Conley index for manifold-valued retarded functional differential equations without uniqueness of solutions

We develop a Conley index theory for retarded functional differential equations $\dot x=f(x_{t})$ with values in a differentiable manifold and (merely) continuous nonlinearities f. We use this index to establish an existence result for nonconstant full solutions of such equations.

LUSTERNIK–SCHNIRELMANN CATEGORY BASED ON THE DISCRETE CONLEY INDEX THEORY

We study Lusternik–Schnirelmann type categories for isolated invariant sets by the use of the discrete Conley index.

The arnold conjecture

This chapter contains a proof of the Arnold conjecture for the standard torus, which is based on the discrete symplectic action. The symplectic part of this proof is very easy. However, for completeness of the exposition, one section is devoted to a fairly detailed discussion of the relevant Conley index theory and of Ljusternik–Schnirelmann theory. Closely related to the problem of finding symplectic fixed points is the Lagrangian intersection problem. The chapter outlines a proof of Arnold’s conjecture for cotangent bundles that again uses the discrete symplectic action, this time to construct generating functions for Lagrangian submanifolds. The chapter ends with a brief outline of the construction and applications of Floer homology.