The Differential Zeta Function for Axiom A Attractors

Abstract

closed orbit of least period X( T). An important problem is to construct the meromorphic extensions for these zeta functions. A detailed knowledge of the domain of '(s) gives information on the long-term behaviour of the For example, this is a standard approach to deriving asymptotic formulae for closed orbits; cf. [9]. For the case of geodesic flows on surfaces of constant negative curvature, the zeta function is known to have a meromorphic extension to the entire complex plane. Unfortunately, an obstruction to a general theory along these lines is the existence of examples of Axiom A flows for which D(s) does not have a meromorphic extension to C. In a recent article, William Parry proposed a modified definition of a zeta function for Axiom A attractors. He defined the differential zeta function by Ns(s) = H(I e-S(T)<l where the least period X(X) of a closed orbit is replaced by the expansion Xu(T) (along the unstable manifold) around the orbit [8]. Parry also defined the useful notion of a synchronized flow (i.e., one for which the measure of maximal entropy and the SRB measure coincide). The following remark is taken from page 273 of his article: When the flow is synchronized ... the zeta functions coincide. This is the case, for example, with a geodesic flow on a surface of constant negative curvature. One wonders whether synchronization has something to do with the very desirable features of the zeta function of such a geodesic flow. In this article we shall give a solution to Parry's question which shows that in many cases DU(s) has a meromorphic extension to the entire complex plane. Our main result can be stated as follows: