On the Integrability of Intermediate Distributions for Anosov Diffeomorphisms

. We study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of C 1 \GammaAnosov diffeomorphism on three dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant. 1. Introduction Let F be a C r \Gammadiffeomorphism of a compact smooth Riemannian manifold M , r = 1; 2; \Delta \Delta \Delta ; 1 and \Gamma 0 (TM) the space of C 0 \Gammavector fields on M . The map F induces an invertible bounded linear operator on \Gamma 0 (TM) by: F v(x) = DFv(F \Gamma1 (x)); v(x) 2 \Gamma 0 (TM); x 2 M: The Mather spectrum oe(F ) is defined to be the spectrum of the complexification of F . In [M], Mather showed that F is an Anosov system, if and only if 1 = 2 oe(F ), and moreover, if the nonperiodic points of F are dense in M ..