Bootstrapping regularity of the Anosov splitting

Abstract

Finite smoothness of the Anosov splitting implies C?c . Definitions. Let M be a compact Riemannian manifold. f E Diffl(M) is called Anosov with Anosov splitting TM = EU E Es if (3C, E > O), (Vp E M) (i1 < 12 < 1-E < 1 + E < V2 < VI) (VV E E(p), UEEU(p), n eN), (11n/C)IIvII < IIDfn(v)II < C? 1nIIVI and (Vn1 C)IluII < IlDf-n(U)II < CV-nIluII f is called a-bunched if SupPEM u2v2 l(min(u1, vV1))< 1. Riemannian metrics with C??-conjugate geodesic flows are called isodynamic. Theorem. If SUpPEMVIIj1 2< 1 and Eu E Cn then Eu 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 fI yields Corollary 1. If supPEM vijlv lv-n < 1 and Es E Cn then Es E CO. Corollary 2. If f preserves volume, codim Eu = 1, and Eu E C2, then Eu E r00 c . Remark. Ghys [GI] proves that a volume preserving Anosov flow on a compact manifold of dimension greater than three is a suspension if codim Eu = 1 and Eu E C2. Corollary 2 suggests that his result might be a rigidity statement. Corollary 3. If dim M = 2 and log IIDfk [Eu(p) II/ log IIDfk IEI() <n 1 -for all k E N, p E M, and EU E Cn, then EU E Coo . Remark. Thus Theorem C of Ghys [G2] about the generalized algebraicity of flows with COO Anosov splitting holds with a C3-assumption. Received by the editors April 4, 1990. 1991 Mathematics Subject Classification. Primary 58F15, 58F17, 58F18; Secondary 53C12, 53C20, 53C35.