Equidistribution of divergent diagonal orbits in positive characteristic
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908273185128448 |
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| author | Dang, Nguyen-Thi Paulin, Frédéric Sayous, Rafael |
| author_facet | Dang, Nguyen-Thi Paulin, Frédéric Sayous, Rafael |
| contents | Given a local field $\widehat K$ with positive characteristic, we study the dynamics of the diagonal subgroup of the linear group $\operatorname{GL}_n(\widehat K)$ on homogeneous spaces of discrete lattices in ${\widehat K}^{\,n}$. We first give a function field version of results by Margulis and Tomanov-Weiss, characterizing the divergent diagonal orbits. When $n=2$, we relate the divergent diagonal orbits with the divergent orbits of the geodesic flow in the modular quotient of the Bruhat-Tits tree of $\operatorname{PGL}_2(\widehat K)$. Using the (high) entropy method by Einsiedler-Lindentraus et al, we then give a function field version of a result of David-Shapira on the equidistribution of a natural family of these divergent diagonal orbits, with height given by a new notion of discriminant of the orbits. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_13995 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equidistribution of divergent diagonal orbits in positive characteristic Dang, Nguyen-Thi Paulin, Frédéric Sayous, Rafael Number Theory Dynamical Systems Group Theory 22F30, 11N45, 20G30, 14G17, 28C10, 11J70, 11P21 Given a local field $\widehat K$ with positive characteristic, we study the dynamics of the diagonal subgroup of the linear group $\operatorname{GL}_n(\widehat K)$ on homogeneous spaces of discrete lattices in ${\widehat K}^{\,n}$. We first give a function field version of results by Margulis and Tomanov-Weiss, characterizing the divergent diagonal orbits. When $n=2$, we relate the divergent diagonal orbits with the divergent orbits of the geodesic flow in the modular quotient of the Bruhat-Tits tree of $\operatorname{PGL}_2(\widehat K)$. Using the (high) entropy method by Einsiedler-Lindentraus et al, we then give a function field version of a result of David-Shapira on the equidistribution of a natural family of these divergent diagonal orbits, with height given by a new notion of discriminant of the orbits. |
| title | Equidistribution of divergent diagonal orbits in positive characteristic |
| topic | Number Theory Dynamical Systems Group Theory 22F30, 11N45, 20G30, 14G17, 28C10, 11J70, 11P21 |
| url | https://arxiv.org/abs/2503.13995 |