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

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          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.

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          Author and article information

          Journal
          09 August 2022
          Article
          2208.04550
          98eeb180-768e-472f-b271-a650612462c7

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          58C40, 37C27
          18 pages, no figure
          math.SP

          Functional analysis
          Functional analysis

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