International Congress of Mathematicians

304 Yiming Longtheory. This index theory was further defined for non-degenerate paths in Sp(2)by E. Zehnder and the author in [33] of 1990. The index theory for degeneratelinear Hamiltonian systems was defined by C. Viterbo in [39] and the author in [20]of 1990 independently. In [25] of 1997, this index was extended to any degeneratesymplectic matrix paths.Motivated by the iteration theories for the Morse type index theories establishedby R. Bott in 1956 and by I. Ekeland in 1980s, in recent years the authorextended the index theory mentioned above, introduced an index function theory forsymplectic matrix paths, and established the iteration theory for the index theoryof symplectic paths. Applying this index iteration theory to nonlinear Hamiltoniansystems, interesting results on periodic solution problems of Hamiltonian systemsare obtained. Here a brief survey is given on these subjects. Readers are referredto the author's recent book [30] for further details.1. Index function theory for symplectic pathsAs usual we define the symplectic group by Sp(2n) = {M £ GL(R 2 ") | M T JM= J}, where J = \ J, J is the identity matrix on R", and M T denotes thetranspose of M. For OJ £ U, the unit circle in the complex plane C, we define thew-singular subset in Sp(2n) by Sp(2n)2, = {M £ Sp(2n) |a! _ "det(7(r) — OJ I) = 0}.Here for any M £ Sp(2n)°, we define the orientation of Sp(2n)2, at M by thepositive direction -f^Mexp(tJ)\t=o- Since the fundamental solution of a generallinear Hamiltonian system with continuous symmetric periodic coefficient 2n x 2nmatrix function B(t),x(t) = JB(t)x(t), Vt€R, (1.1)is a path in Sp(2n) starting from the identity, for r > 0 we define the set ofsymplectic matrix paths by V T (2n) = {7 £ C([0,r],Sp(2n)) | T(0) = I}- For anytwo path £ and n : [0,r] —¥ Sp(2n) with £(r) = n(0), as usual we define n * Ç(t)by Ç(2t) if 0 < t < T/2, and n(2t — r) if r/2 < t < r. We define a special pathC:[0,r]->Sp(2n)bydt) = diag(2 - -,..., 2 - -, (2 - -)-\..., (2 - -)" 1 ), for 0 < t < T.T T T TDefinition 1. (cf. [27]) For any r > 0, OJ £ U, and 7 £ V T (2n), we definethe uj-nullity of 7 byVu(l) = dim c ker c (7(r) -UJI). (1.2)7/7 is OJ non-degenerate, i.e., ^(7) = 0, we define the oj-index of 7 by the intersectionnumberM7) = [Sp(2 0, we let Td) be the set of all open neighborhoods0/7 in V T (2n), and defineiu(l) = sup inf{i u (ß) I ß £ U, v u (ß) = 0}. (1.4)