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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.12412 |
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| _version_ | 1866908656695508992 |
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| author | Barrientos, Pablo G. Díaz, Lorenzo J. Ki, Yuri Lizana, Cristina Pérez, Sebastián A. |
| author_facet | Barrientos, Pablo G. Díaz, Lorenzo J. Ki, Yuri Lizana, Cristina Pérez, Sebastián A. |
| contents | We consider diffeomorphisms $f$ with heterodimensional cycles of co-index two, associated with saddles $P$ and $Q$ having unstable indices $\ell$ and $\ell+2$, respectively. In a partially hyperbolic setting, where a two-dimensional center direction and strong invariant manifolds are defined, we introduce the class of \emph{non-escaping cycles}, where the strong stable manifold of $P$ and the strong unstable manifold of $Q$ are involved in the cycle. This configuration guarantees the existence of orbits that remain in a neighbourhood of the cycle.
We show that such diffeomorphisms $f$ can be $C^1$ approximated by diffeomorphisms exhibiting simultaneously $C^1$ robust heterodimensional cycles of co-indices one and two, encompassing all possible combinations among hyperbolic sets of unstable indices $\ell$, $\ell+1$, and $\ell+2$.
The proof relies on the construction of \emph{split blending machines}. This tool extends Asaoka's blending machines to a partially hyperbolic setting, providing a mechanisms to generate and control robust intersections within a two-dimensional central bundle.
We also present simple dynamical settings where such cycles occur, namely skew product dynamics with surface fiber maps. Non-escaping cycles also appear in contexts such as Derived from Anosov diffeomorphisms and matrix cocycles on $\mathrm{GL}(3,\mathbb{R})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12412 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Robust heterodimensional cycles of co-index two via split blending machines Barrientos, Pablo G. Díaz, Lorenzo J. Ki, Yuri Lizana, Cristina Pérez, Sebastián A. Dynamical Systems We consider diffeomorphisms $f$ with heterodimensional cycles of co-index two, associated with saddles $P$ and $Q$ having unstable indices $\ell$ and $\ell+2$, respectively. In a partially hyperbolic setting, where a two-dimensional center direction and strong invariant manifolds are defined, we introduce the class of \emph{non-escaping cycles}, where the strong stable manifold of $P$ and the strong unstable manifold of $Q$ are involved in the cycle. This configuration guarantees the existence of orbits that remain in a neighbourhood of the cycle. We show that such diffeomorphisms $f$ can be $C^1$ approximated by diffeomorphisms exhibiting simultaneously $C^1$ robust heterodimensional cycles of co-indices one and two, encompassing all possible combinations among hyperbolic sets of unstable indices $\ell$, $\ell+1$, and $\ell+2$. The proof relies on the construction of \emph{split blending machines}. This tool extends Asaoka's blending machines to a partially hyperbolic setting, providing a mechanisms to generate and control robust intersections within a two-dimensional central bundle. We also present simple dynamical settings where such cycles occur, namely skew product dynamics with surface fiber maps. Non-escaping cycles also appear in contexts such as Derived from Anosov diffeomorphisms and matrix cocycles on $\mathrm{GL}(3,\mathbb{R})$. |
| title | Robust heterodimensional cycles of co-index two via split blending machines |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2511.12412 |