A trace formula for foliated flows
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913233315561472 |
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| author | López, Jesús A. Álvarez Kordyukov, Yuri A. Leichtnam, Eric |
| author_facet | López, Jesús A. Álvarez Kordyukov, Yuri A. Leichtnam, Eric |
| contents | Let $F$ be a transversely oriented foliation of codimension 1 on a closed manifold $M$, and let $ϕ=\{ϕ^t\}$ be a foliated flow on $(M,F)$. Assume the closed orbits of $ϕ$ are simple and its preserved leaves are transversely simple. In this case, there are finitely many preserved leaves, which are compact. Let $M^0$ denote their union, $M^1=M\setminus M^0$ and $F^1=F|_{M^1}$. We consider two topological vector spaces, $I(F)$ and $I'(F)$, consisting of the leafwise currents on $M$ that are conormal and dual-conormal to $M^0$, respectively. They become topological complexes with the differential operator $d_{F}$ induced by the de~Rham derivative on the leaves, and they have an $\mathbb{R}$-action $ϕ^*=\{ϕ^{t\,*}\}$ induced by $ϕ$. Let $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$ denote the corresponding leafwise reduced cohomologies, with the induced $\mathbb{R}$-action $ϕ^*=\{ϕ^{t\,*}\}$. We define some kind of Lefschetz distribution $L_{\text{\rm dis}}(ϕ)$ of the actions $ϕ^*$ on both $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$, whose value is a distribution on $\mathbb{R}$. Its definition involves several renormalization procedures, the main one being the b-trace of some smoothing b-pseudodifferential operator on the compact manifold with boundary obtained by cutting $M$ along $M^0$. We also prove a trace formula describing $L_{\text{\rm dis}}(ϕ)$ in terms of infinitesimal data from the closed orbits and preserved leaves. This solves a conjecture of C.~Deninger involving two leafwise reduced cohomologies instead of a single one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06671 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A trace formula for foliated flows López, Jesús A. Álvarez Kordyukov, Yuri A. Leichtnam, Eric Geometric Topology Differential Geometry Dynamical Systems 58A14, 58G11, 57R30 Let $F$ be a transversely oriented foliation of codimension 1 on a closed manifold $M$, and let $ϕ=\{ϕ^t\}$ be a foliated flow on $(M,F)$. Assume the closed orbits of $ϕ$ are simple and its preserved leaves are transversely simple. In this case, there are finitely many preserved leaves, which are compact. Let $M^0$ denote their union, $M^1=M\setminus M^0$ and $F^1=F|_{M^1}$. We consider two topological vector spaces, $I(F)$ and $I'(F)$, consisting of the leafwise currents on $M$ that are conormal and dual-conormal to $M^0$, respectively. They become topological complexes with the differential operator $d_{F}$ induced by the de~Rham derivative on the leaves, and they have an $\mathbb{R}$-action $ϕ^*=\{ϕ^{t\,*}\}$ induced by $ϕ$. Let $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$ denote the corresponding leafwise reduced cohomologies, with the induced $\mathbb{R}$-action $ϕ^*=\{ϕ^{t\,*}\}$. We define some kind of Lefschetz distribution $L_{\text{\rm dis}}(ϕ)$ of the actions $ϕ^*$ on both $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$, whose value is a distribution on $\mathbb{R}$. Its definition involves several renormalization procedures, the main one being the b-trace of some smoothing b-pseudodifferential operator on the compact manifold with boundary obtained by cutting $M$ along $M^0$. We also prove a trace formula describing $L_{\text{\rm dis}}(ϕ)$ in terms of infinitesimal data from the closed orbits and preserved leaves. This solves a conjecture of C.~Deninger involving two leafwise reduced cohomologies instead of a single one. |
| title | A trace formula for foliated flows |
| topic | Geometric Topology Differential Geometry Dynamical Systems 58A14, 58G11, 57R30 |
| url | https://arxiv.org/abs/2402.06671 |