Adapted Metrics for Singular Hyperbolic Flows
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Adapted Metrics for Singular Hyperbolic Flows Vitor Araujo1
· Vinicius Coelho2
· Luciana Salgado3
Received: 10 March 2020 / Accepted: 22 October 2020 © Sociedade Brasileira de Matemática 2020
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. Keywords Adapted metricsA · Singular hyperbolicity Mathematics Subject Classification Primary 37D30; Secondary 37D25 · 58B20
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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Vitor Araujo [email protected]; [email protected] Vinicius Coelho [email protected] Luciana Salgado [email protected]; [email protected]
1
Instituto de Matemática e Estatística, Universidade Federal da Bahia, Av. Adhemar de Barros, S/N , Ondina, Salvador, BA 40170-110, Brazil
2
Centro Multidisciplinar de Bom Jesus da Lapa, Universidade Federal do Oeste da Bahia, Av. Manoel Novais, 1064, Centro, Bom Jesus da Lapa, BA 47600-000, Brazil
3
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Avenida Athos da Silveira Ramos 149 Cidade Universitária, P.O. Box 68530, Rio de Janeiro, RJ 21941-909, Brazil
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V. Araujo et al.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminary Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Statements of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Applications and Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Organization of the Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Linear Multiplicative Cocycles Over Flows . . . . . . . . . . . . . . . . . . 3.2 Exterior Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Exterior Power of Linear Multiplicative Cocycles . . . . . . . . . . . . 3.2.2 Conditions for Sectional Expansion . . . . . . . . . . . . . . . . . . . 3.3 Dominated/Partial/Sectional-Hyperbolic Splittings and Exterior Powers . . . 4 Proofs of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Existence of Singular Adapted Metric . . . . . . . . . . . . . . . . . . . . . 4.1.1 Suba
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