Nonautonomous Conley index theory. The homology index and attractor-repeller decompositions

Abstract

In a previous work, the author established a nonautonomous Conley index based on the interplay between a nonautonomous evolution operator and its skew-product formulation. This index is refined to obtain a Conley index for families of nonautonomous evolution operators. Different variants such as a categorial index, a homotopy index and a homology index are obtained. Furthermore, attractor-repeller decompositions and connecting homomorphisms are introduced for the nonautonomous setting.