$${mathbb{S}}^{2}$上自映射的周期及其同调

Pub Date : 2024-07-30 DOI:10.1007/s11253-024-02308-9
Jaume Llibre
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引用次数: 0

摘要

按照惯例,我们用 \({\mathbb{S}}^{2}\ 表示二维球体。)我们研究映射 f :\({\mathbb{S}}^{2}\to{\mathbb{S}}^{2}\)是连续的或 C1 的,其周期轨道都是双曲的、或横向的、或全态的、或横向全态的。我们首次总结了关于这些不同类型自映射在 \({\mathbb{S}}^{2}\) 上的周期轨道的所有已知结果。我们注意到,每次当一个映射 f :\({\mathbb{S}}^{2}\to{\mathbb{S}}^{2}\)的结构增加时,它对同调的作用所提供的周期轨道的数量也会增加。
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Periods of Self-Maps on $${\mathbb{S}}^{2}$$ Via their Homology

As usual, we denote a 2-dimensional sphere by \({\mathbb{S}}^{2}\). We study the periods of periodic orbits of the maps f : \({\mathbb{S}}^{2}\to {\mathbb{S}}^{2}\) that are either continuous or C1 with all their periodic orbits being hyperbolic, or transversal, or holomorphic, or transversal holomorphic. For the first time, we summarize all known results on the periodic orbits of these distinct kinds of self-maps on \({\mathbb{S}}^{2}\) together. We note that every time when a map f : \({\mathbb{S}}^{2}\to {\mathbb{S}}^{2}\) increases its structure, the number of periodic orbits provided by its action on the homology increases.

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