Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents

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Cocycles with one exponent over partially hyperbolic systems (2011, methodology): “From a different angle, such cocycles over hyperbolic and partially hyperbolic systems were considered in [2,21].”
Extremal Lyapunov exponents: an invariance principle and applications (2010, methodology): “Assuming the base dynamics (g, μ) is fairly “chaotic” (hyperbolic, possibly in a non-uniform fashion), there is now a good understanding of such issues as the existence of non-zero Lyapunov exponents (see Bonatti, Gomez-Mont, Viana [14, 43]) or the simplicity of the Lyapunov spectrum (see Avila, Bon”
Physical measures and absolute continuity for one-dimensional center direction (2013, background): “(See Pugh, Viana and Wilkinson [51].)”
Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows (2011, background): “Also important for our proofs here are the results of Avila, Santamaria and Viana [3] for cocycles over volume-preserving partially hyperbolic systems, both linear and smooth.”
Complex one-frequency cocycles (2013, background): “of cocycles over hyperbolic transformations: Indeed, non-zero Lyapunov exponents tend to appear robustly already for Ho¨lder regularity, even in the presence of topological obstructions to domination [V]. But we will show in this paper that Man˜´e’s picture turns out to hold unexpectedly in very lar”