# Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians

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### "Normal forms, stability and splitti..." refers background or methods or result in this paper

...oncrete examples, but also because we could not ﬁnd any satisfactory abstract deﬁnition (for instance one which would ensure the existence and uniqueness of stable and unstable manifolds, we refer to [LMS03] and [BT00] for some attempts). Now let us consider the setting as described in (G2). Let T0 = Tn ×{0} be the invariant torus, for the integrable system, with frequency ω ∈ Rn \ {0}. Without loss of g...

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...ations (essentially for two degrees of freedom, speciﬁc frequency vectors and speciﬁc perturbations, see [BFGS12] for some recent results and references). Our aim here is to generalize the results of [LMS03] for Hamiltonian systems which are only Gevrey regular. We will also assume the existence of a “hyperbolic” torus, together with the property that its invariant manifolds intersect (that is, the exist...

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...wn as the speed of Arnold diﬀusion). The general principle is that the “splitting” is exponentially small for analytic systems and the literature on the subject is huge. Here we shall closely follows [LMS03], Chapter §2, where an approach to obtain exponentially small upper bounds in the analytic case is given based on normal forms techniques. The results contained in [LMS03] are quite general, as they a...

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...nse that it possesses stable and unstable manifolds. Moreover, such a tori will be isotropic and its asymptotic manifolds will be Lagrangian. If the stable and unstable manifolds intersect, following [LMS03], we can deﬁne a symmetric matrix of size n at a given homoclinic point, called a splitting matrix, the eigenvalues of which are called the splitting angles. Our result is that there exists at least d...

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... of a vector by independent periodic vectors proved in [BF12] and a technique of composition of periodic averaging ﬁrst used in [BN12]. Note that our main result answers a question which was asked in [LMS03], concerning the “interaction of Gevrey conditions with arithmetic properties in normal forms”. In a second paper [Bou12a], we will prove the corresponding result for ﬁnitely diﬀerentiable Hamiltonian...

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