By Celletti A., Chierchia L.
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Let us turn to uniqueness. Let sˆ and C as in Proposition 3 and define k3 = k3 (ξ∗ , κ, τ, η0 , M, ν, α) := 2Cξ∗−ˆs , k4 := k2 k3 , and assume that k4 e ξ∗ ≤1. 91) is satisfied so that, by Proposition 3, u = u and γ = γ. Thus (since k4 is greater than k2 and k3 ), we see that all claims in Theorem 3 follow by taking k := max{k1 , k4 }. 7 Proof of Theorem 1 We now show how Theorem 1 can be obtained as a corollary of Theorem 3. 6) is assumed to be real–analytic, there exists a ξ¯ > 0 such that f ∈ Hξ (point (ii) of Remark 4).
Chierchia, G. Gallavotti: Smooth prime integrals for quasi- integrable Hamiltonian systems, Il Nuovo Cimento, 67 B (1982), 277–295. [8] A. C. M. Correia, J. Laskar: Mercury’s capture into the 3/2 spin–orbit resonance as a result of its chaotic dynamics, Nature 429 (24 June 2004), 848–850 . F. MacDonald, Tidal friction, Rev. Geophys. 2 (1964), 467–541. J. Peale, The free precession and libration of Mercury, Icarus 178 (2005), 4–18. [11] J. P¨oschel: Integrability of Hamiltonian sytems on Cantor sets, Comm.
Fη (u; γ) = 0), satisfying also uθ1 ξ∗ ≤ ν ≤ ξ¯ − ξ∗ , |ρ| ≤ α < 1 , for some ν, α ∈ (0, 1). e. 91) then u ≡ u and γ = γ . 92) where V := 1 + uθ1 and W := V 2 . Thus, (since also Fη (u ; γ ) = 0) 0 = Fη (u ; γ ) = Fη (u; γ) + ∆η w + gx (θ1 + u , θ2 ) − gx (θ1 + u, θ2 ) + γˆ = ∆η w + gxx (θ1 + u, θ2 ) w + γˆ + gx (θ1 + u + w, θ2 ) − gx (θ1 + u, θ2 ) − gxx (θ1 + u, θ2 )w =: Aη,u w + γˆ + Q1 (w) . 93) If yi > 0 and {xi }i≥0 is a sequence of positive numbers satisfying xi+1 ≤ y0 y1i x2i , i then one also has xi ≤ (y0 y1 x0 )2 /(y0 y1i+1 ), as it follows multiplying both sides of the above inequality by y0 y1i+2 so as to obtain zi+1 ≤ zi2 with zi := y0 y1i+1 xi .