Lyapunov exponents with multiplicity 1 for deterministic products of matrices

We exhibit an explicit criterion for the simplicity of the Lyapunov spectrum of linear cocycles, either locally constant or dominated, over hyperbolic (Axiom A) transformations. This criterion is expressed by a geometric condition on the cocycle's behaviour over periodic points and associated homoclinic orbits. It allows us to prove that for an open dense subset of dominated linear cocycles over a hyperbolic transformation and for any invariant probability with continuous local product structure (including all equilibrium states of Hölder continuous potentials), all Oseledets subspaces are one-dimensional. Moreover, the complement of this subset has infinite codimension and, thus, is avoided by any generic family of cocycles described by finitely many parameters. This improves previous results of Bonatti, Gomez–Mont and Viana where it was shown that some Lyapunov exponent is non-zero, in a similar setting and also for an open dense subset.