Smooth Anosov flows: correlation spectra and stability

Abstract

Abstract. By introducing appropriate Banach spaces one can study the spec-tral properties of the generator of the semigroup defined by an Anosov flow. Consequently, it is possible to easily obtain sharp results on the Ruelle reso-nances and the differentiability of the SRB measure. 1. introduction In the last years there has been a growing interest in the dependence of the SRB measures on the parameters of the system. In particular, G.Gallavotti [11] has argued the relevance of such an issue for non-equilibrium statistical mechanics. On a physical basis (linear response theory) one expects that the average be-haviour of an observable changes smoothly with parameters. Yet the related rig-orous results are very limited and the existence of very irregular dependence from parameters (think, for example, to the quadratic family) shows that, in general, smooth dependence must be properly interpreted to have any chance to hold. The only cases in which some simple rigorous results are available are smooth uniformly hyperbolic systems and some partially hyperbolic systems. In particular