Abstract
For a partially hyperbolic splitting \(T_\Gamma M=E\oplus F\) of \(\Gamma \), a \(C^{1}\) vector field X on a m-manifold, we obtain singular-hyperbolicity using only the tangent map DX of X and its derivative \(DX_{t}\) whether E is one-dimensional subspace. We show the existence of adapted metrics for singular hyperbolic set \(\Gamma \) for \(C^{1}\) vector fields if \(\Gamma \) has a partially hyperbolic splitting \(T_\Gamma M=E\oplus F\) where F is volume expanding, E is uniformly contracted and a one-dimensional subspace.
L.S. is partially supported by Fapesb-JCB0053/2013, PRODOC-UFBA/2014 and CNPq 2017 postdoctoral schoolarship at Universidade Federal do Rio de Janeiro, V.C. is supported by CAPES.
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Acknowledgements
L.S. dedicates this article to the memory of Prof. Welington de Melo who taught her the profundity and beauty of hyperbolic dynamics. The authors are deeply grateful to the anonymous referee for the kind analysis of this paper and the excellent suggestions which helped us to improve our work. L.S. is partially supported by Fapesb-JCB0053/2013, PRODOC-UFBA/2014 and CNPq 2017 postdoctoral schoolarship at Universidade Federal do Rio de Janeiro, V.C. is supported by CAPES.
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Salgado, L., Coelho, V. (2019). Adapted Metrics for Codimension One Singular Hyperbolic Flows. In: Pacifico, M., Guarino, P. (eds) New Trends in One-Dimensional Dynamics. Springer Proceedings in Mathematics & Statistics, vol 285. Springer, Cham. https://doi.org/10.1007/978-3-030-16833-9_14
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