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On the simplicity of the Lyapunov spectrum of products of random matrices


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.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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