A characterization of pseudo-Anosov mapping classes via tropical cluster transformations
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arXiv
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| Format: | Preprint |
| Published: |
2020
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| _version_ | 1866929646897987584 |
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| author | Ishibashi, Tsukasa Kano, Shunsuke |
| author_facet | Ishibashi, Tsukasa Kano, Shunsuke |
| contents | We give a characterization of generic pseudo-Anosov mapping classes purely in terms of their expressions in the shear coordinates, thus giving an answer to a problem raised by Papadopoulos--Penner [PP93]. This characterization has a cluster algebraic generalization called the sign stability introduced in [IK21]. By combining with the results in [IK21], we see that the algebraic entropies of the cluster $\mathcal{A}$- and $\mathcal{X}$-transformations induced by a generic pseudo-Anosov mapping class both coincide with its topological entropy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2010_05214 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A characterization of pseudo-Anosov mapping classes via tropical cluster transformations Ishibashi, Tsukasa Kano, Shunsuke Geometric Topology Combinatorics Dynamical Systems We give a characterization of generic pseudo-Anosov mapping classes purely in terms of their expressions in the shear coordinates, thus giving an answer to a problem raised by Papadopoulos--Penner [PP93]. This characterization has a cluster algebraic generalization called the sign stability introduced in [IK21]. By combining with the results in [IK21], we see that the algebraic entropies of the cluster $\mathcal{A}$- and $\mathcal{X}$-transformations induced by a generic pseudo-Anosov mapping class both coincide with its topological entropy. |
| title | A characterization of pseudo-Anosov mapping classes via tropical cluster transformations |
| topic | Geometric Topology Combinatorics Dynamical Systems |
| url | https://arxiv.org/abs/2010.05214 |