Pietro Majer

A Morse complex for infinite dimensional manifolds, Part I

Abstract In this paper and in the forthcoming Part II, we introduce a Morse complex for a class of functions f defined on an infinite dimensional Hilbert manifold M , possibly having critical points of infinite Morse index and co-index. The idea is to consider an infinite dimensional subbundle—or more generally an essential subbundle—of the tangent bundle of M , suitably related with the gradient flow of f . This Part I deals with the following questions about the intersection W of the unstable manifold of a critical point x and the stable manifold of another critical point y : finite dimensionality of W , possibility that different components of W have different dimension, orientability of W and coherence in the choice of an orientation, compactness of the closure of W , classification, up to topological conjugacy, of the gradient flow on the closure of W , in the case dim W = 2 .

LECTURES ON THE MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

After reviewing some classical results about hyperbolic dynamics in a Banach setting, we describe the Morse complex for gradient-like flows on an infinite-dimensional Banach manifold M, under the assumption that rest points have finite Morse index. Then we extend these ideas to rest points with infinite Morse index and co-index, by using a suitable subbundle of the tangent bundle of M as a comparison object.

A non-squeezing theorem for convex symplectic images of the Hilbert ball

We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by duality methods of a symplectic capacity for bounded convex neighbourhoods of the origin. We also discuss the role of infinite-dimensional non-squeezing results in the study of Hamiltonian PDEs and show some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions.

When the Morse index is infinite

Let f be a smooth Morse function on an infinite-dimensional separable Hilbert manifold, all of whose critical points have infinite Morse index and coindex. For any critical point x, choose an integer a(x arbitrarily. Then there exists a Riemannian structure on M such that the corresponding gradient flow of f has the following property: for any pair of critical points x,y, the unstable manifold of x and the stable manifold of y have a transverse intersection of dimension a(x)−a(y.