International Congress of Mathematicians

306 Yiming Longform matrices, which we call basic normal forms. Correspondingly by the homotopyinvariance and symplectic additivity of the index theory, the computations in (2.3)are reduced to iterations of those paths in Sp(2) or Sp(4) whose end points areone of the 10 basic normal form matrices. The study of the index for iterationsof any symplectic paths is carried out for paths in Sp(2) via the R 3 -cylindricalcoordinate representation of Sp(2), then for hyperbolic and elliptic paths in Sp(2n).This yields the precise iteration formula obtained in [29] of the index theory for anysymplectic path 7 £ V T (2n) in terms of the basic norm form decomposition of 7(7-),(z(7, l),i/( 7 ,1)), and the iteration time TO.For any M £ Sp(2n), its splitting numbers at an OJ £ U is defined in [27] byS M (OJ)= Hm t u (±v=Te) (7)-t u (7), (2.5)via any 7 £ V T (2n) satisfying 7(7-) = M. Then it is proved that the splittingnumbers of M at OJ can be characterized algebraically.Motivated by the precise iteration formulae of [29], the following second indexiteration formula of any symplectic path is established by C. Zhu and the author.Here we denote by (z( 7 ,TO), 1/(7, TO)) = (ii(7), ^1(7)).Theorem 3 (cf. [34]). For any r > 0, 7 £ V T (2n), and m £ N, there holds:*(7,m) = m(*(7,l) + S+(l)-C(M))n,2 52 EAs M (e^=Tl> )^(S M (i) + C(Mj), (2.6)9G(0,2TT)2TTwhere M = 7(r), C(M) = X^O