1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 1. PROBLEMS, BASIC CONCEPTS AND OVERVIEW 49

Theorem 1.37. An area preserving self map of the open annulus having one peri-

odic point has infinitely many periodic points.

Applying this theorem of course implies the desired result. See figure 4 for

the open book decomposition associated to >. 0. Being dynamically convex even has

topological implications.

Theorem 1.38. A dynamically convex contact form on a closed three manifold

M is tight. Moreover, if M admits a dynamically convex contact form, we have

7r2(M) = 0.

Let us introduce for a contractible periodic orbit (x, T) another interesting

number. Let u: D -t M be as described before and let Z be a nowhere vanishing

section of u*~ -t M. Push xr into the direction of Z to obtain a loop x'r such

that x'r(ffi./TZ) n xr(ffi./TZ) = 0. Give D c C the induced orientation and define
the self linking number of (x, T) with respect to [u] by

sl(x,T,[u]) =int(u,x'r).

Here int denotes the (oriented) intersection number.
We have the following, see [48]:

Theorem 1.39. Let M be a closed three-manifold equipped with a dynamically
convex contact form>.. Assume the Reeb vector field admits a contractible periodic
orbit (x, T) such that

• T is the minimal period of x

• xr is unknotted and sl ( xr) = -1

Then M is diff eomorphic to S^3.
The hypotheses allow to construct an embedded finite energy plane asymptotic
to xr. Then making use of the dynamical convexity one can construct a finite
energy foliation of JR. x M by planes asymptotic to xr, see definition 1.47. On the
other hand, see [55].

Theorem 1.40. Every Reeb vector field on S^3 admits a periodic orbit (x, T) with

T the minimal period such that xr is unknotted and sl ( xr) = -1.

Figure 4. The figure shows the trace of the open book decomposition on a

two-dimensional plane. The dotted lines are the Hopf circles. The three-sphere

is viewed as ~^3 U {oo}.