Conley index and stable sets for flows on flag bundles

TL;DR

This paper investigates the stable sets and Conley indices of Morse components in flows on flag bundles associated with noncompact semi-simple Lie groups, extending previous Morse decomposition results.

Contribution

It characterizes stable sets and computes Conley indices for Morse components in flows on flag bundles of semi-simple Lie groups, under certain assumptions.

Findings

Stable sets of Morse components are described.

Conley indices are computed under additional assumptions.

Extends Morse decomposition analysis to new settings.

Abstract

Consider a continuous flow of automorphisms of a G-principal bundle which is chain transitive on its compact Hausdorff base. Here G is a connected noncompact semi-simple Lie group with finite center. The finest Morse decomposition of the induced flows on the associated flag bundles were obtained in previous articles. Here we describe the stable sets of these Morse components and, under an additional assumption, their Conley indices.