Abstract
We give sharp regularity results for the invariant subbundles of hyperbolic dynamical systems and give open dense sets of codimension one systems where this regularity is not exceeded as well as open dense sets of symplectic, geodesic, and codimension one systems where the analogous regularity results of Pugh, Shub, and Wilkinson are optimal. We produce open sets of symplectic Anosov diffeomorphisms and flows with low transverse Hölder regularity of the invariant foliations almost everywhere. Prevalence of low regularity of conjugacies on large sets is a corollary. We also establish a new connection between the transverse regularity of foliations and their tangent subbundles.
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