Circle packings, renormalizations and subdivision rules
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912642816278528 |
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| author | Luo, Yusheng Zhang, Yongquan |
| author_facet | Luo, Yusheng Zhang, Yongquan |
| contents | In this paper, we use iterations of skinning maps on Teichmüller spaces to study circle packings and develop a renormalization theory for circle packings whose nerves satisfy certain subdivision rules. We characterize when the skinning map has bounded image. Under the corresponding condition, we prove that the renormalization operator $\mathfrak{R}$ is uniformly contracting. This allows us to give complete answers for the existence and moduli problems for such circle packings. The exponential contraction of $\mathfrak{R}^n$ means that despite the non-rigidity of such circle packings, they are geometrically inflexible. As an application, we show that any geometrically finite Kleinian circle packing is combinatorially rigid. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_13151 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Circle packings, renormalizations and subdivision rules Luo, Yusheng Zhang, Yongquan Geometric Topology Dynamical Systems In this paper, we use iterations of skinning maps on Teichmüller spaces to study circle packings and develop a renormalization theory for circle packings whose nerves satisfy certain subdivision rules. We characterize when the skinning map has bounded image. Under the corresponding condition, we prove that the renormalization operator $\mathfrak{R}$ is uniformly contracting. This allows us to give complete answers for the existence and moduli problems for such circle packings. The exponential contraction of $\mathfrak{R}^n$ means that despite the non-rigidity of such circle packings, they are geometrically inflexible. As an application, we show that any geometrically finite Kleinian circle packing is combinatorially rigid. |
| title | Circle packings, renormalizations and subdivision rules |
| topic | Geometric Topology Dynamical Systems |
| url | https://arxiv.org/abs/2308.13151 |