Infinitely Many Attracting Periodic Circles in Higher Dimensions
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918438782369792 |
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| author | Tomizawa, Shuntaro |
| author_facet | Tomizawa, Shuntaro |
| contents | We study $C^r$ ($5 \le r \le \infty$) diffeomorphisms on closed manifolds of dimension at least three with a heteroclinic cycle between two hyperbolic periodic points. At each point, the unstable direction is one dimensional, and the stable and unstable eigenvalues closest to $1$ in modulus are real and simple. One heteroclinic connection is transverse and the other is non-transverse, and the product of those two eigenvalues is less than $1$ at one point and greater than $1$ at the other. Arbitrarily close to such a map, there are open sets in which a residual subset of diffeomorphisms has infinitely many attracting normally hyperbolic periodic circles. The proof uses a rescaling to the standard Hénon map and a corrected formula for the Lyapunov coefficient on its Neimark-Sacker (Andronov-Hopf) line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09441 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Infinitely Many Attracting Periodic Circles in Higher Dimensions Tomizawa, Shuntaro Dynamical Systems We study $C^r$ ($5 \le r \le \infty$) diffeomorphisms on closed manifolds of dimension at least three with a heteroclinic cycle between two hyperbolic periodic points. At each point, the unstable direction is one dimensional, and the stable and unstable eigenvalues closest to $1$ in modulus are real and simple. One heteroclinic connection is transverse and the other is non-transverse, and the product of those two eigenvalues is less than $1$ at one point and greater than $1$ at the other. Arbitrarily close to such a map, there are open sets in which a residual subset of diffeomorphisms has infinitely many attracting normally hyperbolic periodic circles. The proof uses a rescaling to the standard Hénon map and a corrected formula for the Lyapunov coefficient on its Neimark-Sacker (Andronov-Hopf) line. |
| title | Infinitely Many Attracting Periodic Circles in Higher Dimensions |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2604.09441 |