A note on the theorem of Johnson, Palmer and Sell
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A note on the theorem of Johnson, Palmer and Sell Davor Dragiˇcevi´c1
© Akadémiai Kiadó, Budapest, Hungary 2016
Abstract The well-known theorem of Johnson, Palmer and Sell asserts that the endpoints of the Sacker–Sell spectrum of a given cocycle 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 A in terms of the nonvanishing of Lyapunov exponents for 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 A . Keywords Sacker–Sell spectrum · Lyapunov exponents · Invariant measures · Stability Mathematics Subject Classification Primary 37C40 · 37C60
1 Introduction In their landmark paper [10], Sacker and Sell introduced the notion of (what is now called) the Sacker–Sell spectrum for cocycles over topological dynamical systems and they described all possible structures of the spectrum. Furthermore, they indicated a strong relationship between their spectral theory and the theory of Lyapunov exponents which plays a central role in the stability of dynamical systems. A deeper connection between those two theories was discovered in a remarkable paper [7] where the authors proved that the endpoints of the Sacker–Sell spectrum of a given cocycle are realized as Lyapunov exponents of the cocycle with respect to some ergodic probability measure which is invariant for the base on which the cocycle acts. The arguments in [7] heavily rely on the new proof of the celebrated Oseledets
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Davor Dragiˇcevi´c [email protected] School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
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multiplicative ergodic theorem presented in the same paper and on the study of the induced flow on the projective bundle. The main purpose of this note is to derive the theorem of Johnson, Palmer and Sell as a simple consequence of a more recent and independent result of Cao [5] who gave a beautiful criterion for the uniform hyperbolicity of cocycles in terms of nonvanishing Lyapunov exponents (see Theorem 3.1). We note that in general the cocycles with nonzero Lyapunov exponents exibit only a weaker form of the hyperbolicity which is known in the litrature as nonuniform hyperbolicity. We refer to [3] for the detailed exposition of this theory which goes back to the landmark works of Oseledets [8] and Pesin [9]. Furthermore, using the recent results on the continuity of Lyapunov exponents, we obtain some simple consequences regarding the stability of the Sacker–Sell spectrum with respect to perturbations of the cocycle. Although those observations are far from satisfactory, we hope that they could lead to new directions in the research on the stability of spectrum.
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