Symplectic blenders near whiskered tori and persistence of saddle-center homoclinics
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915880520122368 |
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| author | Li, Dongchen Turaev, Dmitry |
| author_facet | Li, Dongchen Turaev, Dmitry |
| contents | A blender is a hyperbolic basic set such that the projection of its stable/unstable set onto some center subspace has a higher topological dimension than the set itself. We prove that, for any $C^s$ symplectic diffeomorphism (where $s=2,\dots\infty,ω$), if it has a one-dimensional whiskered torus with a homoclinic orbit, then a symplectic blender can be created by an arbitrarily $C^s$-small perturbation. Using this result, we show that the non-transverse homoclinic intersection between the invariant manifolds of a saddle-center periodic point is persistent, in the sense that the original system lies in the $C^s$-closure of a $C^1$-open set of symplectic diffeomorphisms where those having saddle-center homoclinics are dense. Our results also hold in the corresponding continuous-time settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20830 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symplectic blenders near whiskered tori and persistence of saddle-center homoclinics Li, Dongchen Turaev, Dmitry Dynamical Systems 37C29, 37G25, 37J11, 37D30 A blender is a hyperbolic basic set such that the projection of its stable/unstable set onto some center subspace has a higher topological dimension than the set itself. We prove that, for any $C^s$ symplectic diffeomorphism (where $s=2,\dots\infty,ω$), if it has a one-dimensional whiskered torus with a homoclinic orbit, then a symplectic blender can be created by an arbitrarily $C^s$-small perturbation. Using this result, we show that the non-transverse homoclinic intersection between the invariant manifolds of a saddle-center periodic point is persistent, in the sense that the original system lies in the $C^s$-closure of a $C^1$-open set of symplectic diffeomorphisms where those having saddle-center homoclinics are dense. Our results also hold in the corresponding continuous-time settings. |
| title | Symplectic blenders near whiskered tori and persistence of saddle-center homoclinics |
| topic | Dynamical Systems 37C29, 37G25, 37J11, 37D30 |
| url | https://arxiv.org/abs/2603.20830 |