Non-isometric pairs of Riemannian manifolds with the same Guillemin-Ruelle zeta function

Abstract

In 1985, T. Sunada constructed a vast collection of non-isometric Laplace-isospectral pairs $(M_1,g_1)$, resp. $(M_2,g_2)$ of Riemannian manifolds. He further proves that the Ruelle zeta functions $Z_g(s):= \prod_{\gamma}(1 - e^{-sL(\gamma)})^{-1}$ of $(M_1,g_1)$, resp. $(M_2,g_2)$ coincide, where $\{\gamma\}$ runs over the primitive closed geodesics of $(M,g)$ and $L(\gamma)$ is the length of $\gamma$. In this article, we use the method of intertwining operators on the unit cosphere bundle to prove that the same Sunada pairs have identical Guillemin-Ruelle dynamical L-functions $L_G(s) = \sum_{\gamma\in \mathscr{G}}\frac{L_\gamma^\# e^{-sL_\gamma}}{|\det(I -\mathbf{P}_\gamma)|}$, where the sum runs over all closed geodesics.