A note on the theorem of Johnson, Palmer and Sell

Abstract

The well-known theorem of Johnson, Palmer and Sell asserts that the endpoints of the Sacker–Sell spectrum of a given cocycle $\mathcal {A}$ over a topological dynamical system (M, f) are realized as Lyapunov exponents with respect to some ergodic invariant probability measure for f. The main purpose of this note is to give an alternative proof of this theorem which uses a more recent and independent result of Cao which formulates sufficient conditions for the uniform hyperbolicity of a given cocyle $\mathcal {A}$ in terms of the nonvanishing of Lyapunov exponents for $\mathcal {A}$. We also discuss the possibility of obtaining positive results related to the stability of the Sacker–Sell spectra under the perturbations of the cocycle $\mathcal {A}$.