Dynamical Zeta Functions and Closed Orbits for Geodesic and Hyperbolic Flows

Statistical properties of zeta functions’ zeros

“…One particularly important example of a flow is the geodesic flow on a smooth manifold M. In the case when M has a constant negative curvature, the corresponding dynamical zeta function is closely related to the Selberg zeta function. Namely, where Z(s) is as defined in (22) (see Remark 2.5 in [45]). Another important example is the geodesic flow for billiards on polygons.…”

“…On the other hand one can study the location of the pole with the maximal real part and the results of this study give valuable information about the distribution of closed geodesics. In this way, one can study the distribution of closed geodesics on spaces of variable curvature (see Corollary 6.11 in [45]).…”

“…At the same time, there is a connection to number-theoretical zeta functions, namely, to the zeta functions of curves over finite fields. The main sources for this section are reviews by Ruelle ([48] and [49]) and Pollicott ([44] and [45]).…”

“…where ω denotes a periodic orbit of f, and l(ω) is its length. It is a more general case than the case of maps since there is a construction ("suspension") that allows us to realize maps as flows (see Example 5 on p. 70 in [46] or Section 4.3 in [45]) but not vice versa. Unsurprisingly, it turns out that zeta functions for flows are more difficult to investigate than zeta functions for maps.…”

“…where Z(s) is as defined in (22) (see Remark 2.5 in [45]). Another important example is the geodesic flow for billiards on polygons.…”