Fredholm determinants, Anosov maps and Ruelle resonances

I show that the dynamical determinant, 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 determinant describe the eigenvalues of the transfer operator and the Ruelle resonances and that, for ${\text{Co}}^{\mathrm{\infty }}$ Anosov diffeomorphisms, the dynamical determinant is an entire function.