Differentiability of Lyaupnov Exponents

TL;DR

This paper proves the differentiability of specific linear combinations of Lyapunov exponents for flows on principal bundles of semi-simple Lie groups, based on Morse decompositions and gauge group structures.

Contribution

It establishes the differentiability of Lyapunov spectra linear combinations in a new setting involving principal bundles and Morse decompositions.

Findings

Differentiability of certain Lyapunov spectra linear combinations is proven.

The differentiability depends on the Morse decomposition of the flag bundles.

Results are obtained within the framework of Banach-Lie gauge groups.

Abstract

We prove differentiability of certain linear combinations of the Lyapunov spectra of a flow on a principal bundle of a semi-simple Lie group. The specific linear combinations that yield differentiability are determined by the finest Morse decomposition on the flag bundles. Differentiability is taken with respect to a differentiable structure on the gauge group, which is a Banach-Lie group.