Topologically Conjugate Classifications Of The Translation Actions On Lie Groups SU(2)×T~n

In the 1960s,S.Smale raised an open question,he hoped to give the smooth conjugate classification of all smooth self-maps of any smooth manifold,and then studied the dynamical properties of some smooth self-maps better.In the latest 50 years,Mathematicians have done a lot of works about the issue raised by S.Smale in many ways.We focus on those compact connected Lie groups and the translation actions on them.For the moment,we have already given the topologically conjugate classifications of the translation actions on low-dimensional compact connected Lie groups.In this paper,we investigate a class of non-commutative compact connected Lie groups with higher topological dimension,i.e.,SU(2)× Tm,and for any n ?Z+,give the topologically conjugate classification of the left actions on SU(2)× Tn(the relevant results of the right actions on SU(2)× Tn are completely the same as left actions).Firstly,for any set of real numbers {?}in=1,we give the definitions of its rank and reductive rank.Definition 1 Regard R as an infinite dimensional vector space over rational field Q.Let {?i}in=1 be n points in vector space R.Define the rank of {?i}in=1 over Q by Moreover,define the reductive rank of{ai}in=1 over Q bySecondly,under the background of the rotation vectors of the rotations on Tn,we give three important applications of the definitions of rank and reductive rank as follows.Proposition 1 Assume that {?i}i=1n,{?'j}j=1n(?)R satisfy Then Furthermore,if f,g are two rotations on Tn satisfying and where A'? G GLn(Z),thenProposition 2 Let f be a rotation on Tn satisfying If R?(f)=m,then there exists some rotation g on Tn with such that f and g are topologically conjugate.Furthermore,if R?(f)=R?(f)=m,then?1,?2,…,?m are m rationally independent irrational numbers.IfR?(f)=R?(f)-1=m-1,then there exists some rotation g' on Tn with where ?1,?2,…,?m-1 are m-1 rationally independent irrational numbers,but ?m=k/d is a rational number,and d,k?Z,gcd {d,k)=1 such that f and g' topologically conjugate.Proposition 3 Assume that f is a rotation on Tn satisfying?(f)=(?1 ?2…?n)T,?i?[0,1),i=1,2,…,n,and R?(f)=m.Then Orbf(e)is homeomorphic to the disjoint union of a group of Tm,where e is the identity element of Tn.Next,we utilize the relevant knowledge of cobordism theory,Whitehead torsion theory,fiber bundle theory,lens space theory and number theory to prove the followings four important lemmas.Lemma 1 Suppose that f1 is a rotation of Tn,f2 is a rotation of Tm,f:Tn?Tm is a continuous subjective map,and f1,f2,f satisfy fof1=f2of,i.e.,f is a topological semi-conjugacy form f1 to f2.For any if f*:?1(Tn)? ?1(Tm)induced by f satisfies f*:k?Ak,a?Mm×n(Z),we haveLemma 2 Define a composite map f:M?M by where n? 1,M is a 3-dimensional lens space satisfying that the self-homeomorphisms on it are all orientation-preserving self-homeomorphisms,i is a natural inclusion map,h is a self-homeomorphism on M × Tn,? is a projection.Then deg f=1 and f is homotopic to some orientation-preserving self-homeomorphism on M.Lemma 3 Let p1,p2,…,Pn ?Z,and gcd(p1,p2,…,pn)=1.Then there exists some matrix G ? GLn(Z)such that(p1,p2,…,pn)is just some row of G.Lemma 4 Assume that k0,k1,…·,km ?Z,gcd(k0,k1,…·,km)=1,and where zj ? C,|zj|=1,j=0,1,…,m.Then G(?)Tm is a subgroup of SU(2)x Tn,and SU 2)? Tn/G(?)L(k0,-1)× Tn-m,where L(k0,-1)is a 3-dimensional lens space.The above propositions and lemmas are all important for us to prove the main conclusion in this article.Next,we define the rotation vectors of the left actions induced by the elements in a maximum torus of SU(2)x Tn,and then give the main result of this paper.Fix one maximum torus of SU(2)× Tn as Obviously,we have TSU(2)×Tn(?)TSU(2)×Tm(?)Tn+1.Define a map ?:Tsu(2)×Tn?Tn byIt is easy to know that ? is a isomorphism from TSU(2)×Tn to Tn+1.For any Lg ?MTSU(2)×Tn,set f=?oLgTSU(2)×Tn o ?-1.Then one can see that f:Tn+1?Tn+1 is a rotation satisfyingTherefore,we define?(Lg)(?)P(f)=(?0 ?1…?n)T,?i?[0,1),i=0,1,…,n.Theorem 1 For the left actions Lg,Lg'? MTSU(2)×Tn with where ?i,?i?[0,1),i=0,1,…,n,Lg and Lg' are topologically conjugate if and only if where l=(l1 l2…ln)is a 1 × n integer matrix,and A ? GLn(Z).In fact,the equivalence relation in the above theorem gives a topologically con-jugate classification of the left actions induced by the elements in TSU(2)×Tn.Then according to the properties of maximum tori,we could utilize this equivalence relation to give a topologically conjugate classification of all left actions on SU(2)×Tn.Finally,we prove that for any positive integer n,the topologically conjugate clas-sification of the left actions on SU(2)x Tn is equivalent to their smooth conjugate classification,but not equivalent to their algebraically conjugate classification.