Abstract
Singular and sectional-hyperbolic sets are the objects of the extension of the classical Smale Hyperbolic Theory to flows having invariant sets with singularities accumulated by regular orbits within the set. It is by now well-known that (partially) hyperbolic sets admit adapted metrics. We show the existence of singular-adapted metrics for any singular-hyperbolic set with respect to a \(C^{1}\) vector field on finite dimensional compact manifolds. Moreover, we obtain sectional-adapted metrics for certain open classes of sectional-hyperbolic sets and also for any hyperbolic set.
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Notes
For example, in \({{\mathbb {R}}}^4\) the following 2-vector (a symplectic form) is not decomposable: \(e_{1}\wedge e_{2}+e_{3}\wedge e_{4}.\)
Here \({\text {m}}(A) = \inf _{\Vert u\Vert =1} \Vert A (u) \Vert \) for a morphism A of normed vector spaces.
Set \({\tilde{v}}=v-hu\) so that \(h\cdot DX_{-t}u\) is the orthogonal projection of \(DX_{-t}v\) along \(DX_{-t}u\).
We write \({\text {m}}_*\) for the conorm associated to the norm \(\Vert \cdot \Vert _*\).
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Acknowledgements
This work is part of the PhD. Thesis of V. Coelho developed at the Mathematics Department of the Federal University of Bahia (UFBA) under the supervision of L. Salgado. The authors would like to thank the facilities provided by the Mathematics Institute for the PhD. Program and the partial financial support from several federal and state agencies to the Faculty and Students of this Program.
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V.A. is partially supported by CNPq-Brazil (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Grant 301392/2015-3) and FAPESB-Brazil (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Grand PIE0034/2016), Grant (Demanda Social). L.S. is partially supported Fapesb-JCB0053/2013, PRODOC-UFBA/2014 and CNPq. V.C. is supported by CAPES.
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Araujo, V., Coelho, V. & Salgado, L. Adapted Metrics for Singular Hyperbolic Flows. Bull Braz Math Soc, New Series 52, 815–840 (2021). https://doi.org/10.1007/s00574-020-00233-6
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DOI: https://doi.org/10.1007/s00574-020-00233-6