Every finite set of natural numbers is realizable as algebraic periods of a Morse$\unicode{x2013}$Smale diffeomorphism
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916763517583360 |
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| author | Graff, Grzegorz Marzantowicz, Wacław Michalak, Łukasz Patryk Myszkowski, Adrian |
| author_facet | Graff, Grzegorz Marzantowicz, Wacław Michalak, Łukasz Patryk Myszkowski, Adrian |
| contents | A given self-map $f\colon M\to M$ of a compact manifold determines the sequence $(L(f^n))$ of the Lefschetz numbers of its iterations. We consider its dual sequence $(a_n(f))$ given by the Möbius inversion formula. The set ${\mathcal AP}(f)=\{ n\in \mathbb N\ \colon\ a_n(f)\neq 0\}$ is called the set of algebraic periods.
We solve an open problem existing in literature by showing that for every finite subset ${\mathcal A}$ of natural numbers there exist an orientable surface $S_{\rm g}$, as well as a non-orientable surface $N_{\rm g}$, of genus ${\rm g}$, and a Morse$\unicode{x2013}$Smale diffeomorphism $f$ of this surface such that $\mathcal{AP}(f)=\mathcal{A}$. For such a map it implies the existence of points of a minimal period $n$ for each odd $n \in \mathcal{A}$. For the orientation-reversing Morse$\unicode{x2013}$Smale diffeomorphisms of $S_{\rm g}$, we identify strong restrictions on ${\mathcal AP}(f)$. Our method also provides an estimate of the number of conjugacy classes of mapping classes containing Morse$\unicode{x2013}$Smale diffeomorphisms, which is exponential in ${\rm g}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12372 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Every finite set of natural numbers is realizable as algebraic periods of a Morse$\unicode{x2013}$Smale diffeomorphism Graff, Grzegorz Marzantowicz, Wacław Michalak, Łukasz Patryk Myszkowski, Adrian Dynamical Systems Primary: 37C25 37E30, 57K20, Secondary: 55M20, 37D15 A given self-map $f\colon M\to M$ of a compact manifold determines the sequence $(L(f^n))$ of the Lefschetz numbers of its iterations. We consider its dual sequence $(a_n(f))$ given by the Möbius inversion formula. The set ${\mathcal AP}(f)=\{ n\in \mathbb N\ \colon\ a_n(f)\neq 0\}$ is called the set of algebraic periods. We solve an open problem existing in literature by showing that for every finite subset ${\mathcal A}$ of natural numbers there exist an orientable surface $S_{\rm g}$, as well as a non-orientable surface $N_{\rm g}$, of genus ${\rm g}$, and a Morse$\unicode{x2013}$Smale diffeomorphism $f$ of this surface such that $\mathcal{AP}(f)=\mathcal{A}$. For such a map it implies the existence of points of a minimal period $n$ for each odd $n \in \mathcal{A}$. For the orientation-reversing Morse$\unicode{x2013}$Smale diffeomorphisms of $S_{\rm g}$, we identify strong restrictions on ${\mathcal AP}(f)$. Our method also provides an estimate of the number of conjugacy classes of mapping classes containing Morse$\unicode{x2013}$Smale diffeomorphisms, which is exponential in ${\rm g}$. |
| title | Every finite set of natural numbers is realizable as algebraic periods of a Morse$\unicode{x2013}$Smale diffeomorphism |
| topic | Dynamical Systems Primary: 37C25 37E30, 57K20, Secondary: 55M20, 37D15 |
| url | https://arxiv.org/abs/2408.12372 |