Mathers regions of instability for annulus diffeomorphisms
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866929274257145856 |
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| author | Addas-Zanata, Salvador Tal, Fabio Armando |
| author_facet | Addas-Zanata, Salvador Tal, Fabio Armando |
| contents | Let $f$ be a $C^{1+\varepsilon}$ diffeomorphism of the closed annulus $A$ that preserves orientation and the boundary components, and $\widetilde{f}$ be a lift of $f$ to its universal covering space. Assume that $A$ is a Birkhoff region of instability for $f$, and the rotation set of $\widetilde{f}$ is a non-degenerate interval. Then there exists an open $f$-invariant annulus $A^*$ whose boundary intersects both boundary components of of $A$, and points $z^+$ and $z^-$ in $A^*$, such that the positive (resp. negative) orbit of $z^+$ converges to a set contained in the upper (resp. lower) boundary component of $A^*$ and the positive (resp. negative) orbit of $z^-$ converges to a set contained in the lower (resp. upper) boundary component of $A^*$. This extends a celebrated result originally proved by Mather for area-preserving twist diffeomorphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_13102 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Mathers regions of instability for annulus diffeomorphisms Addas-Zanata, Salvador Tal, Fabio Armando Dynamical Systems 37E30 Let $f$ be a $C^{1+\varepsilon}$ diffeomorphism of the closed annulus $A$ that preserves orientation and the boundary components, and $\widetilde{f}$ be a lift of $f$ to its universal covering space. Assume that $A$ is a Birkhoff region of instability for $f$, and the rotation set of $\widetilde{f}$ is a non-degenerate interval. Then there exists an open $f$-invariant annulus $A^*$ whose boundary intersects both boundary components of of $A$, and points $z^+$ and $z^-$ in $A^*$, such that the positive (resp. negative) orbit of $z^+$ converges to a set contained in the upper (resp. lower) boundary component of $A^*$ and the positive (resp. negative) orbit of $z^-$ converges to a set contained in the lower (resp. upper) boundary component of $A^*$. This extends a celebrated result originally proved by Mather for area-preserving twist diffeomorphisms. |
| title | Mathers regions of instability for annulus diffeomorphisms |
| topic | Dynamical Systems 37E30 |
| url | https://arxiv.org/abs/2112.13102 |