The authors study the correlation spectra and stability for smooth Anosov flows. Recently it has become clear that it is possible to construct appropriate functional spaces such that the statistical properties of the systems are accurately described by the spectral date of the so-called transfer operator acting on such spaces. The authors present an application of the methods of using the transfer operators acting on certain Banach spaces (functional spaces) to study the differentiability properties of the SRB measure for Anosov flows.
More precisely, by introducing appropriate Banach spaces the authors analyze the operator (the approach is based on the study of the resolvent). As a result, not only are the formulae in D. Ruelle [Commun. Math. Phys. 187, No. 1, 227–241 (1997; Zbl 0895.58045)] easily recovered, but higher differentiability of the SRB measure is obtained as well, making rigorous some of the results in D. Ruelle [Nonlinearity 11, No. 1, 5–18 (1998; Zbl 0896.58071)]. In addition, the authors’ method naturally yields precise information on the structure of the Ruelle resonances, extending the results in M. Pollicott [Invent. Math. 81, 413–426 (1985; Zbl 0591.58025)] and H. H. Rugh [Ergodic Theory Dyn. Syst. 16, No. 4, 805–819 (1996; Zbl 0948.37003)].