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Periodica Mathematica Hungarica

, Volume 75, Issue 2, pp 167–171 | Cite as

A note on the theorem of Johnson, Palmer and Sell

  • Davor DragičevićEmail author
Article

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 (Mf) 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 \(\mathcal {A}\).

Keywords

Sacker–Sell spectrum Lyapunov exponents Invariant measures Stability 

Mathematics Subject Classification

Primary 37C40 37C60 

Notes

Acknowledgements

I would like to thank Gary Froyland for carefully reading this manuscript and for many useful comments. Furthermore, I am thankful to my collaborators Luis Barreira and Claudia Valls for introducing me to Sacker–Sell spectral theory. The author was supported by an Australian Research Council Discovery Project DP150100017 and in part by Croatian Science Foundation under the project IP-2014-09-2285. The paper is dedicated to the memory of P. R. Nelson (1958–2016).

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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