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

TL;DR

This paper demonstrates that certain non-isometric Riemannian manifolds constructed by Sunada share the same Guillemin-Ruelle zeta function, extending the understanding of spectral invariants beyond the Laplace spectrum.

Contribution

It proves that Sunada's non-isometric manifolds also have identical Guillemin-Ruelle dynamical L-functions using intertwining operators, revealing new spectral invariants.

Findings

Sunada pairs have identical Guillemin-Ruelle zeta functions

The method of intertwining operators is effective in comparing dynamical L-functions

Spectral invariants extend beyond Laplace spectra to dynamical zeta functions

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.