On the number of periodic orbits in Hamiltonian
dynamics
Yoshihiro Sugimoto
Tokyo Metropolitan University

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

1 / 29

Table of contents
1

Contractible periodic orbits
Notations(C ∞ -generic) Conley conjecture
Main theorem 1
Sketch of the proof
Further directions2

Non-contactible periodic orbits
Hofer-Zehnder conjecture
Main theorem 2
Sketch of the proof
Further directions

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On the number of periodic orbits in Hamiltonian dynamics

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Notations
Let (M, ω) be a (closed) symplectic manifold and H : M → R be a
Hamiltonian function.

Hamiltonian vector field
The Hamiltonian vector field XH is defined by
ω(XH , ·) = −dH.
In the standard Eucledean space (R2n , ω0 ), XH is written as follows.
XH (x1 , · · · , xn , y1 , · · · , yn ) =

n n
X
i=1

−

∂H ∂
∂H ∂ o+∂yi ∂xi∂xi ∂yiThis is a local model of Hamiltonian vector fields (Darboux theorem).

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

3 / 29We use S 1 -dependent (1-periodic) Hamiltonian functions and Hamiltonian
vector fields. Let {ϕtH } be a flow of a periodic Hamiltonian vector field.
H : S1 × M → R
ϕ0H (x) = xd t
ϕ (x) = XHt (ϕtH (x))
dt H
The Hamiltonian diffeomorphism group is the set of time 1 flow of suchvector fields.

Hamiltonian diffeomorphim groupnHam(M, ω) = ϕ1H ∈ Symp0 (M, ω)

H : S1 × M → R

oSymp 0 (M, ω) is the identity component of the symplectic diffeomorphismgroup Symp(M, ω) = {ϕ ∈ Diff(M) | ϕ∗ ω = ω}.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

4 / 29Periodic orbits
A l-periodic orbit (l ∈ N) of a Hamiltonian flow {ϕtH } is a loop which
satisfies the following conditions.
x : R/l · Z −→ Mdx(t) = XH (x(t)).
dtNote that any iteration of a periodic orbit is a periodic orbit.
y : R/kl · Z −→ M
y (t) = x(t)is a kl-periodic orbit. We study the number of periodic orbits, so we haveto exclude iterated periodic orbits of lower periods.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

5 / 29(simple) periodic orbits

A l-periodic orbit x(t) is called ”not simple” if there is another l ′ -periodic
orbit y (t) such that l ′ is a divisor of l andx(t) = y (t)(∀t ∈ R)holds. In other words, x(t) is a iteration of y (t). Non-simple periodic
orbits are not essential.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

6 / 29homotopy class of periodic “points”
Let H and G be periodic Hamiltonian functions so that their time one
flows coincide (ϕ1H = ϕ1G ). There is one to one correspondence between
periodic orbits of {ϕtH } and {ϕtG }. Assume that x(t) and y (t) bel-periodic orbits of their flows such that x(0) = y (0) holds. Then a loop(x(t) 0 ≤ t ≤ lγ(t) =y (l − t) l ≤ t ≤ 2l
is contractible. So x(t) and y (t) are homotopic.This is a consequence of
the Arnold conjecture. If {ψ t }0≤t≤1 is a loop of Hamiltonian
diffeomorphisms, then every loop {ψ t (x)}0≤t≤1 (x ∈ M) is contactible.

RemarkA homotopy class of periodic points of a Hamiltonian diffeomorphim
ϕ ∈ Ham(M, ω) is well-defined. It does not depend on the choice of a
Hamiltonian function H and its flow such that ϕ = ϕ1H holds.
Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

7 / 29Conley conjecture
Hamiltonian diffeomorphisms tend to have infinitely many simple
contractible periodic orbits.

