Wild Sets with Collet-Eckmann Points and Infinitely Many Sinks: Stability and Coexistence
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| Format: | Preprint |
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2025
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| _version_ | 1866917301483208704 |
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| author | Martens, Marco Palmisano, Liviana |
| author_facet | Martens, Marco Palmisano, Liviana |
| contents | In two-dimensional unfoldings of homoclinic tangencies, the parameter space contains codimension one laminations whose leaves consist of maps with invariant non-hyperbolic Cantor sets. These Cantor sets are wild both in the sense of Hofbauer-Keller and in the sense of Newhouse, and they contain Collet-Eckmann points with dense orbits. Hence, wildness and non-uniform chaotic hyperbolicity can coexist on a single invariant set, while persisting along codimension one manifolds. In addition, each leaf of the lamination contains a map with infinitely many sinks accumulating on the Cantor set containing the Collet-Eckmann point. This surprising symbiosis of contraction and expansion may not, in fact, be pathological. Along the way, we introduce a generalized renormalization scheme for two-dimensional systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12926 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wild Sets with Collet-Eckmann Points and Infinitely Many Sinks: Stability and Coexistence Martens, Marco Palmisano, Liviana Dynamical Systems In two-dimensional unfoldings of homoclinic tangencies, the parameter space contains codimension one laminations whose leaves consist of maps with invariant non-hyperbolic Cantor sets. These Cantor sets are wild both in the sense of Hofbauer-Keller and in the sense of Newhouse, and they contain Collet-Eckmann points with dense orbits. Hence, wildness and non-uniform chaotic hyperbolicity can coexist on a single invariant set, while persisting along codimension one manifolds. In addition, each leaf of the lamination contains a map with infinitely many sinks accumulating on the Cantor set containing the Collet-Eckmann point. This surprising symbiosis of contraction and expansion may not, in fact, be pathological. Along the way, we introduce a generalized renormalization scheme for two-dimensional systems. |
| title | Wild Sets with Collet-Eckmann Points and Infinitely Many Sinks: Stability and Coexistence |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2511.12926 |