Mostly nonuniformly sectional expanding systems
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arXiv
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| Format: | Preprint |
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2025
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| author | Araújo, Vitor Salgado, Luciana |
| author_facet | Araújo, Vitor Salgado, Luciana |
| contents | We introduce the notion of \emph{mostly nonuniform sectional expanding} (MNUSE) for singular flows which encompasses the notions of sectional hyperbolicity, asymptotically sectional and multisingular hyperbolicity. We exhibit an example of a vector field of class $C^r, r > 1$, whose flow exhibits a nonuniformly sectional hyperbolic set satisfying MNUSE, which is neither sectional hyperbolic nor asymptotically sectional hyperbolic.
We obtain sufficient conditions for the existence of physical/SRB measures for asymptotically sectionally hyperbolic attracting sets with any finite co-dimension, extending the co-dimension two case.
We provide examples of such attractors, either with non-sectional hyperbolic equilibria, or with sectional-hyperbolic equilibria of mixed type, i.e., with a Lorenz-like singularity together with a Rovella-like singularity in a transitive set. These are higher-dimensional versions of contracting Lorenz-like attractors (also known as Rovella-like attractors) to which we apply our criteria to obtain a physical/SRB measure with full ergodic basin.
We also adapt the previous examples to obtain higher co-dimensional (i.e. with central direction of dimension greater than $2$) non-uniformly sectional expanding attractors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_06233 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mostly nonuniformly sectional expanding systems Araújo, Vitor Salgado, Luciana Dynamical Systems 37C10, 37D25, 37D30, 37C40 We introduce the notion of \emph{mostly nonuniform sectional expanding} (MNUSE) for singular flows which encompasses the notions of sectional hyperbolicity, asymptotically sectional and multisingular hyperbolicity. We exhibit an example of a vector field of class $C^r, r > 1$, whose flow exhibits a nonuniformly sectional hyperbolic set satisfying MNUSE, which is neither sectional hyperbolic nor asymptotically sectional hyperbolic. We obtain sufficient conditions for the existence of physical/SRB measures for asymptotically sectionally hyperbolic attracting sets with any finite co-dimension, extending the co-dimension two case. We provide examples of such attractors, either with non-sectional hyperbolic equilibria, or with sectional-hyperbolic equilibria of mixed type, i.e., with a Lorenz-like singularity together with a Rovella-like singularity in a transitive set. These are higher-dimensional versions of contracting Lorenz-like attractors (also known as Rovella-like attractors) to which we apply our criteria to obtain a physical/SRB measure with full ergodic basin. We also adapt the previous examples to obtain higher co-dimensional (i.e. with central direction of dimension greater than $2$) non-uniformly sectional expanding attractors. |
| title | Mostly nonuniformly sectional expanding systems |
| topic | Dynamical Systems 37C10, 37D25, 37D30, 37C40 |
| url | https://arxiv.org/abs/2508.06233 |