Conley conjecture (Conley 1984)
Any Hamiltonian diffeomorphism on the standard torus (T2n , ω0 ) has
infinitely many simple contractible periodic orbits.
This original conjecture was already proved and it is known that this isalso true for weakly exact symplectic manifolds (ω|π2 (M) = 0), symplectic
manifolds with vanishing c1 (c1 |π2 (M) = 0) and negatively monotone
symplectic manifolds. Today, it is believed that this theorem holds for”almost all” (not ”all”) symplectic manifolds. This is today’s Conley
conjecture.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

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Counterexample
An irrational rotation of the 2-sphere ϕθ only has 2 simple periodic orbits(the North pole and the South pole). So Conley conjecture is false on S 2 .
For (x, y , z) ∈ S 2 and θ ∈ R\Q, the irrational rotation is defined asfollows.
ϕtθ (x, y , z) = (cos

θt
θt
θt
θt
x + sin y , − sin x + cos y , z)
2π
2π
2π
2π

The periodic points of ϕθ are N = (0, 0, 1) and S = (0, 0, −1).

RemarkIn dimension two, the Conley conjecture holds for Σg ≥1 .

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

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Generic Conley conjecture

We also have another variant of the Conley conjecture. That is thefollowing generic Conley conjecture.

C ∞ -generic Conley conjecture
For any closed symplectic manifold (M, ω), C ∞ -generic Hamiltonian
diffeomorphisms have infinitely many simple contractible periodic orbits.
In dimension 2, the following strong results is known.

C ∞ -generic density theorem (Asaoka-Irie)
Periodic points are dense C ∞ -generically in dimension two.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

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Main theorem 1
We proved C ∞ -generic Conley conjecture for ”almost all” symplectic
manifolds.

Main theorem 1
Let (M 2n , ω) be a closed symplectic manifold which satisfies at least oneof the following three conditions.
1n is odd

2

Hodd (M : R) ̸= 0

3

N ̸= 1 (N: minimum Chern number)

Then, there exists a C ∞ -generic subset U ⊂ Ham(M, ω) such that any
element of U has infinitely many simple contractible periodic orbits.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

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The tangent bundle π : TM −→ M has (comtatible) complex structures.
Let c1 (TM) ∈ H 2 (M : Z) be the uniquely determined first Chern class.
The minimum Chern number N ∈ N ∪ {+∞} is the positive generator of
the image of the following map.
c1 : π2 (M) −→ Z[u] 7→ ⟨c1 (TM), [u]⟩If this map is trivial, N is +∞.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

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Sketch of the proof
We apply the following local theorem.

Theorem (Birkhoff, Moser)
Let ϕ be a symplectic map defined in an open neighborhood of the originin (R2n , ω0 ) and the origin is a fixed point. Let−1{λ1 , · · · , λm , λ−1
1 , · · · , λm } be the all engenvalues of the differential map
dϕ(0,··· ,0) : R2n −→ R2n
on the unit circle in C (|λi | = 1). Assume that ϕ satisfies the following
conditions.
1

m≥1

2

jk
Πm
k=1 λk ̸= 1 for 1 ≤3Pm

k=1 |jk | ≤ 4

The Taylor coefficient of ϕ up to order three satisfies a”non-degenerate” condition.

Then ϕ possesses infinitely many periodic orbits in any neighborhood of 0.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

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1

Let U ⊂ R2 and ϕ(x, y ) = (u, v ) be as follows.
u = x cos Φ − y cos Φ + f1
v = x sin Φ + y cos Φ + f2
Φ = α + β(x 2 + y 2 )The orders of the error terms fi are ≥ 4. ”non-degenerate” meansβ ̸= 0.

2

Let U ⊂ R2n and ϕ(x1 , , xn , y1 , · · · , yn ) = (u1 , · · · , un , v1 · · · , vn ) be
as follows.
uk = xk cos Φk − yk sin Φk + fk
vk = xk sin Φk − yk cos Φk + fn+kPΦk = αk + nl=1 βkl (xl2 + yl2 )”Non-degenerate” means the matrix {βkl } is non-singular.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

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Our proof is based on Birkhoff-Moser theorem and Hamiltonian Floer
homology. We divide Ham(M, ω) into three subsets.
)(
contractible periodic orbits are finite(1)
H = ϕ ∈ Ham(M, ω)
all periodic orbits are hyperbolic

(
H

H

(2)

(3)

)
contractible periodic orbits are finite
there is non-hyperbolic orbit=

ϕ ∈ Ham(M, ω)ϕ ∈ Ham(M, ω) contractible periodic orbits are infinite=

We first use Floer theory to prove that H(1) is empty under ourassumptions. Next we apply Birkhoff-Moser theorem to construct a
generic subset U ⊂ Ham(M, ω).
Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

15 / 29

Further directions
Question 1
Let ϕ be a Hamiltonian diffeomorphism with finite contractible periodic
orbits. Is there at least non-hyperbolic orbit? Are all periodic orbits
elliptic?
If this is correct, H(1) is empty and generic Conley conjectre holds for all
symplectic manifolds.

