Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness

Given a map and a function , a `Fredholm determinant' can be defined as a formal power series . The coefficients are related to the periodic points of . Assume that is a hyperbolic diffeomorphism of class , and belongs to . Then the Fredholm determinant is analytic in the disc of radius , where is a hyperbolicity index of (roughly speaking, is -contracting in one direction and -expanding in the other direction). In the case, the Fredholm determinant is an entire function.