Relative train tracks and generalized endperiodic graph maps
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912445361029120 |
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| author | He, Yan Mary Wu, Chenxi |
| author_facet | He, Yan Mary Wu, Chenxi |
| contents | Motivated by the work of Cantwell-Conlon-Fenley on endperiodic homeomorphisms of infinite type surfaces, we define and study endperiodic and generalized endperiodic maps of an infinite graph with finitely many ends. Adapting the work of Bestvina-Handel to the infinite type setting, we define endperiodic relative train track maps. We prove that any generalized endperiodic map is homotopic to a generalized endperiodic relative train track map, via a combinatorially bounded homotopy equivalence. We show that the (largest) Perron-Frobenius eigenvalue of a relative train track representation of a generalized endperiodic map $f$ is a canonical quantity associated to $f$ as it admits a canonical group theoretic interpretation. Moreover, the (largest) Perron-Frobenius eigenvalue and the topological entropy of a relative train track map is the smallest among its proper homotopy equivalence class. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_13401 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative train tracks and generalized endperiodic graph maps He, Yan Mary Wu, Chenxi Geometric Topology Dynamical Systems 20F65, 20E36, 57M60 Motivated by the work of Cantwell-Conlon-Fenley on endperiodic homeomorphisms of infinite type surfaces, we define and study endperiodic and generalized endperiodic maps of an infinite graph with finitely many ends. Adapting the work of Bestvina-Handel to the infinite type setting, we define endperiodic relative train track maps. We prove that any generalized endperiodic map is homotopic to a generalized endperiodic relative train track map, via a combinatorially bounded homotopy equivalence. We show that the (largest) Perron-Frobenius eigenvalue of a relative train track representation of a generalized endperiodic map $f$ is a canonical quantity associated to $f$ as it admits a canonical group theoretic interpretation. Moreover, the (largest) Perron-Frobenius eigenvalue and the topological entropy of a relative train track map is the smallest among its proper homotopy equivalence class. |
| title | Relative train tracks and generalized endperiodic graph maps |
| topic | Geometric Topology Dynamical Systems 20F65, 20E36, 57M60 |
| url | https://arxiv.org/abs/2408.13401 |