1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

38 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS

preserving the volume form:

LxD = 0.

Here L xD. denotes the Lie derivative of D. into the direction of X.

Conjecture 1.4. Given a smooth volume form D. on S^3 there exists a smooth, non

singular volume preserving vector field X having no periodic orbits.

Are there classes of vector fields for which one can establish the existence of

periodic orbits? We know for example since 1978 by results of P. Rabinowitz and

A. Weinstein that on a smooth compact regular energy surface for a Hamiltonian

system in IR^2 n , which bounds a star-shaped or convex domain, respectively, there

exists a periodic orbit, see [76, 77, 86]. Turther research then showed that in

Hamiltonian dynamics, the abundance of periodic orbits can be a real phenomenon.

However, also the opposite can be the case, see [37].

The abundance of periodic orbits hold for example for autonomous Hamiltonian

systems in IR^2 n. Indeed, let IR^2 n be equipped with the standard symplectic form w

defined by

n

w = ~ dqj /\dpj,

j=l

where (q1,P1, .. ., qn,Pn) are the coordinates on IR^2 n. Assume H : IR^2 n ---+ IR is a

smooth map, called the Hamiltonian. Define the associated Hamiltonian vector

field XH by

ixHw = -dH.

The associated Hamiltonian system is then

(1)

Recall that H is a constant of motion, i.e. for every solution x of (1) the map

t---+ H(x(t)) is constant.

A periodic solution for (1) is a solution x : IR ---+ IR^2 n of (1) such that x(O) = x(T)

for some T > 0.

Define L,H C IR to be the set consisting of all E E IR such that there exists

no periodic orbit (x , T) of x = XH(x) with H(x) = E. Surprisingly the following

result holds:

Theorem 1.5. (Hofer-Zehnder, Struwe) Assume that H : IR^2 n ---+ IR is a smooth

Hamiltonian satisfying

H(x)---+ oo for lxl---+ oo.

Then meas(L,H) = 0.

This result can be used to prove a result by C. Viterbo, which is one of the mile
stones in the field. It gives a partial answer to the Weinstein conjecture, formulated
by A. Weinstein in 1978 , see [87]. This conjecture was the result of Weinstein's
analysis of the proofs in [76, 77, 86].

Corollary 1.6. (Viterbo) Let M := H -^1 (E) be a compact smooth energy surface
for some regular value E E IR of the Hamiltonian H : IR^2 n ---+ IR. If there exists

a one-form A on M satisfying d.A = wlM and .A(XH(x)) =I- 0 for all x E M, then

there exists a periodic orbit on M.

Exercise 1. 7. Show the following: