Bootstrapping regularity of the Anosov splitting

Finite smoothness of the Anosov splitting implies C°° .Definitions.Let M be a compact Riemannian manifold./ e Diff°°(M) is called Anosov with Anosov splitting TM = Eu © Es if (3C, e >0), (Vp 6 M) (3px < p2 < I -e < I + e < v2 < vx) (Vt; eEs(p), ueEu(p), «eN), (p"/C)\\\ < \(v)\\ < Cpn2\\\ and (v-nIC)\\\<\-"(u)\\<Cu-n\\\.f is called a-bunched if suppeM p2v2~x (min(px -vxX))~a < 1 • Riemannian metrics with C°°-conjugate geodesic flows are called isodynamic.Theorem.If s€Mvxp~x$ < 1 and Eu e C then E" e C°° .The same holds for the weak unstable distribution of Anosov flows.Katok obtained a similar result via nonstationary normal forms (oral communication).Since our technique is standard [HK, LMM], we only sketch the proof.We first list corollaries.Considering f~x yields Corollary 1.If suppeM vxp2n < 1 and Es e C" then Es e C°° .Corollary 2. If f preserves volume, codim E" = 1 , and E" e C2, then Eu e C°°.Remark.Ghys [Gl] proves that a volume preserving Anosov flow on a compact manifold of dimension greater than three is a suspension if codim E" = 1 and E" eC2.Corollary 2 suggests that his result might be a rigidity statement.Corollary 3. If dim M = 2 and -log \ \ ||/ log \ ^ || < n -1 -Ô for all keW, peM, and Eu eCn , then E" e C°° .Remark.Thus Theorem C of Ghys [G2] about the "generalized algebraicity" of flows with C°° Anosov splitting holds with a C3-assumption._