On robust expansiveness for sectional hyperbolic attracting sets
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arXiv
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| Format: | Preprint |
| Published: |
2019
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| _version_ | 1866909546033709056 |
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| author | Araujo, Vitor Cerqueira, Junilson |
| author_facet | Araujo, Vitor Cerqueira, Junilson |
| contents | We prove that sectional-hyperbolic attracting sets for $C^1$ vector fields are robustly expansive (under an open technical condition of strong dissipative for higher codimensional cases). This extends known results of expansiveness for singular-hyperbolic attractors in $3$-flows even in this low dimensional setting. We deduce some converse results taking advantage of recent progress in the study of star vector fields: a robustly transitive attractor is sectional-hyperbolic if, and only if, it is robustly expansive. In a low dimensional setting, we show that an attracting set of a $3$-flow is singular-hyperbolic if, and only if, it is robustly chaotic (robustly sensitive to initial conditions). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1910_12095 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On robust expansiveness for sectional hyperbolic attracting sets Araujo, Vitor Cerqueira, Junilson Dynamical Systems 37C10, 37D30, 37D50 We prove that sectional-hyperbolic attracting sets for $C^1$ vector fields are robustly expansive (under an open technical condition of strong dissipative for higher codimensional cases). This extends known results of expansiveness for singular-hyperbolic attractors in $3$-flows even in this low dimensional setting. We deduce some converse results taking advantage of recent progress in the study of star vector fields: a robustly transitive attractor is sectional-hyperbolic if, and only if, it is robustly expansive. In a low dimensional setting, we show that an attracting set of a $3$-flow is singular-hyperbolic if, and only if, it is robustly chaotic (robustly sensitive to initial conditions). |
| title | On robust expansiveness for sectional hyperbolic attracting sets |
| topic | Dynamical Systems 37C10, 37D30, 37D50 |
| url | https://arxiv.org/abs/1910.12095 |