Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

we deduce that Γ 5 is submersive (“covering”) from a small neighborhood of(x1 , a 1 , 0, −a 1 , −x 1)in K 5 onto a neighborhood of the origin in M .10.26. Covering a neighborhood of the origin in( M . For (z 0 , c 0 ) ∈ Mfixed and close to the origin, we denote by Γ k [xa]k ; (z 0 , c 0 ) ) and byΓk( ∗ [ax]k ; (z 0 , c 0 ) ) the (dual) chains issued from (z 0 , c 0 ). For given parameters[xa] k = (x 1 , a 1 , x 2 , . . .), we denote by [−xa] k the collection(· · · , −x 2 , −a 1 , −x 2 ) with minus signs and reverse order; similarly, we introduce[−ax] k . Notably, we have L −x1 (L x1 (0)) = 0 (because L −x1 +x 1(·) =L 0 (·) = Id), and also L −x1 (L ∗ −a 1(L ∗ a 1(L x1 (0)))) = 0 and generally:(10.27) Γ k([−xa]k ; Γ k ([xa] k ; 0) ) ≡ 0.Geometrically speaking, by following backward the k-th chain Γ k , we comeback to 0.Theorem 10.28. ([Me2005a, Me2005b], [∗]) The two maps Γ 2µ+1 andΓ ∗ 2µ ∗ +1 are submersive onto a neighborhood of the origin in M . Precisely,there exist two points [xa] 0 2µ+1 ∈ K (µ+1)n+µp and [ax] 0 2µ ∗ +1 ∈K µ∗ n+(µ ∗ +1)p arbitrarily close to the origin with Γ 2µ+1 ([xa] 0 2µ+1) = 0 andΓ ∗ 2µ ∗ +1 ([ax]0 2µ ∗ +1 ) = 0 such that the two maps{ ( )K (µ+1)n+µp ∋ [xa] 2µ+1 ↦−→ Γ 2µ+1 [xa]2µ+1 ∈ M and(10.29)K µ∗ n+(µ ∗ +1)p ∋ [ax] 2µ ∗ +1 ↦−→ Γ ∗ 2µ ∗ +1([ax]2µ ∗ +1)∈ M295are of rank n + m + p = dim K M at the points [xa] 0 2µ and [ax] 0 2µ ∗ respectively.In particular, the ranges of the two maps Γ 2µ+1 and Γ ∗ 2µ ∗ +1 cover aneighborhood of the origin in M .Let π z (z, c) := z and π c (z, c) := c be the two canonical projections. Thenext corollary will be useful in Section 12. In the example above, it alsofollows that the map(10.30) [xa] 4 ↦→ π c(Γ4 ([xa 4 ]) ) = ( a 1 + a 2 , −x 1 a 1 − x 1 a 2 − x 2 a 2)∈ K2is of rank two at all points [xa] 0 4 := ( x 0 1 , a0 1 , 0, −a0 1)with a01 ≠ 0.Corollary 10.31. ([Me2005a, Me2005b], [∗]) There exist two points[xa] 0 2µ ∈ Kµ(n+p) and [ax] 0 2µ ∈ (n+p) arbitrarily close to the origin with( ∗ Kµ∗π c (Γ 2µ ([xa] 0 2µ)) = 0 and π z Γ∗2µ ∗([ax] 0 2µ ∗)) = 0 such that the two maps{ (K µ(n+p) ∋ [xa] 2µ ↦−→ π c Γ2µ ([xa] 2µ ) ) ∈ K m+p and(10.32)K µ∗ (n+p) ∋ [ax] 2µ ∗ ↦−→ π z(Γ∗2µ ∗([ax] 2µ ∗) ) ∈ K n+mare of rank m + p at the point [xa] 0 2µ ∈ Kµ(n+p) and of rank n + m at thepoint [ax] 0 2µ ∗ ∈ Kµ∗ (n+p) .