Fredholm pseudo-gradients for the action functional on a sub-manifold of dual Legendrian curves of a three dimensional contact manifold (M3, α)

A Bahri

doi:10.1007/s40065-013-0089-7

Abstract

We prove in this paper that the intersection numbers between periodic orbits have an intrinsic meaning for the variational problem (J,C β ) {Bahri (Pseudo-Orbits of Contact Forms Pitman Research Notes in Mathematics Series No. 173, 1984), Bahri (C R Acad Sci Paris 299, Serie I 15:757–760, 1984), Bahri (Classical and Quantic periodic motions of multiply polarized spin-manifolds. Pitman Research Notes in Mathematics Series No. 378, 1998)}, corresponding to the periodic orbit problem on a sub-manifold of the loop space of a three dimensional compact contact manifold (M, α).

Highlights

1 Introduction Given a compact finite dimensional manifold without boundary N n and a C2 function f : N n −→ R, with non-degenerate critical points, the intersection number i(xm, xm−1) of a critical point xm of index m with a critical point xm−1 of index (m − 1) is defined, for a Morse–Smale [29,30] pseudo-gradient Z for f, to be the algebraic number of flow-lines in the intersection Wu(xm) ∩ Ws(xm−1) of the unstable manifold of xm with the stable manifold of xm−1
If there are in f −1([ f, f]) critical points of index m ym such that i is non zero or if there are, in the same set, critical points ym−1 such that i(xm, ym−1) is non zero, this intersection number is not intrinsic; it depends on the choice of Z
We prove in the present paper that there is a pseudo-gradient flow for (J, Cβ ), that can be continuously tracked along deformations of contact forms, for which the Fredholm assumption at the periodic orbits is verified and that, for this pseudo-gradient, the intersection number between two periodic orbits of consecutive indexes is defined intrinsically

Summary

Introduction

Given a compact finite dimensional manifold without boundary N n and a C2 function f : N n −→ R, with non-degenerate critical points, the intersection number i(xm, xm−1) of a critical point xm of index m with a critical point xm−1 of index (m − 1) is defined, for a Morse–Smale [29,30] pseudo-gradient Z for f , to be the algebraic number of flow-lines in the intersection Wu(xm) ∩ Ws(xm−1) of the unstable manifold of xm with the stable manifold of xm−1.If there are in f −1([ f (xm−1), f (xm)]) critical points of index m ym such that i (ym, xm−1) is non zero or if there are, in the same set, critical points ym−1 such that i(xm, ym−1) is non zero, this intersection number is not intrinsic; it depends on the choice of Z.

Results

Conclusion