Morse and Lyapunov Spectra and Dynamics on Flag Bundles

TL;DR

This paper investigates the spectral properties of flows on principal bundles with semi-simple groups, focusing on Lyapunov and Morse exponents, their symmetry, and their geometric placement within Weyl chambers, with applications to linear flows on vector bundles.

Contribution

It establishes symmetry properties of spectral sets and their localization in Weyl chambers for flows on flag bundles, extending understanding of their dynamical and geometric structure.

Findings

Lyapunov and Morse exponents are invariant under the Weyl group.

Spectral sets are contained within specific Weyl chambers.

Results apply to linear flows on vector bundles.

Abstract

This paper studies characteristic exponents of flows in relation with the dynamics of flows on flag bundles. The starting point is a flow on a principal bundle with semi-simple group $G$. Projection against the Iwasawa decomposition $G = KAN$ defines an additive cocycle over the flow with values in $\frak{a} = \log A$. Its Lyapunov exponents (limits along trajectories) and Morse exponents (limits along chains) are studied. It is proved a symmetric property of these spectral sets, namely invariance under the Weyl group. It is proved also that these sets are located in certain Weyl chambers, defined from the dynamics on the associated flag bundles. As a special case linear flows on vector bundles are considered.