Fredholm determinants and Anosov maps

ABSTRACT:
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.