Regularity of the Anosov splitting and of horospheric foliations

Abstract ‘Bunching’ conditions on an Anosov system guarantee the regularity of the Anosov splitting up to C 2−ε . Open dense sets of symplectic Anosov systems and geodesic flows do not have Anosov splitting exceeding the asserted regularity. This is the first local construction of low-regularity examples.