Index Iteration Theory for Symplectic Paths and Periodic Solution Orbits

It is well known that the geodesic, i.e., the shortest curve, connecting two prescribed points in the Euclidean plane is the line segment connecting them. But the geodesic, especially the closed geodesic, problem on the earth is very difficult. In fact, the closed geodesic problem is a very important subject in both dynamical systems and differential geometry, and has stimulated many creative ideas and new developments in mathematics. For closed geodesics on spheres with Riemannian structures or Finsler structures, modern mathematical studies can be traced back at least to the work of J. Hadamard, H. Poincaré, G. D. Birkhoff, M. Morse, L. Lyusternik, L. Schnirlmann, and many other famous mathematicians. In this short survey, I can only introduce some of the vast literature which is related to closed geodesics on 2-spheres and to our current interests. This paper is organized as follows: §1 A partial and certainly not complete history of the studies of closed geodesics mainly on 2-spheres. §2 The multiplicity result obtained by Victor Bangert and the author, and the stability result obtained by Wei Wang and the author about closed geodesics on Finsler 2-spheres. §3 Main ideas in the proof of the multiplicity theorem of V. Bangert and the author. §4 Open problems. ∗Partially supported by the 973 Program of MOST, Yangzi River Professorship, NNSF, MCME, RFDP, LPMC