Question 2 (generic density theorem?)Is there C ∞ -generic subset U ⊂ Ham(M, ω) such that periodic points of∀ϕ ∈ U are dense in M?
This is a generalization of Asaoka-Irie’s theorem for higher dimensions.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

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Non-contractible periodic orbits
There is a big difference between contractible periodic orbits andnon-contractible periodic orbits.

RemarkThe set of contractible periodic orbits of a Hamiltonian diffeomorphism isnever empty. This is a consequence of the Arnold conjecture. However,the set of non-contractible periodic orbits may be empty.
For example, there is no non-contractible periodic orbits if M is simply
connected. If the Hamiltonian function H is C 2 -small, every one peroidic
orbits of ϕ1H are contractible. However, it seems that there are infinitely
many simple non-contractible periodic orbits if there is at least one
non-contractible periodic orbits.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

17 / 29

Hofer-Zehnder conjecture
Theorem (Frank)
Any area preserving homeomorphism on the 2-sphere S 2 with more than
two fixed points has infinitely many periodic points.
Hofer-Zehnder conjecture is a generalization of Frank’s theorem.

Hofer-Zehnder conjecture
Every Hamiltonian map on a compact symplectic manifold (M, ω)possessing more fixed points than necessarily required by the V. Arnold
conjecture possesses always infinitely many periodic orbits.
This conjecture was stated in their famous book ”Symplectic invariantsand Hamiltonian dynamics”.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

18 / 29

Hofer-Zehnder conjecture implies that a Hamiltonian diffeomorphism
possesses infinitely many contractible periodic orbits if it possesses morethan the sum of the Betti numbers. Non-contractible version of
Hofer-Zehnder conjecture can be written as follows.

Conjecture
Any Hamiltonian diffeomorphism possesses infinitely many non-contractible
periodic orbits if it possesses at least one non-contractible periodic orbit.
Note that the required number of non-contractible periodic orbits is zero
because the Floer homology of non-contractible orbits is trivial. So thesum of the Betti numbers is replaced by zero.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

19 / 29

Previous research

1

A symplectic manifold (M, ω) is called symplectically atoroidal ifZu ∗ ω = 0 ( ∀u : T2 → M )T2holds. Gurel proved the conjecture for atoroidal symplectic manifolds.
2

Orita proved the conjecture for the standard torus (T2n , ω0 ). Notethat (T2n , ω0 ) is not symplectically atoroidal. (T2n , ω0 ) is the simplest
example of non-atoroidal and non simply connected symplectic
manifold.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

20 / 29

Weakly monotone symplectic manifold

A symplectic manifold is called weakly monotone if it satisfies one of thefollowing conditions.
1(M, ω) is a monotone symplectic manifold. There is a constant λ ≥ 0such that
Z
Z
∗
u ω=λ
u ∗ c1 ( ∀u : S 2 → M )S2

S2

holds.
2c1 (A) = 0 holds for ∀A ∈ π2 (M)3N ≥n−2

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On the number of periodic orbits in Hamiltonian dynamics

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Every symplectic manifold (M 2n , ω) (2n ≤ 6) is a weakly monotone
symplectic manifold.

RemarkIf (M, ω) is weakly monotone, We can define the Floer homology without
using so-called virtual technique. We can avoid ”bubbling phenomena”. In
particular, Floer homology can be defined over Z-coefficient. The virtualtechnique works over Q-coefficient.

