Generic Dynamics of 4-Dimensional C 2 Hamiltonian Systems

Bessa, Mário; Dias, João Lopes

doi:10.1007/s00220-008-0500-y

Cited by 21 publications

(20 citation statements)

References 10 publications

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“…Here µ E is the natural induced invariant measure on the energy level sets (see [4]). Denote by Z ⊂ E the full measure subset of such points with zero Lyapunov exponents.…”

Section: Proof Of Theorem 1 Takementioning

confidence: 99%

“…For that we rely on recently available results, viz. Vivier's Hamiltonian version of Franks' lemma [14] and the authors' theorem on the Bochi-Mañé dichotomy for Hamiltonians [4] (see also [6,3]). …”

Section: Introductionmentioning

confidence: 99%

“…Notation and basic definitions. We refer to [4] for the basic notions in Hamiltonian dynamical systems and the notation used in this paper. In particular, we recall that given a differentiable Hamiltonian H : M → R we denote X H as the Hamiltonian vector field, ϕ …”

Section: Introductionmentioning

confidence: 99%

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Hamiltonian elliptic dynamics on symplectic $4$-manifolds

Bessa¹,

Dias²

2008

Proc. Amer. Math. Soc.

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Abstract. We consider C 2 -Hamiltonian functions on compact 4-dimensional symplectic manifolds to study the elliptic dynamics of the Hamiltonian flow, namely the so-called Newhouse dichotomy. We show that for any open set U intersecting a far from Anosov regular energy surface, there is a nearby Hamiltonian having an elliptic closed orbit through U . Moreover, this implies that, for far from Anosov regular energy surfaces of a C 2 -generic Hamiltonian, the elliptic closed orbits are generic.

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“…Here µ E is the natural induced invariant measure on the energy level sets (see [4]). Denote by Z ⊂ E the full measure subset of such points with zero Lyapunov exponents.…”

Section: Proof Of Theorem 1 Takementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hamiltonian elliptic dynamics on symplectic $4$-manifolds

Bessa¹,

Dias²

2008

Proc. Amer. Math. Soc.

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“…Later, in [2, Theorem A], a global version for vector fields with equilibrium points was obtained. In [8], a version was proved for linear differential systems with conservative properties, and in [9] a similar result was obtained in the Hamiltonian setting.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 69%

Contributions to the geometric and ergodic theory of conservative flows

Bessa

Rocha

2012

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385712001101How to cite this article: MÁRIO BESSA and JORGE ROCHA (2013). Contributions to the geometric and ergodic theory of conservative ows.Abstract. We prove the following dichotomy for vector fields in a C 1 -residual subset of volume-preserving flows: for Lebesgue-almost every point, either all of its Lyapunov exponents are equal to zero or its orbit has a dominated splitting. Moreover, we prove that a volume-preserving and C 1 -stably ergodic flow can be C 1 -approximated by another volume-preserving flow which is non-uniformly hyperbolic. Introduction and statement of the resultsLet M be a d-dimensional (with d ≥ 3), compact, connected and boundaryless Riemannian manifold endowed with a volume form ω, and let µ denote the Lebesgue measure associated to it. We denote by X 1 µ (M) the space of C 1 vector fields X over M which are divergence-free, that is, the associated flow X t of each X preserves the measure µ. We consider X 1 µ (M) endowed with the usual Whitney C 1 -topology. Given a flow X t , one usually deduces properties of it by studying its linear approximation. One way to do that is by considering the Lyapunov exponents, which, in broad terms, detect whether there is any exponential behavior of the linear tangent map along orbits. Given X ∈ X 1 µ (M), the existence of Lyapunov exponents for almost every point is guaranteed by Oseledets' theorem [31]. Positive (or negative) exponents ensure, on average, an exponential rate of divergence (or convergence) of two neighboring trajectories, whereas zero exponents tell us there is a lack of any kind of average exponential behavior. A flow is said to be non-uniformly hyperbolic if its Lyapunov exponents are all different from zero. In [25], Hu et al gave examples of non-uniformly hyperbolic flows in any compact manifold. Non-zero exponents, together with some smoothness assumptions on the flow, allow us to obtain invariant manifolds that are dynamically defined (see [32]). Since this stable/unstable manifold theory forms the basis

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“…We remark that the proof makes use of some results that are only available in dimension four (see [6,7]). …”

Section: 2 the Star Systemsmentioning

confidence: 97%