# A uniform exponential spectrum for linear flows on vector bundles

L. Grüne:
**A uniform exponential spectrum for linear flows on vector bundles**

*Journal of Dynamics and Differential Equations*
**12**,
435 - 448,
2000

DOI:
10.1023/A:1009024610394

Keywords: linear flow; uniform; exponential spectrum; Lyapunov exponent; accumulation point

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## Abstract:

For a linear flow on a vector bundle we define a uniform exponential spectrum. For a compact invariant set for the projected flow we obtain this spectrum by taking all accumulation points for the time tending to infinity of the union over the finite time exponential growth rates for all initial values in this set. Using direct arguments we show that for a connected compact invariant set this spectrum is a closed interval whose boundary points are Lyapunov exponents. For a compact invariant set on which the flow is chain transitive we show that this spectrum coincides with the Morse spectrum. In particular this approach admits a straightforward analytic proof for the regularity and continuity properties of the Morse spectrum without using cohomology or ergodicity results.