And there is no clear evidence that Z ( s ) is better than the others. Each of such generalized dynamical zeta functions can be regarded as a generalization of Selberg zeta function in their own rights and their analytic property will be different when we consider them in more general cases. In fact, there are variety of generalized dynamical zeta functions which coincide with Z ( s ) in the cases of geodesic flows on closed hyperbolic surfaces, since some dynamical exponents coincide. We devote this paper to the study of the extension of the claim (c).īefore proceeding with the problem of generalizing the claim (c), we would like to pose a question whether the zeta function Z ( s ) introduced by Smale is the ”right” one to be studied. The argument in should be true for more general type of dynamical zeta functions.) However, to the best of authors’ understanding, not much is known about the extension of the claims (c) or (d), or more generally on the distributions of singularity of the (generalized) dynamical zeta functions. Z ( s ) for a C ∞ Anosov flow f t : N → N is known to have meromorphic extension to the whole complex plane C. For the extensions of the claim (a) and (b) above, we already have rather satisfactory results: for instance, the dynamical zeta function We refer the papers for the development in the early stage, and the recent paper by Giulietti, Liverani and Pollicott (and the references therein) for the recent state of the art. In dynamical system theory, the dynamical zeta function Z ( s ) and its variants are related to the semi-group of transfer operators associated to the flow through the so-called Atiyah-Bott(-Guillemin) trace formula, as we will explain later. Later, the dynamical zeta function Z ( s ) is interpreted and generalized from the view point of dynamical system theory and studied extensively by many people not only in dynamical system theory but also in the fields of physics related to ”quantum chaos”. However the main part of the ”wild ” idea was left as a question. In, he showed that Z ( s ) has meromorphic extension to the whole complex plane if the flow is a suspension flow of an Anosov diffeomorphism with a constant roof function. This should be the reason why Smale described his idea ”wild”. should have been whether the claims above could be generalized to more general types of flows, such as the geodesic flows on manifolds with negative variable curvature or, more generally, to general Anosov flows.īut it was not clear whether this idea was reasonable, since the results of Selberg were based on the so-called Selberg trace formula for the heat kernel on the surface and depended crucially on the fact that the surface was of negative constant curvature. For instance the relation of special values of the dynamical zeta function to the geometric properties of the underlying manifolds should be an interesting problem. The main question 2 2 2There are many other related problems. Smale’s idea was to study the dynamical zeta function Z ( s ) defined as above in more general context.
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