On the simplicity of the Lyapunov spectrum of productsof random matrices

Abstract

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

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