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By M. Aizenman (Chief Editor)

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4. . 6) we need a parameter space disc of radius 100 δ All we have seen allows us to define the constant K R as being precisely what is needed. That is, to find t ∗ , and to lose a little bit of radius to obtain the estimates with respect to the parameter in the previous preparative process. Thus we have to choose 2400(wn ∗ )−1 . (151) δ To finish this previous preparative process and to start with the algorithm, it is enough to notice that the following estimate: K R = 2L(1 + 4Wn ∗ ) + 96000L Wn ∗ + 0 − ∂t T1 n¯ ρ1,t ≤ 0 − ∂t n−1 ¯ ∗ T1 in + ∂t n=n ∗ i=0 n ρ1,t ∂z ρtn,i (·, u n,i 0,t ) ≤ T ε 2L(1 + 4Wn ∗ ) + ≤ L −1 1000 2400(wn ∗ )−1 δ + L −1 2000 Persistence of Invariant Curves for Fibered Holomorphic Transformations 41 holds inside the parameter space disc centered at t0 with radious ε (2L(1 + 4Wn ∗ )+ 2400(wn ∗ )−1 .

130) For such a disc, we also verify that the parameter t¯ proving the existence of the invariant curve is located at a distance ε times a constant from the origin in the complex plane, as desired. If ε goes to 0, then the level n¯ of the first stage grows and (123) allows to conclude that the size of the invariant curve tends to 0. 36 M. 1 holds under the stronger hypothesis ρ0 ≤ wn ∗ (or more precisely under hypothesis (125)). In the next section we will see that we only need to impose the hypothesis ρ0 ≤ ε.

2) 164(3), 911–940 (2006) 2. : Tores invariantes des systèmes dynamiques hamiltoniens. Asterisque, Séminaire Bourbaki 639(133–134), 113–157 (1986) 3. : On Melnikov’s persistency problem. Math. Res. Lett. 4(4), 445–458 (1997) 4. : Analytic form of differential equations I. Trans. Moscow Math. Soc. 25, 131–288 (1971) 5. : An Introduction to Diophantine Approximation. Volume 45 of Cambridge tracts. Cambridge: Cambridge University Press, 1957 6. : Perturbations of stable invariant tori for Hamiltonian systems.

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