Fredholm determinants, Anosov maps and Ruelle resonances

I show that the Ruelle dynamical zeta function, associated to an Anosov diffeomorphism, is the Fredholm determinant of the corresponding Ruelle-Perron-Frobenius transfer operator acting on appropriate Banach spaces. As a consequence it follows, for example, that the zeroes of the dynamical zeta function describe the eigenvalues of the operator and that, for $\Co^\infty$ Anosov diffeomorphisms, the zeta function is entire.