Asymptotic distribution of the pseudo-orbits and the generalized Euler constantγΔfor a family of strongly chaotic systems

Dynamical zeta functions, defined as Euler products over classical periodic orbits, have recently received enhanced attention as an important tool for the quantization of chaos. Their representation as a Dirichlet series over pseudo-orbits has proven to be particularly useful, since these series seem to possess in the general case much better convergence properties than the original Euler product. The convergence of the Dirichlet series depends crucially on the asymptotic distribution of the pseudo-orbits and thus on the ergodicity of the underlying dynamical system. It is shown that the lengths ${\mathit{l}}_{\mathit{n}}$ (or rather exp${\mathit{l}}_{\mathit{n}}$) of the classical periodic orbits play mathematically the role of generalized prime numbers. Based on the theory of Beurling’s generalized prime numbers, we derive an exact law for the proliferation of psuedo-orbits for the Hadamard-Gutzwiller model, which is one of the main testing grounds of our ideas about quantum chaos. The strength of growth of the pseudo-orbits is determined by the ratio ZETA(2)/ZETA’(1), where ZETA(s) denotes the Selberg zeta function. Two explicit, complementary representations are given that allow the computation of this ratio solely from the length spectrum {${\mathit{l}}_{\mathit{n}}$} of the classical periodic orbits, or from the quantal energy spectrum {${\mathit{E}}_{\mathit{n}}$}. One of these relations depends exponentially on the generalized Euler constant ${\gamma }_{\mathrm{\Delta }}$, which is therefore also studied. The formulas are applied to two strongly chaotic systems. It turns out that our asymptotic law describes the mean proliferation of pseudo-orbits very well not only in the asymptotic region, but also surprisingly well down to the shortest pseudo-orbit.