Regularized determinant formulas for the zeta functions of 3-dimensional Riemannian foliated dynamical systems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929562822115328 |
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| author | López, Jesús A. Álvarez Kim, Junhyeong Morishita, Masanori |
| author_facet | López, Jesús A. Álvarez Kim, Junhyeong Morishita, Masanori |
| contents | We prove a regularized determinant formula for the zeta functions of certain 3-dimensional Riemannian foliated dynamical systems, in terms of the infinitesimal operator induced by the flow acting on the reduced leafwise cohomologies. It is the formula conjectured by Deninger. The proof is based on relating the dynamical spectral $ξ$-functions, analogues of the $ξ$-function in analytic number theory, with the zeta function, by applying the distributional dynamical Lefschetz trace formula. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_20758 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularized determinant formulas for the zeta functions of 3-dimensional Riemannian foliated dynamical systems López, Jesús A. Álvarez Kim, Junhyeong Morishita, Masanori Dynamical Systems Differential Geometry 37C25, 37C30, 53C12 We prove a regularized determinant formula for the zeta functions of certain 3-dimensional Riemannian foliated dynamical systems, in terms of the infinitesimal operator induced by the flow acting on the reduced leafwise cohomologies. It is the formula conjectured by Deninger. The proof is based on relating the dynamical spectral $ξ$-functions, analogues of the $ξ$-function in analytic number theory, with the zeta function, by applying the distributional dynamical Lefschetz trace formula. |
| title | Regularized determinant formulas for the zeta functions of 3-dimensional Riemannian foliated dynamical systems |
| topic | Dynamical Systems Differential Geometry 37C25, 37C30, 53C12 |
| url | https://arxiv.org/abs/2410.20758 |