Converse Sturm-Hurwitz-Kellogg theorem and related results

The classical Sturm-Hurwitz-Kellogg theorem asserts that a function, orthogonal to an n-dimensional Chebyshev system on a circle, has at least n+1 sign changes. We prove the converse: given an n-dimensional Chebyshev system on a circle and a function with at least n+1 sign changes, there exists an orientation preserving diffeomorphism of the circle that takes this function to a function, orthogonal to the Chebyshev system. We also prove that if a function on the real projective line has at least four sign changes then there exists an orientation preserving diffeomorphism of the projective line that takes this function to the Schwarzian derivative of some function. These results extend the converse four vertex theorem of H. Gluck and B. Dahlberg: a function on a circle with at least two local maxima and two local minima is the curvature of a closed plane curve

On Lagrangian tangent sweeps and Lagrangian outer billiards

Given a Lagrangian submanifold in linear symplectic space, its tangent sweep is the union of its (affine) tangent spaces, and its tangent cluster is the result of parallel translating these spaces so that the foot point of each tangent space becomes the origin. This defines a multivalued map from the tangent sweep to the tangent cluster, and we show that this map is a local symplectomorphism (a well known fact, in dimension two). We define and study the outer billiard correspondence associated with a Lagrangian submanifold. Two points are in this correspondence if they belong to the same tangent space and are symmetric with respect to its foot pointe. We show that this outer billiard correspondence is symplectic and establish the existence of its periodic orbits. This generalizes the well studied outer billiard map in dimension two.Comment: revision as requested by the refere