Shades of hyperbolicity for Hamiltonians

We prove that a Hamiltonian system is globally hyperbolic if any of the following statements hold: H is robustly topologically stable; H is stably shadowable; H is stably expansive; and H has the stable weak specification property. Moreover, we prove that, for a C2-generic Hamiltonian H, the union o...

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Bibliographic Details
Published in:Nonlinearity 2013-10, Vol.26 (10), p.2851-2873
Main Authors: Bessa, Mário, Rocha, Jorge, Torres, Maria Joana
Format: Article
Language:English
Subjects:
Expansion
Geometry
Nonlinearity
Orbits
Shades
Specifications
Unions
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Summary:We prove that a Hamiltonian system is globally hyperbolic if any of the following statements hold: H is robustly topologically stable; H is stably shadowable; H is stably expansive; and H has the stable weak specification property. Moreover, we prove that, for a C2-generic Hamiltonian H, the union of the partially hyperbolic regular energy hypersurfaces and the closed elliptic orbits, forms a dense subset of M. As a consequence, any robustly transitive regular energy hypersurface of a C2-Hamiltonian is partially hyperbolic. Finally, we prove that stable weakly-shadowable regular energy hypersurfaces are partially hyperbolic.
ISSN:0951-7715
1361-6544
DOI:10.1088/0951-7715/26/10/2851