Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems

We consider a generalized Fredholm determinant d ( z ) and a generalized Selberg zeta function ζ(ω) −1 for Axiom A diffeomorphisms of a surface and Axiom A flows on three-dimensional manifolds, respectively. We show that d ( z ) and ζ(ω) −1 extend to entire functions in the complex plane. That the functions are entire and not only meromorphic is proved by a new method, identifying removable singularities by a change of Markov partitions.