Spectra of analytic hyperbolic maps and flows: Correlation functions, Fredholm determinants and zeta-functions

In [13] we defined a class of so-called ‘Hyperbolic Analytic Maps’. Given a map in this class one associates a Banach space and a family of transfer operators with analytic weights on the space. These operators are nuclear in the sense of Grothendieck. An elementary proof was given in the case of 1+1 dimensional maps [Fried has extended the proof to higher dimensional systems]. In this case such an operator admits a Fredholm determinant which is an entire function in the complex plane. Applying a judicious choice of weights we may relate the zeroes of a determinant to resonances for certain ergodic measures on the underlying dynamical system. In particular, we consider real-analytic Anosov maps or Axiom A attractors (still in 1+1 dimensions) with their SRB (or natural) measure and show that the SRB-resonances form a discrete subset of the complex plane and are localized in the zero-set of a Fredholm determinant. Regarding an Axiom A flow as a suspension of a ditto map we prove similar results for the SRB-resonances in the flow case.