Zeta functions and the Fried conjecture for smooth pseudo-Anosov flows
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917786816610304 |
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| author | Jézéquel, Malo Zung, Jonathan |
| author_facet | Jézéquel, Malo Zung, Jonathan |
| contents | To a transitive pseudo-Anosov flow $φ$ on a $3$-manifold $M$ and a representation $ρ$ of $π_1(M)$, we associate a zeta function $ζ_{φ,ρ}(s)$ defined for $\Re s \gg 1$, generalizing the Anosov case. For a class of ``smooth pseudo-Anosov flows'', we prove that $ζ_{φ,ρ}(s)$ has a meromorphic continuation to $\mathbb{C}$. We also prove a version of the Fried conjecture for smooth pseudo-Anosov flows which, under some conditions on $ρ$, relates $ζ_{φ,ρ}(0)$ to the Reidemeister torsion of $M$. Finally we prove a topological analogue of the Dirichlet class number formula. In order to deal with singularities, we use $C^\infty$ versions of the approaches of Rugh and Sanchez--Morgado, based on Markov partitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17014 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zeta functions and the Fried conjecture for smooth pseudo-Anosov flows Jézéquel, Malo Zung, Jonathan Dynamical Systems 37C30 To a transitive pseudo-Anosov flow $φ$ on a $3$-manifold $M$ and a representation $ρ$ of $π_1(M)$, we associate a zeta function $ζ_{φ,ρ}(s)$ defined for $\Re s \gg 1$, generalizing the Anosov case. For a class of ``smooth pseudo-Anosov flows'', we prove that $ζ_{φ,ρ}(s)$ has a meromorphic continuation to $\mathbb{C}$. We also prove a version of the Fried conjecture for smooth pseudo-Anosov flows which, under some conditions on $ρ$, relates $ζ_{φ,ρ}(0)$ to the Reidemeister torsion of $M$. Finally we prove a topological analogue of the Dirichlet class number formula. In order to deal with singularities, we use $C^\infty$ versions of the approaches of Rugh and Sanchez--Morgado, based on Markov partitions. |
| title | Zeta functions and the Fried conjecture for smooth pseudo-Anosov flows |
| topic | Dynamical Systems 37C30 |
| url | https://arxiv.org/abs/2409.17014 |