Conditions for Equality between Lyapunov and Morse Decompositions

TL;DR

This paper establishes necessary and sufficient conditions for the Lyapunov and Morse decompositions of automorphism flows on principal bundles to coincide, ensuring continuity of Lyapunov spectra under certain perturbations.

Contribution

It provides a characterization of when the Morse and Lyapunov decompositions are equal in the context of reductive group principal bundles with automorphism flows.

Findings

Necessary and sufficient conditions for decomposition equality.

Equality implies continuity of Lyapunov spectra under perturbations.

Bridges Morse and Lyapunov decomposition theories.

Abstract

Let $Q\rightarrow X$ be a continuous principal bundle whose group $G$ is reductive. A flow $\phi $ of automorphisms of $Q$ endowed with an ergodic probability measure on the compact base space $X$ induces two decompositions of the flag bundles associated to $Q$. A continuous one given by the finest Morse decomposition and a measurable one furnished by the Multiplicative Ergodic Theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under pertubations leaving unchanged the flow on the base space.