Partial hyperbolicity or dense elliptic periodic points for 𝐶¹-generic symplectic diffeomorphisms

DOI10.1090/S0002-9947-06-04171-7zbMath1210.37014MaRDI QIDQ3420358

Radu Saghin, Zhihong Xia

Publication date: 1 February 2007

Published in: Transactions of the American Mathematical Society (Search for Journal in Brave)

Mathematics Subject Classification ID

Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics (37C25) Generic properties, structural stability of dynamical systems (37C20) Partially hyperbolic systems and dominated splittings (37D30)

Related Items (13)

Hofer growth of \(C^1\)-generic Hamiltonian flows ⋮ Generic Hamiltonian Dynamical Systems: An Overview ⋮ Area-Preserving Diffeomorphisms from theC 1Standpoint ⋮ The 𝐶¹ density of nonuniform hyperbolicity in 𝐶^{𝑟} conservative diffeomorphisms ⋮ Billiards in generic convex bodies have positive topological entropy ⋮ Local perturbations of conservativeC¹diffeomorphisms ⋮ Realization of tangent perturbations in discrete and continuous time conservative systems ⋮ Abundance of elliptic dynamics on conservative three-flows ⋮ Billiards: A singular perturbation limit of smooth Hamiltonian flows ⋮ Hamiltonian elliptic dynamics on symplectic $4$-manifolds ⋮ Continuum-wise expansive symplectic diffeomorphisms ⋮ Stability in high dimensional steep repelling potentials ⋮ C¹-generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents

Cites Work

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• Diffeomorphisms with infinitely many sinks
• Homoclinic points in conservative systems
• The C¹ Closing Lemma, including Hamiltonians
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• Connexions hétéroclines et généricité d'une infinité de puits et de sources
• 𝐶¹ Connecting Lemmas
• Difféomorphismes symplectiques de classe C1 en dimension 4
• The generic symplectic C^{1}-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point
• Genericity of zero Lyapunov exponents
• The Closing Lemma
• Generic Properties of Conservative Systems II
• Necessary Conditions for Stability of Diffeomorphisms
• Homoclinic points near elliptic fixed points

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