We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if Ω is any open set in C d , and L is a suitable transfer operator acting on Bergman space A 2 (Ω), its eigenvalue sequence {λn(L)} is bounded by |λn(L)| ≤ A exp(−an 1/d ), where a, A > 0 are explicitly given.

Using analytic properties of Blaschke factors we construct a family of analytic hyperbolic diffeomorphisms of the torus for which the spectral properties of the associated transfer operator acting on a suitable Hilbert space can be computed explicitly. As a result, we obtain explicit expressions for the decay of correlations of analytic observables without resorting to any kind of perturbation argument.

For certain analytic data, we show that the eigenvalue sequence of the associated transfer operator L is insensitive to the holomorphic function space on which L acts. Explicit bounds on this eigenvalue sequence are established. LEMMA 2.9. Let D and D be domains in C d such that D ⊂ ⊂ D. Let A and A be favourable Banach spaces of holomorphic functions on D and D , respectively. Then A ⊂ A , and the natural embedding J : A → A , defined by J f = f | D , is strongly nuclear. Proof. Choose D ∈ D d with D ⊂ ⊂ D ⊂ ⊂ D, and consider the natural embeddings A J 1 → Hol(D ) J 2 → U (D ) J 3 → A . Clearly J = J 3 J 2 J 1 . The unit ball of U (D ) is a neighbourhood in Hol(D ), so the map J 2 is bounded. But the Fréchet space Hol(D ) is nuclear [Gro, II, Corollaire on p. 56], † The embedding U (D) → A is automatically continuous: continuity of point evaluation on both A and U (D) implies that it has closed graph; cf. the proof of Lemma 2.9. ‡ See [Gro, II, Définition 1, p. 3] for the generalization to locally convex spaces.

Abstract. Chaotic dynamics with sensitive dependence on initial conditions may result in exponential decay of correlation functions. We show that for one-dimensional interval maps the corresponding quantities, that is, Lyapunov exponents and exponential decay rates, are related. For piecewise linear expanding Markov maps observed via piecewise analytic functions we provide explicit bounds of the decay rate in terms of the Lyapunov exponent. In addition, we comment on similar relations for general piecewise smooth expanding maps.

Abstract. We show that for any λ ∈ C with |λ| < 1 there exists an analytic expanding circle map such that the eigenvalues of the associated transfer operator (acting on holomorphic functions) are precisely the nonnegative powers of λ and λ. As a consequence we obtain a counterexample to a variant of a conjecture of Mayer on the reality of spectra of transfer operators.