1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 1. PROBLEMS, BASIC CONCEPTS AND OVERVIEW 53

By μ( z ), z E r , we denote, of course, the Conley-Zehnder index of the corre-

sponding asymptotic limit.

The main result concerning stable finite foliations of 83 is [56]:

Theorem 1.51. For every choice off E 31 there exists a Baire set of admissable

complex multiplications J admitting a stable finite energy foliation :F of (8^3 , f >.. 0 , J).

Given a stable finite energy energy foliation of 83 the projected surfaces es-

tablish a singular foliation of 83 , which gives a smooth foliation transversal to the

flow in the complement of a finite number of periodic orbits. Using this system of

surfaces one can prove [56]:

Theorem 1.52. Let f E 32. Then the Reeb flow on 83 associated to>..= f >..o has

the following properties.

• Either X>-. has precisely two geometrically distinct periodic orbits or infinitely
  many.


• If X does not admit a disk-like global surface of section there exists a hy-
  perbolic periodic orbit with orientable stable manifold and a homoclinic orbit
  converging in forward and backward time to the hyperbolic orbit. In partic-
  ulat there are infinitely many geometrically distinct periodic orbits and the
  topological entropy of the flow is positive.
  This gives the following corollary
  Corollary 1.53. Let f E 3 2. If the associated Reeb flow admits a periodic orbit

(x, T), with T the minimal period, so that xr : IR/(TZ) ~ 83 is knotted, then there

exist infinitely many geometrically distinct periodic orbits.
Exercise 1.54. Prove the previous corollary as consequence of theorem 1.52.
We shall not give a proof of the results concerning finite energy foliations in this
paper, but refer the reader to the forthcoming paper [56]. Figure 5 can be viewed
as depicting a stable finite energy foliation with two spanning orbits of index 3 and
one of index 2. Moreover the leaves are either finite energy planes or cylinders, i.e.
they have 1 and 2 punctures respectively.