. Then we have a C*-algebra C(M) ×αZ. The classification of C*-algebras has always been a frontier theory of modern Mathematics. People apply the classification theory of C*-algebras to the classification of diffeomorphism and find that it is not satisfying. The essential reason is that C*-algebras fail to reflect information of grading. Professor Gong raises the following two questions:1 Is smooth crossed product good enough to reflect information of grad-ing?2 If so, is cyclic cohomology good enough to demonstrate this fact?In our thesis, we prove that smooth crossed product algebras and cyclic cohomology are what we want.We arrange our thesis as follows:1 In prologue, we introduce the background of our problem. The C*-crossed product algebras of two diffeomorphisms. which are obviously not smooth flip conjugate to each other, C(S3) ×α3 Z= C(S3) × α3 Z im-pulses us to find a better algebra. The interesting example discovered by Elliott and Gong inspires us to consider smooth crossed product algebras ad cyclic cohomology.2 In chapter 2, we introduce some necessary notions to our theory. Utiliz-ing spectral sequence and some other algebraic devices, we claim that to compute the grading structure is to compute groups E^. Also we indict the method to compute E∞n. The work of Alain Connes and Nest are vital to this chapter.3 In chapter 3,We construct some examples whose existence rely heavily on the structure of odd dimensional spheres to make the point.First we consider minimal unique ergodic diffeomorphism αn on S2n+1.We prove that C(S2n+1)×αnZ≌C(S2m+1)×αmZ.Then we prove that the cyclic co-homology of C∞(S2n+1)×αn and C∞(S2m+1)×αmZ bear different grading structure:Theorem 1We also construct two minimal unique ergodic diffeomorphisms α and β of S3×S6×S8,where α reverses the orientation of S6,β reverses the orientation of S8.We prove that C(S3×S6×S8)×αZ ≌C(S3×S6×S8)×βZ. However,HP1 of the corresponding smooth crossed product algebras are consist of direct summand of different gradesTheorem 2 Hcoeq0(S3×S6×S8)=C,Hcoeq0(S3×S6×S8,α)=C: Heq3(S3×S6×S8,α)=C,Heq9(S3×S6×S8,α)=C, All ther Heq*(S3×S6×S8,α)and Hcoeq(S3×S6×S8,α)are{0),i.e. E∞1(C∞(S3×S6×S8)×αZ) E∞3(C∞(S3×S6×S8)×αZ) E∞9(C∞(S3×S6×S8)×αZ) E∞11(C∞(S3×S6×S8)×αZ) are consist of one C direct summand,with all the others being{0}.Theorem 3 Hcoeq0(S3×S6×S8,β)=C,Hcoeq8(S3×S6×S8,β)=C: Heq3(S3×S6×S8,β)=C,Heq1(S3×S6×S8,β)=C, All other Heq*(S3×S6×S8,β)andHcoeq*(S3×S6×S8,β)are {0}.i.e. E∞1(C∞(S3×S6×S8)×βZ) E∞3(C∞(S3×S6×S8)×βZ) E∞7(C∞(S3×S6×S8)×βZ) E∞9(C∞(S3×S6×S8)×βZ) are consist of a C direct summand,with all the others being{0}.4 Chapter 4 are dedicated to the computation of cyclic cohomology theory of diffeomorphism of torus.We prove two diffeomorphism are not smooth conjugate to each other by cyclic cohomology. In the last we compute the grading structure of Furstenberg transformation:Theorem 4...