1Symplectic manifolds
III Symplectic Geometry
1.4 Periodic points of symplectomorphisms
Symplectic geometers have an unreasonable obsession with periodic point prob-
lems.
Definition
(Periodic point)
.
Let (
M, ω
) be a symplectic manifold, and
ϕ
:
M → M
a symplectomorphism. An
n
-periodic point of
ϕ
is an
x ∈ M
such that
ϕ
n
(x) = x. A periodic point is an n-periodic point for some n.
We are particularly interested in the case where
M
=
T
∗
X
with the canonical
symplectic form on
M
, and
ϕ
is generated by a function
f
:
X → R
. We begin
with 1-periodic points, namely fixed points of f.
If ϕ(x
0
, ξ
0
) = (x
0
, ξ
0
), this means we have
ξ
0
= ∂
x
f
(x
0
,x
0
)
ξ
0
= −∂
y
f
(x
0
,x
0
)
In other words, we need
(∂
x
+ ∂
y
)f
(x
0
,x
0
)
= 0.
Thus, if we define
ψ : X → R
x 7→ f(x, x),
then we see that
Proposition.
The fixed point of
ϕ
are in one-to-one correspondence with the
critical points of ψ.
It is then natural to consider the same question for
ϕ
n
. One way to think
about this is to consider the symplectomorphism ˜ϕ : M
n
→ M
n
given by
˜ϕ(m
1
, m
2
, . . . , m
n
) = (ϕ(m
n
), ϕ(m
1
), . . . , ϕ(m
n−1
)).
Then fixed points of ˜ϕ correspond exactly to the n-periodic points of ϕ.
We now have to find a generating function for this
˜ϕ
. By inspection, we see
that this works:
˜
f((x
1
, . . . , x
n
), (y
1
, . . . , y
n
)) = f(x
1
, y
2
) + f(x
2
, y
3
) + ··· + f(x
n
, y
1
).
Thus, we deduce that
Proposition.
The
n
-periodic points of
ϕ
are in one-to-one correspondence with
the critical points of
ψ
n
(x
1
, . . . , x
n
) = f(x
1
, x
2
) + f(x
2
, x
3
) + ··· + f(x
n
, x
1
).
Example.
We consider the problem of periodic billiard balls. Suppose we have
a bounded, convex region Y ⊆ R
2
.
Y
We put a billiard ball in
Y
and set it in motion. We want to see if it has periodic
orbits.
Two subsequent collisions tend to look like this:
x
θ
We can parametrize this collision as (
x, v
=
cos θ
)
∈ ∂Y ×
(
−
1
,
1). We can think of
this as the unit disk bundle of cotangent bundle on
X
=
∂Y
∼
=
S
1
with canonical
symplectic form d
x ∧
d
v
, using a fixed parametrization
χ
:
S
1
∼
=
R/Z → X
. If
ϕ
(
x, v
) = (
y, w
), then
v
is the projection of the unit vector pointing from
x
to
y
onto the tangent line at x. Thus,
v =
χ(y) − χ(x)
kχ(y) − χ(x)k
·
dχ
ds
s=x
=
∂
∂x
(−kχ(x) − χ(y)k),
and a similar formula holds for
w
, using that the angle of incidence is equal to
the angle of reflection. Thus, we know that
f(x) = −kχ(x) −χ(y)k
is a generating function for ϕ.
The conclusion is that the
N
-periodic points are given by the critical points
of
(x
1
, . . . , x
N
) 7→ −|χ(x
1
) − χ(x
2
)| − |χ(x
2
) − χ(x
3
)| − ··· − |χ(x
N
) − χ(x
1
)|.
Up to a sign, this is the length of the “generalized polygon” with vertices
(x
1
, . . . , x
N
).
In general, if
X
is a compact manifold, and
ϕ
:
T
∗
X → T
∗
X
is a symplecto-
morphism generated by a function
f
, then the number of fixed points of
ϕ
are
the number of fixed points of
ψ
(
x
) =
f
(
x, x
). By compactness, there is at least
a minimum and a maximum, so ϕ has at least two fixed points.
In fact,
Theorem
(Poincar´e’s last geometric theorem (Birkhoff, 1925))
.
Let
ϕ
:
A →
A
be an area-preserving diffeomorphism such that
ϕ
preserves the boundary
components, and twists them in opposite directions. Then
ϕ
has at least two
fixed points.