\(\mathcal{C}^{1}\) self-maps on \(\mathbb {S}^{n}\), \(\mathbb {S}^{n} \times \mathbb {S}^{m}\), \(\mathbb {C}\mathbf P^{n}\) and \(\mathbb {H}\mathbf P^{n}\) with all their periodic orbits hyperbolic (Q763738)
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scientific article; zbMATH DE number 6019762
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | \(\mathcal{C}^{1}\) self-maps on \(\mathbb {S}^{n}\), \(\mathbb {S}^{n} \times \mathbb {S}^{m}\), \(\mathbb {C}\mathbf P^{n}\) and \(\mathbb {H}\mathbf P^{n}\) with all their periodic orbits hyperbolic |
scientific article; zbMATH DE number 6019762 |
Statements
\(\mathcal{C}^{1}\) self-maps on \(\mathbb {S}^{n}\), \(\mathbb {S}^{n} \times \mathbb {S}^{m}\), \(\mathbb {C}\mathbf P^{n}\) and \(\mathbb {H}\mathbf P^{n}\) with all their periodic orbits hyperbolic (English)
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29 March 2012
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Assume that \(M\) is one of the manifolds \(\mathbb{S}^n\), \(\mathbb{S}^n\times \mathbb{S}^m\), \(\mathbb {C}\mathbf P^{n}\) or \(\mathbb {H}\mathbf P^{n}\) and let \(f\) be a \(C^1\) self-map on \(M\). Under some special conditions (too technical to be presented here in detail), including the assumption that all periodic orbits of \(f\) are hyperbolic, the authors provide a partial characterization of the structure of the set of least periods of periodic orbits of \(f\) (e.g., by showing that this set is infinite, contains \(1\) or \(2\), etc.).
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Lefschetz zeta function
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sphere
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complex projective space
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hyperbolic periodic orbit
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least period
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