1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 1. PROBLEMS, BASIC CONCEPTS AND OVERVIEW 47

Given J and (x, T) this interval only depends on the homotopy class [u] of u with

boundary fixed. Hence we shall write I(x, T, [u], J). Sometimes it is independent

even of the choice of [u]. Namely in the following case. Define a group homomor-

phism 'PC. by

'PC. : 7r2 ( M) ---7 z: u ---7 ( u. C1 ( ~' J)) [ S^2 ].

If 'PC. = 0 then I(x, T, [u], J) = I(x, T, J).

Exercise 1.28. Show that 'PE. is a well-defined group homorphism.

The winding interval I(x, T, [u], J) in fact depends on J. But certain informa-

tion does not depend on J.

Some properties of the winding interval are easily established, see [41]. For

example its length is smaller than ~:

1

[I(x, T, [u], J)[ <

2

.

Exercise 1.29. Prove the bound on the length of the winding interval.

We define the Conley Zehnder index of the contractible periodic orbit (x, T)
with respect to [u] by

{

2k+ 1

μ( x, T, [u]) =

2

k

if I(x, T , [u], J) C (k, k + 1)

if k E J(x, T, [u], J)

Exercise 1.30. Show that the definition of μ is independent of the choice of an
admissable J.

Definition 1.31. A contact form ,\ on the closed three manifold M is called dy-
namically convex if

• 'PC.= 0


• For every contractible periodic orbit (x, T) of the associated Reeb vector
  field X we have μ(x, T) 2: 3.

The following exercise is not difficult.

Exercise 1.32. The standard contact form Ao on S^3 is dynamically convex.

The above exercise is only a very particular case of a more general fact.

Proposition 1.33. Let M be a closed hypersurface in JR^4 = C^2 enclosing a strictly

convex domain G containing 0. Let,\ be the restriction of the 1-form ~[q·dp-p·dq]

to M. Then ,\ is a dynamically convex contact farm.

The proof of this proposition is not trivial and can be found in [49]. Neverthe-
less the reader might enjoy to try to prove it!
For dynamically convex contact forms on S^3 much more can be said about the
dynamics; see [51, 48]:

Theorem 1.34. Let ,\ be a dynamically convex contact form on S^3. Then there

exists an embedded disk V c S^3 such that av is a periodic orbit for X and iJ is

transversal to X. Moreover, every orbit for X other than av hits iJ in forward and

backward time. Further there are precisely two periodic orbits or infinitely many.