The topology of the monodromy map of a second order ODE

Abstract

We consider the following question: given A ∈ SL(2, R), which potentials q for the second order SturmLiouville problem have A as its Floquet multiplier? More precisely, define the monodromy map μ taking a potential q ∈ L 2 ([0, 2π ]) to μ(q) = Φ(2π), the lift to the universal cover G = SL(2, R) of SL(2, R) of the fundamental matrix map Φ : [0, 2π ] → SL(2, R),Let H be the real infinite-dimensional separable Hilbert space: we present an explicit diffeomorphism Ψ : G 0 × H → H 0 ([0, 2π ]) such that the composition μ • Ψ is the projection on the first coordinate, where G 0 is an explicitly given open subset of G diffeomorphic to R 3 . The key ingredient is the correspondence between potentials q and the image in the plane of the first row of Φ, parametrized by polar coordinates, which we call the Kepler transform. As an application among others, let C 1 ⊂ L 2 ([0, 2π ]) be the set of potentials q for which the equation −u + qu = 0 admits a nonzero periodic solution: C 1 is diffeomorphic to the disjoint union of a hyperplane and Cartesian products of the usual cone in R 3 with H.

The homotopy type of spaces of locally convex curves in the sphere

“…We may keep θ 0 fixed and change θ 1 and the Bruhat cell will cycle. If φ 0 + φ 1 < 0 the cell will go through P (13);7 , P (123);5 , P (13);1 and P (123);3 (and back); notice that the cells Bru (13); 7 and Bru (13);1 are open and correspond to intervals while the cells Bru (123); 5 and Bru (123);3 have dimension 2 and correspond to transition points. If φ 0 + φ 1 > 0 the cell will go through P (13);7 , P (132);6 , P (13); 4 and P (132);5 .…”

“…For A 1 , each of the four 2-cells included is sandwiched between two of the included 3-cells: Bru (123);3 between Bru (13);1 and Bru (13);7 ; Bru (123);5 between Bru (13);1 and Bru (13);7 ; Bru (132);5 between Bru (13); 4 and Bru (13);7 ; Bru (132);6 between Bru (13); 4 and Bru (13);7 . For A 2 , since all four 3-dimensional cells are included we need only check that each of the two 1-dimensional cells included is surrounded by 2-and 3-dimensional cells which are also included: Bru (23);1 is surrounded by the four 3-dimensional cells plus Bru (132);0 , Bru (123);6 , Bru (123); 3 and Bru (132);6 ; Bru (12);4 is surrounded by the four 3-dimensional cells plus Bru (123);6 , Bru (132);0 , Bru (132); 5 and Bru (123);5 . This completes the proof of the claim By hypothesis, if t 0 (p) ≤ t a < t b ≤ t 1 (p) then F f (p) (t a ; t b ) ∈ A 2 .…”

The geometry of the critical set of nonlinear periodic Sturm–Liouville operators

“…For the linear case, the level sets of the monodromy map μ were explicitly parametrized in [8]. Indeed, for a smoothing Banach space X , let Lev X (g) ⊂ X be the level set μ −1 ({g}), where μ : X → G 0 ⊂ G is the monodromy map.…”

“…Now, for X = H 1 (S 1 ), given a loop γ : S k → Lev H 1 f (g), define γ Lin : S k → H 1 (S 1 ) by γ Lin (s) = f • γ (s) so that μ(γ Lin (s)) = g for all s ∈ S k . From [8], the loop γ Lin admits an extension Γ Lin : B k+1 → H 1 (S 1 ) with μ(Γ Lin (s)) = g for all s ∈ B k+1 . In order to obtain Γ : B k+1 → Lev H 1 f (g) such that μ f (Γ (s)) = g, we might want to define Γ Lin (s) = f • Γ (s): such a construction will not respect H 1 (S 1 ) if f is not invertible.…”

“…The present paper can also be considered a continuation of [8], where Theorem 1 is proved for the linear case f (x) = x 2 /2: the phrasing is justified by the fact that f (x) = x and therefore D F (u)v = −v + uv. In this case we trivially have C * = C. Notice that in the periodic case, unlike the Dirichlet case, the critical set C has singular points and is not a Hilbert manifold.…”