Knots and Dynamics Étienne Ghys Unité de Mathématiques Pures et Appliquées CNRS - ENS Lyon

Knots and Dynamics

Étienne GhysUnité de Mathématiques Pures et Appliquées

CNRS - ENS Lyon

Lorenz equation (1963)

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dx

dt=10 (y − x)

dy

dt= 28x − y − xz

dz

dt= xy −

8

3z

Birman-Williams: Periodic orbits are knots.

Topological description of the Lorenz attractor

41 Figure eight

Birman-Williams: Lorenz knots and links are very peculiar• Lorenz knots are prime

• Lorenz links are fibered• non trivial Lorenz links have positive signature

819 =T(4,3)

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10124=T(5,3)

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51Cinquefoil31 trefoil

10132

71

91

01 Unknot







Schwarzman-Sullivan- Thurston etc. One should think of a measure preserving vector field as « an asymptotic cycle »€

k(T , x) = x φ orbit ⏐ → ⏐ ⏐ φT (x) segment ⏐ → ⏐ ⏐ ⏐ x{ }

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Flow ≈ " limT →∞"1

Tk(T , x) dμ (x)∫

Example : helicity as asymptotic linking number

+1

-1

Linking # = +1

Linking # = 0

Theorem (Arnold) Consider a flow in a bounded domain in R3 preserving an ergodic probability measure (not concentrated on a periodic orbit). Then, for -almost every pair of points x1,x2, the following limit exists and is independent of x1,x2

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Helicity = limT1 ,T2 →∞1

T1T2

Link(k(T1, x1),k(T2, x2))

Example : helicity as asymptotic linking number

Open question (Arnold): Is Helicity a topogical invariant ?

Let t and t be two smooth flows preserving and . Assume there is an orientation preserving homeomorphism

h such that h o t = t o h and h* .

Does it follow that

Helicity(t) = Helicity(t) ?

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http://www.math.rug.nl/~veldman/movies/dns-large.mpg

Suspension: an area preserving diffeomorphim of the disk defines a volume preserving vector field in a solid torus.

Invariants on Diff (D2,area)?

xf(x)

Theorem (Banyaga): The Kernel of Cal is a simple group.

Theorem (Gambaudo-G): Cal( f ) = Helicity(Suspension f )

Theorem (Gambaudo-G): Cal is a topological invariant.

Corollary : Helicity is a topological invariant for those flows which are suspensions.

Theorem (Calabi): There is a non trivial homomorphism

€

Cal : Diff∞ (D2 ,∂D2 ,area) → R

A definition of Calabi’s invariant (Fathi)

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f ∈ Diff∞ (D2 ,∂D2,area)

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f0 = Id

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f1 = f

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Cal( f ) = Vart= 0t=1∫∫ Arg f t (x) − f t (y)( ) dx dy

Open question (Mather):

Is the group Homeo(D2, ∂D2, area) a simple group?

• Can one extend Cal to homeomorphisms?

• Good candidate for a normal subgroup: the group of « hameomorphisms » (Oh). Is it non trivial?

Abelian groupsor

SL(n,Z) for n>2)

No non trivial (Trauber, Burger-Monod)

More invariants on the group Diff(D2, ∂D2, area)? No homomorphism besides Calabi’s…

Quasimorphisms

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χ :Γ → R

€

χ γ1γ 2( ) − χ γ1( ) − χ γ 2( ) ≤ Const

Homogeneous if

€

χ γn( ) = nχ γ( )

non abelian free group

Many non trivial homogeneous quasimorphisms (Gromov, …)

Theorem (Gambaudo-G) The vector space of homogeneous quasimorphisms on Diff(D2, ∂D2, area) is infinite dimensional.

One idea: use braids and quasimorphisms on braid groups.

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(x1, x2,..., xn ) a b(x1, x2 ,..., xn ) ∈ Bnχ ⏐ → ⏐ R

Average over n-tuples of points in the disk

Example: n=2. Calabi homomorphism.

More quasimorphims…Theorem (Entov-Polterovich) : There exists a « Calabi quasimorphism »such thatwhen the support of f is in a disc D with area < 1/2 area(S2).

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χ( f ) = Cal( f D )

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χ :Diff0 (S2 ,area) → R

Theorem (Py) : If is a compact orientable surface of genus g > 0,there is a « Calabi quasimorphism » such thatwhen the support of f is contained in some disc D .

