Accademia Nazionale Dei Lincei

Zeta functions considered here are functions of one complex variable arising from the problem of studying the distribution of the periods of the periodic orbits generated on a manifold M by the action of groups of maps homomorphic to R or N (the continuous and discrete, respectively, cases). The functions describe the distribution of the periodic orbits periods in the same sense in which the Riemann zeta function describes the distribution of the logarithms of the prime numbers. In the case in which the group of maps is a group of diffeomorphisms of a Riemannian manifold homomorphic to N and, furthermore, the group satisfies the Axiom A (Cf. Appendix 2), it can be shown that the associated zeta function is meromorphic. Our purpose here is to show, by providing an example, that such property does not extend, in general, to the case of a group of maps which, although satisfying the axiom A, is homomorphic to R.