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On Hölder‐continuity of Oseledets subspaces

For Hölder cocycles over a Lipschitz base transformation, possibly non‐invertible, we show that the subbundles given by the Oseledets Theorem are Hölder‐continuous on compact sets of measure arbitrarily close to 1. The results extend to vector bundle automorphisms, as well as to the Kontsevich–Zorich cocycle over the Teichmüller flow on the moduli space of abelian differentials. Following a recent result of Chaika–Eskin, our results also extend to any given Teichmüller disk.