Banach spaces adapted to Anosov system      
                         by
         S\'ebastien Gou\"ezel and Carlangelo Liverani
          



                       ABSTRACT

We study the spectral properties of the Ruelle-Perron-Frobenius
operator associated to an Anosov map on classes of functions with
high smoothness. To this end we construct anisotropic Banach
spaces of distributions on which the transfer operator has a large
spectral gap. In the $\Co^\infty$ case, the spectral radius is
arbitrarily small, which yields a description of the correlations
with arbitrary precision. Moreover, we obtain sharp spectral
stability results for deterministic and random perturbations. In
particular, we obtain differentiability results for spectral data
(which imply differentiability of the SRB measure, the variance
for the CLT, the rates of decay for smooth observable, etc.).