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On the number of periodic orbits in Hamiltonian dynamics

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Main theorem 2

Main theorem 2
Let (M, ω) be a closed weakly monotone symplectic manifold. Letϕ ∈ Ham(M, ω) be a Hamiltonian diffeomorphism such that 1-periodic
orbits in the class γ ̸= 0 ∈ H1 (M : Z)/Tor is finite, not empty and the
local Floer homology HF loc (ϕ, x) of at least one of them is not zero. Then,for sufficiently large prime p, ϕ possesses p-periodic or p ′ -periodic orbit in
the class p · γ. Here, p ′ is the first prime number greater than p. In
particular, there are infinitely many simple non-contractible periodic orbits.
The assumption ”weakly monotone” is a purely technical assumption. If
we can define Floer theory over Z-coefficient, we can remove thisassumption.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

23 / 29

We use the local Floer homology of periodic orbits to define ”reasonablehomological count of periodic orbits”. If a periodic orbit x is”non-degenerate” (generic case),
dimHF loc (ϕ, x) = 1
This suggests that the reasonable homological count of periodic orbit x is
dimHF loc (ϕ, x) (not 1 in general). So the condition ”the local Floer
homology HF loc (ϕ, x) of at least one of them is not zero” means that”there is at least one non-contractible periodic orbit”. If the local Floer
homology of all periodic orbits are trivial, they should be regarded as
empty.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

24 / 29

Sketch of the proof
We fix ϕ ∈ Ham(M, ω) and γ ̸= 0 ∈ H1 (M : Z)/Tor. Let P(ϕ, γ) be theset of 1-periodic orbits in γ.
P(ϕ, γ) = x ∈ P(ϕ) [x] = γ
We define the Floer homology for such periodic orbits. We use so-called
Novikov ring Λ and its subring Λ0 over the ground field Zp .
Λ=

l
nX

ai T λ ioai ∈ Zp , λi ∈ R, λi ↗ ∞ (if l = ∞)

i=1

Λ0 =

l
nX

ai T λi ∈ Λ

λi ≥ 0oi=1

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

25 / 29

The Floer chain complex is a module generated by periodic orbits P(ϕ, γ).
M
CF (ϕ, γ : Λ0 ) =Λ0 · xx∈P(ϕ,γ)There is a boundary operator dFloer and the Floer homology is thehomology of this chain complex.
HF (ϕ, γ : Λ0 ) = H(CF (ϕ, γ : Λ0 ), dFloer )RemarkWe use the Floer homology over Λ0 (not over Λ). HF (ϕ, γ : Λ) is always
trivial and not interesting. However, HF (ϕ, γ : Λ0 ) is not necessarily
trivial. In general, HF (ϕ, γ : Λ0 ) has a torsion.

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On the number of periodic orbits in Hamiltonian dynamics

26 / 29

There is a sequence 0 < β1 ≤ · · · ≤ βm so that
HF (ϕ, γ : Λ0 ) ∼=m
MΛ0 /T βi Λ0

i=1holds. The behavior of β1 under iterations is important for our purpose. β1stands for the ”minimum energy” of the Floer boundary operator dFloer .
We also fix a prime number p and a sequence 0 < δ1 ≤ · · · ≤ δm′ so that′

HF (ϕ , pγ : Λ0 ) ∼
=
p

m
M

Λ0 /T δi Λ0

i=1

holds. Our next purpose is to compare p · β1 and δ1 .

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

27 / 29Assume that there is no simple p-periodic orbits in pγ. This means every
element of P(ϕp , pγ) is an iteration of an element of P(ϕ, γ). Then wecan prove the following proposition.

Proposition
Under this assumption, p · β1 ≤ δ1 holds.
Combining this proposition with ”filtered” Floer homology, we can prove′′that P(ϕp , pγ) is not empty. Note that any element of P(ϕp , pγ) is
simple.

RemarkIn the proof of this proposition, we used the Zp -equivariant Floer
homology. Here we used the Floer theory over Zp -coefficient.

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

28 / 29

Furhter directions

We assumed that (M, ω) is a weakly monotone symplectic manifold. This
was a purely technical assumption. In general, Floer homology is definedover Q. If we can define the Floer thoery over Z for every closed
symplectic manifold, we can remove this assumption.

Question
Can we construct Z or Zp -coefficient Floer homology and Zp -equivariant
Floer homology for every closed symplectic manifold?
Constructions of such theories could be very important for other problems(e.g. Hofer-Zehnder conjecture).

Yoshihiro Sugimoto (Tokyo Metropolitan University)
On the number of periodic orbits in Hamiltonian dynamics

29 / 29