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 $Z_g(s):= \prod_γ(1 - e^{-sL(γ)})^{-1}$ of $(M_1,g_1)$, resp. $(M_2,g_2)$ coincide, where $\{γ\}$ runs over the primitive closed geodesics of $(M,g)$ and $L(γ)$ is the length of $γ$. 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 $L_G(s) = \sum_{γ\in \mathscr{G}}\frac{L_γ^\# e^{-sL_γ}}{|\det(I -\mathbf{P}_γ)|}$, where the sum runs over all closed geodesics.