Some Results and Examples on The Relation of Leray Equivalent and (Strong) Shift Equivalent

Using the concept of Leray functor and the way it is used to define the Cohomological Conley index, we define Leray equivalent, Leray shift equivalent, Leray elementary strong shift equivalent and Leray strong shift equivalent. We established their relationship with shift equivalent, strong shift equivalent and elementary strong shift equivalent. Then, we show that in the setting of Artinian and Noetherian module Leray equivalent implies strong shift equivalent.

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.

The E-Cohomological Conley Index, Cup-Lengths and the Arnold Conjecture on T 2n

We show that the E-cohomological Conley index, that was introduced by the first author recently, has a natural module structure. This yields a new cup-length and a lower bound for the number of critical points of functionals on Hilbert spaces. When applied to the setting of the Arnold conjecture, this paves the way to a short proof on tori, where it was first shown by C. Conley and E. Zehnder in 1983.