(Joint work with M. Tsujii) The Ruelle transfer operator is a powerful tool in ergodic theory, which involves composition with the dynamics. Many relevant dynamical systems are hyperbolic, i.e. they involve contracting and expanding directions. Composition with a contraction

improves regularity - but composing with an expanding map "worsens" regularity: It has been an open problem for many years to find a space

of distributions on which composition by a hyperbolic diffeomorphism (of finite smoothness) can be well understood. Last year we constructed such a space and estimated the essential spectral radius of the transfer operator on this space. After recalling this result, we shall describe more recent progress including spectral interpretation of zeroes of dynamical determinants.