Fine properties of Lp-cocycles which allow abundance of simple and trivial spectrum

TL;DR

This paper extends previous results on Lp-cocycles, showing that a broad class has simple or trivial spectrum, with implications for linear differential systems and spectrum analysis.

Contribution

It generalizes prior work to include a wider class of Lp-cocycles, establishing conditions for simple and trivial spectra using new approaches.

Findings

Accessible and saddle-conservative cocycles have simple spectrum.

Residual subsets of accessible cocycles exhibit one-point spectrum.

Linear differential systems also display these spectral properties.

Abstract

In this paper we generalize [3] and prove that the class of accessible and saddle-conservative cocycles (a wide class which includes cocycles evolving in GL(d,R), SL(d,R) and Sp(d,R) Lp-densely have a simple spectrum. We also generalize [3, 1] and prove that for an Lp-residual subset of accessible cocycles we have a one-point spectrum, by using a different approach of the one given in [3]. Finally, we show that the linear differential system versions of previous results also hold and give some applications.