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χ( f ) = Cal( f D )

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χ :Ham(Σ,area) → R

Questions:

• Can one define similar invariants for volume preservingflows in 3-space, in the spirit of « Cal( f ) = Helicity(suspension f ) »?

• Does that produce topological invariants for smooth volume preserving flows?

• Higher dimensional topological invariants in symplectic dynamics?

• etc.

Example : the modular flow on

SL(2,R)/SL(2,Z)

• Space of lattices in R2 of area 1

€

Λ ≈Z2 ⊂R2

€

area( R2 / Λ) = 1

Topology: SL(2,R)/SL(2,Z) is homeomorphic to the complement of the trefoil knot in the 3-sphere

€

Λ ⊂ R2 ≈ C a g2 (Λ),g3 (Λ)( ) ∈C2 − g23 = 27g3

2{ }

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g2 (Λ) = 60 ω−4

ω∈Λ− 0{ }∑ ∈ C

€

g3 (Λ) = 140 ω−6

ω∈Λ− 0{ }∑ ∈ C

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area( R2 / Λ) = 1

g = 0 g = 27 g323 23

S3

g = 02

Dynamics

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φt (Λ) =et 0

0 e−t

⎛

⎝ ⎜

⎞

⎠ ⎟(Λ)On lattices

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A ∈ SL(2,Z)

• Conjugacy classes of hyperbolic elements in PSL(2,Z)• Closed geodesics on the modular surface D/PSL(2,Z)• Ideal classes in quadratic fields• Indefinite integral quadratic forms in two variables.• Continuous fractions etc.

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A(Z2) = Z2

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Λ =P(Z2)

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PAP −1 = ±et 0

0 e−t

⎛

⎝ ⎜

⎞

⎠ ⎟

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et 0

0 e−t

⎛

⎝ ⎜

⎞

⎠ ⎟(Λ) = Λ

Periodic orbits

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Each hyperbolic matrix A in PSL(2,Z) defines a periodic orbit in SL(2,R)/SL(2,Z), hence a closed curve kA in the complement of the trefoil knot.

Questions :

1) What kind of knots are the « modular knots » kA ?

2) Compute the linking number between kA and the trefoil.

Theorem: The linking number between kA and the trefoil is equal to R(A) where R is the « Rademacher function ».

€

η aτ + b

cτ + d

⎛

⎝ ⎜

⎞

⎠ ⎟24

= η (τ )24 (cτ + d)12 ;a b

c d

⎛

⎝ ⎜

⎞

⎠ ⎟∈ SL(2,Z)

Dedekind eta-function :

€

η(τ ) = exp(iπτ /12) 1− exp(2iπnτ )( )n≥1

∏ ; ℑ (τ ) > 0

This is a quasimorphism.

€

R : SL(2,Z) → Z

€

24(logη)aτ + b

cτ + d

⎛

⎝ ⎜

⎞

⎠ ⎟= 24(logη)(τ ) + 6 log(−(cτ + d)2) + 2iπ R (A)

€

log(z) = log z + i Arg(z)

€

(g23 − 27g3

2)(Λ) ∈ R+is a Seifert surface

« Proof » that R(A) = Linking(kA, )

is the trefoil knot

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g23 − 27g3

2 = 0

Jacobi proved that

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(g23 − 27g3

2)(Z +τ Z) = (2π )12η (τ )24





Theorem:Modular knots (and links) are the same as Lorenz knots (and links)

Step 1: find some template inside SL(2,R)/SL(2,Z) which looks like the Lorenz template.

Make it thicker by pushing along the unstable direction.

Look at « regular hexagonal lattices »

and lattices with horizontal rhombuses as fundamental domain with angle between 60 and 120 degrees.

Step 2 : deform lattices to make them approach the Lorenz template.

•From modular dynamics to Lorenz dynamics and vice versa?

• For instance, « modular explanation » of the fact that all Lorenz links are fibered?

Further developments?

Many thanks to Jos Leys !

Mathematical Imagery : http://www.josleys.com/

« A mathematical theory is not to be considered complete until you made it so clear that you can explain it to the man you meet on the street »

« For what is clear and easily comprehended attracts and the complicated repels us »

D. Hilbert



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Knots and Dynamics Étienne Ghys Unité de Mathématiques Pures et Appliquées CNRS - ENS Lyon