Lagrangian Curve Flows on Symplectic Spaces

A smooth map γ in the symplectic space R2n is Lagrangian if γ,γx,…, γx(2n−1) are linearly independent and the span of γ,γx,…,γx(n−1) is a Lagrangian subspace of R2n. In this paper, we (i) construct a complete set of differential invariants for Lagrangian curves in R2n with respect to the symplectic group Sp(2n), (ii) construct two hierarchies of commuting Hamiltonian Lagrangian curve flows of C-type and A-type, (iii) show that the differential invariants of solutions of Lagrangian curve flows of C-type and A-type are solutions of the Drinfeld-Sokolov’s C^n(1)-KdV flows and A^2n−1(2)-KdV flows respectively, (iv) construct Darboux transforms, Permutability formulas, and scaling transforms, and give an algorithm to construct explicit soliton solutions, (v) give bi-Hamiltonian structures and commuting conservation laws for these curve flows.

Sturm theory with applications in geometry and classical mechanics

Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase plane of the equation. In the higher dimensional symplectic version of these theorems, lines are replaced by Lagrangian subspaces and intersections with a given line are replaced by non-transversality instants with a distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe some properties of the Maslov index. Starting from the celebrated paper of Arnol’d on symplectic Sturm theory for optical Hamiltonians, we provide a generalization of his results to general Hamiltonians. We finally apply these results for detecting some geometrical information about the distribution of conjugate and focal points on semi-Riemannian manifolds and for studying the geometrical properties of the solutions space of singular Lagrangian systems arising in Celestial Mechanics.

An elementary proof of the symplectic spectral theorem

The classical spectral theorem completely describes self-adjoint operators on finite-dimensional inner product vector spaces as linear combinations of orthogonal projections onto pairwise orthogonal subspaces. We prove a similar theorem for self-adjoint operators on finite-dimensional symplectic vector spaces over perfect fields. We show that these operators decompose according to a polarization, i.e. as the product of an operator on a Lagrangian subspace and its dual on a complementary Lagrangian. Moreover, if all the eigenvalues of the operator are in the base field, then there exists a Darboux basis such that the matrix representation of the operator is [Formula: see text] blocks block-diagonal, where the first block is in Jordan normal form and the second block is the transpose of the first one.

Ribbon graphs and bialgebra of Lagrangian subspaces

To each ribbon graph we assign a so-called [Formula: see text]-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of [Formula: see text]-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of [Formula: see text]-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

Principal Pivot Transforms of Quasidefinite Matrices and Semidefinite Lagrangian Subspaces

Lagrangian subspaces are linear subspaces that appear naturally in control theory applications, and especially in the context of algebraic Riccati equations. We introduce a class of semidefinite Lagrangian subspaces and show that these subspaces can be represented by a subset I ⊆ {1, 2, . . . , n} and a Hermitian matrix X ∈ C n×n with the property that the submatrix X II is negative semidefinite and the submatrix X I c I c is positive semidefinite. A matrix X with these definiteness properties is called I-semidefinite and it is a generalization of a quasidefinite matrix. Under mild hypotheses which hold true in most applications, the Lagrangian subspace associated to the stabilizing solution of an algebraic Riccati equation is semidefinite, and in addition we show that there is a bijection between Hamiltonian and symplectic pencils and semidefinite Lagrangian subspaces; hence this structure is ubiquitous in control theory. The (symmetric) principal pivot transform (PPT) is a map used by Mehrmann and Poloni [SIAM J. Matrix Anal. Appl., 33(2012), pp. 780–805] to convert between two different pairs (I, X) and (J , X 0 ) representing the same Lagrangian subspace. For a semidefinite Lagrangian subspace, we prove that the symmetric PPT of an I-semidefinite matrix X is a J -semidefinite matrix X 0 , and we derive an implementation of the transformation X 7→ X 0 that both makes use of the definiteness properties of X and guarantees the definiteness of the submatrices of X 0 in finite arithmetic. We use the resulting formulas to obtain a semidefiniteness-preserving version of an optimization algorithm introduced by Mehrmann and Poloni to compute a pair (I opt , X opt ) with M = max i,j |(X opt ) ij | as small as possible. Using semidefiniteness allows one to obtain a stronger inequality on M with respect to the general case.

SELF-INVERSES, LAGRANGIAN PERMUTATIONS AND MINIMAL INTERVAL EXCHANGE TRANSFORMATIONS WITH MANY ERGODIC MEASURES

Thanks to works by Kontsevich and Zorich followed by Boissy, we have a classification of all Rauzy Classes of any given genus. It follows from these works that Rauzy Classes are closed under the operation of inverting the permutation. In this paper, we shall prove the existence of self-inverse permutations in every Rauzy Class by giving an explicit construction of such an element satisfying the sufficient conditions. We will also show that self-inverse permutations are Lagrangian, meaning any suspension has its vertical cycles span a Lagrangian subspace in homology. This will simplify the proof of a lemma in a work by Forni. Veech proved a bound on the number of distinct ergodic probability measures for a given minimal interval exchange transformation. We verify that this bound is sharp by constructing examples in each Rauzy Class.