Lyapunov exponents with multiplicity 1 for deterministic products of matrices
Lyapunov Exponents | linear cocycles
It has been shown by Bonatti, Gomez-Mont, Viana that generic dominated linear cocycles over a hyperbolic (Axiom A) transformation $f:M\to M$ have some non-zero Lyapunov exponent, for every equilibrium state of $f$ associated to a Hólder continuous potential. Here we prove a stronger fact: For an open dense subset of dominated linear cocycles over a hyperbolic transformation, all the Oseledets subspaces are $1$-dimensional. This subset is defined by an explicit geometric condition on the cocycle's behaviour over periodic orbits and associated homoclinic points. Its complement has infinite codimension and, thus, is avoided by any generic family of cocycles described by finitely many parameters.