Dold coefficients of quasi-unipotent homeomorphisms of orientable surfaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909625711853568 |
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| author | Graff, Grzegorz Marzantowicz, Wacław Michalak, Łukasz Patryk |
| author_facet | Graff, Grzegorz Marzantowicz, Wacław Michalak, Łukasz Patryk |
| contents | The sequence of Dold coefficients $(a_n(f))$ of a self-map $f\colon X \to X$ forms a dual sequence to the sequence of Lefschetz numbers $(L(f^n))$ of iterations of $f$ under the Möbius inversion formula. The set ${\mathcal AP}(f) = \{ n \,\colon\, a_n(f) \neq 0 \}$ is called the set of algebraic periods of $f$. Both the set of algebraic periods and sequence of Dold coefficients play an important role in dynamical systems and periodic point theory.
In this work we provide a description of surface homeomorphisms with bounded $(L(f^n))$ (quasi-unipotent maps) in terms of Dold coefficients. We also discuss the problem of minimization of the genus of a surface for which one can realize a given set of natural numbers as the set of algebraic periods. Finally, we compute and list all possible Dold coefficients and algebraic periods for a given orientable surface with small genus and give some geometrical applications of the obtained results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19313 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dold coefficients of quasi-unipotent homeomorphisms of orientable surfaces Graff, Grzegorz Marzantowicz, Wacław Michalak, Łukasz Patryk Dynamical Systems Primary: 37C25, Secondary: 55M20, 37E30, 37D15 The sequence of Dold coefficients $(a_n(f))$ of a self-map $f\colon X \to X$ forms a dual sequence to the sequence of Lefschetz numbers $(L(f^n))$ of iterations of $f$ under the Möbius inversion formula. The set ${\mathcal AP}(f) = \{ n \,\colon\, a_n(f) \neq 0 \}$ is called the set of algebraic periods of $f$. Both the set of algebraic periods and sequence of Dold coefficients play an important role in dynamical systems and periodic point theory. In this work we provide a description of surface homeomorphisms with bounded $(L(f^n))$ (quasi-unipotent maps) in terms of Dold coefficients. We also discuss the problem of minimization of the genus of a surface for which one can realize a given set of natural numbers as the set of algebraic periods. Finally, we compute and list all possible Dold coefficients and algebraic periods for a given orientable surface with small genus and give some geometrical applications of the obtained results. |
| title | Dold coefficients of quasi-unipotent homeomorphisms of orientable surfaces |
| topic | Dynamical Systems Primary: 37C25, Secondary: 55M20, 37E30, 37D15 |
| url | https://arxiv.org/abs/2411.19313 |