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On the simplicity of the Lyapunov spectrum of productsof random matrices

Published online by Cambridge University Press:  12 April 2001

LUDWIG ARNOLD
Affiliation:
Institut für Dynamische Systeme, Universität Bremen, Postfach 330 440, 28334 Bremen, Germany
NGUYEN DINH CONG
Affiliation:
Institut für Dynamische Systeme, Universität Bremen, Postfach 330 440, 28334 Bremen, Germany

Abstract

Assuming that the underlying probability space is non-atomic, we prove that products of random matrices (linear cocycles) with simple Lyapunov spectrum form an $L^p$-dense set ($1 \leq p < \infty$) in the space of all cocycles satisfying the integrability conditions of the multiplicative ergodic theorem. However, the linear cocycles with one-point spectrum are also $L^p$-dense. Further, in any $L^\infty$-neighborhood of an orthogonal cocycle there is a diagonalizable cocycle.

For products of independent identically distributed random matrices (with distribution $\mu$), simplicity of the Lyapunov spectrum holds on a set of $\mu$'s which is open and dense in both the topology of total variation and the topology of weak convergence, hence is generic in both topologies. For products of matrices which form a Markov chain, the spectrum is simple on a set of transition functions dense in the topology of weak convergence.

Type
Research Article
Copyright
1997 Cambridge University Press

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