A version of the theorem of Johnson, Palmer and Sell for quasicompact cocycles

TL;DR

This paper extends a classical theorem relating spectrum endpoints to Lyapunov exponents from invertible matrix cocycles to quasicompact cocycles of operators on Banach spaces, broadening its applicability.

Contribution

It establishes a version of Johnson, Palmer, and Sell's theorem for quasicompact cocycles on Banach spaces, generalizing previous results beyond finite-dimensional matrices.

Findings

Endpoints of the Sacker--Sell spectrum are realized as Lyapunov exponents for quasicompact cocycles.

The result applies to operators on arbitrary Banach spaces, not just finite-dimensional cases.

The theorem broadens the understanding of spectral and dynamical properties of operator cocycles.

Abstract

The well-known theorem of Johnson, Palmer and Sell asserts that the endpoints of the Sacker--Sell spectrum of a given cocycle of invertible matrices over a topological dynamical system $(M, f)$ are realized as Lyapunov exponents with respect to some ergodic invariant probability measure for $f$. In this note we establish the version of this result for quasicompact cocycles of operators acting on an arbitrary Banach space.