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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2406.06250 |
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| _version_ | 1866918232488673280 |
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| author | Sambarino, Andrés |
| author_facet | Sambarino, Andrés |
| contents | Inspired by Benoist, we study objects linked to integrable tangent vectors on the character variety of a semi-group $Γ$ with values in a semi-simple real-algebraic group $\mathsf G$. We prove the \emph{cone of Jordan variations} has non-empty interior and, when $\mathsf G$ is split, establish non-empty interior of the set of \emph{length-normalized variations}. We apply these techniques to pressure forms on Anosov representations and higher-rank Teichmüller spaces. We identify an explicit functional $φ\in\mathfrak a^*$ whose pressure form is compatible with Goldman's symplectic form at Fuchsian points in the Hitchin component. Finally, we show the degeneration of the Hausdorff dimension of \emph{higher}-quasi-circles is governed by a Diophantine equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic properties of infinitesimal characters and applications Sambarino, Andrés Group Theory Differential Geometry Dynamical Systems Inspired by Benoist, we study objects linked to integrable tangent vectors on the character variety of a semi-group $Γ$ with values in a semi-simple real-algebraic group $\mathsf G$. We prove the \emph{cone of Jordan variations} has non-empty interior and, when $\mathsf G$ is split, establish non-empty interior of the set of \emph{length-normalized variations}. We apply these techniques to pressure forms on Anosov representations and higher-rank Teichmüller spaces. We identify an explicit functional $φ\in\mathfrak a^*$ whose pressure form is compatible with Goldman's symplectic form at Fuchsian points in the Hitchin component. Finally, we show the degeneration of the Hausdorff dimension of \emph{higher}-quasi-circles is governed by a Diophantine equation. |
| title | Asymptotic properties of infinitesimal characters and applications |
| topic | Group Theory Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2406.06250 |