We study spectral properties of transfer operators for diffeomorphisms T : X → X on a Riemannian manifold X: Suppose that Ω is an isolated hyperbolic subset for T , with a compact isolating neighborhood V ⊂ X. We first introduce Banach spaces of distributions supported on V , which are anisotropic versions of the usual space of C p functions C p (V ) and of the generalized Sobolev spaces W p,t (V ), respectively. Then we show that the transfer operators associated to T and a smooth weight g extend boundedly to these spaces, and we give bounds on the essential spectral radii of such extensions in terms of hyperbolicity exponents. * (T, V ), for 1 < t < ∞, of distributions supported on V , such that, for any C r−1 function g : X → C supported on V ,(1) (Hölder spaces) L T,g extends boundedly to L T,g : C p,q * (T, V ) and r ess (L T,g | C p,q * (T,V ) ) ≤ R p,q,∞ (T, g, Ω). (2) (Sobolev spaces) L T,g extends boundedly to L T,g : W p,q,t *

Dedicated to Prof. Michael Brin on the occasion of his 60th birthday.Abstract. For smooth hyperbolic dynamical systems and smooth weights, we relate Ruelle transfer operators with dynamical Fredholm determinants and dynamical zeta functions: First, we establish bounds for the essential spectral radii of the transfer operator on new spaces of anisotropic distributions, improving our previous results [7]. Then we give a new proof of Kitaev's [17] lower bound for the radius of convergence of the dynamical Fredholm determinant. In addition we show that the zeroes of the determinant in the corresponding disc are in bijection with the eigenvalues of the transfer operator on our spaces of anisotropic distributions, closing a question which remained open for a decade.

We study dynamical systems generated by skew productswhere 2, 1/ < λ < 1 and f is a C 2 function on S 1 . We show that the SBR measure for T is absolutely continuous for almost every f .

We consider suspension semi-flows of angle-multiplying maps on the circle for Cr ceiling functions with r≥3. Under a Crgeneric condition on the ceiling function, we show that there exists a Hilbert space (anisotropic Sobolev space) contained in the L2 space such that the Perron–Frobenius operator for the time-t-map acts naturally on it and that the essential spectral radius of that action is bounded by the square root of the inverse of the minimum expansion rate. This leads to a precise description of decay of correlations. Furthermore, the Perron–Frobenius operator for the time-t-map is quasi-compact for a Cr open and dense set of ceiling functions.

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