Knots and Dynamics

Étienne GhysUnité de Mathématiques Pures et Appliquées

CNRS - ENS Lyon

Lorenz equation (1963)

!

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) 10124=T(5,3)

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" orbit

# $ # # "T (x)segment

# $ # # # x{ }

!

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 abounded domain in R3 preserving an ergodicprobability measure µ (not concentrated on aperiodic orbit). Then, for µ-almost every pairof points x1,x2, the following limit exists andis independent of x1,x2

!

Helicity = limT1 ,T2"#

1

T1T2Link(k(T1, x1),k(T2, x2))

Example : helicity as asymptotic linking number

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

Let φ1t and φ2

t be two smooth flows preserving µ1 and µ1. Assume there is an orientation preserving homeomorphism h such that h o φ1

t = φ2t o h and h* µ1= µ2.

Does it follow that

Helicity(φ1t) = Helicity(φ2

t) ?

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

Suspension: an area preserving diffeomorphimof the disk defines a volume preserving vectorfield in a solid torus.

Invariants on Diff (D2,area)?

x f(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 thoseflows which are suspensions.

Theorem (Calabi): There is a non trivial homomorphism

!

Cal : Diff"(D

2,#D

2,area)$ R

A definition of Calabi’s invariant (Fathi)

!

f " Diff#(D

2,$D

2,area)

!

f0

= Id

!

f1

= f

!

f t (t " [0,1])

!

Cal( f ) = Vart= 0t=1"" Arg ft (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

!

" :#$ R

!

" #1#2( ) $ " #

1( ) $ " #2( ) % Const

Homogeneous if

!

" # n( ) = n" #( )

Γ non abelian free group

Many non trivial homogeneousquasimorphisms (Gromov, …)

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

One idea: use braids and quasimorphisms on braid groups.

!

(x1, x2,..., xn ) a b(x1, x2,..., xn ) " Bn#

$ % $ 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).

!

"( f ) =Cal( fD)

!

" :Diff0 (S2,area)# R

Theorem (Py) :If Σ 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 ⊂ Σ.

!

"( f ) =Cal( fD)

!

" :Ham(#,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 symplecticdynamics?

• etc.

Example : the modular flow on SL(2,R)/SL(2,Z)

• Space of lattices in R2 of area 1

!

" # Z2$ R

2

!

area( R2/ ") = 1

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

!

" # R2$Ca g2 ("),g3 (")( ) % C2 & g2

3 = 27g32{ }

!

g2 (") = 60 #$4

#%"$ 0{ }& % C

!

g3 (") = 140 #$6

#%"$ 0{ }& % C

!

area( R2/ ") = 1

g = 0g = 27 g32

3 23

S3

g = 02

Dynamics

!

" t (#) =et

0

0 e$t

%

& '

(

) * (#)On lattices

!

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.

!

A(Z2) = Z

2

!

" = P(Z2)

!

PAP"1

= ±et

0

0 e"t

#

$ %

&

' (

!

et

0

0 e"t

#

$ %

&

' ( ()) = )

Periodic orbits

Each hyperbolic matrix A in PSL(2,Z) defines aperiodic orbit in SL(2,R)/SL(2,Z), hence a closed curvekA 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 andthe trefoil is equal to R(A) where R is the« Rademacher function ».

!

"a# + b

c# + d

$

% &

'

( )

24

= "(# )24 (c# + d)12

;a b

c d

$

% &

'

( ) * SL(2,Z)

Dedekind eta-function :

!

"(# ) = exp(i$# /12) 1% exp(2i$n# )( )n&1

' ; ((# ) > 0

This is a quasimorphism.

!

R : SL(2,Z)" Z

!

24(log")a# + b

c# + d

$

% &

'

( ) = 24(log")(# ) + 6 log(*(c# + d)2) + 2i+ R(A)

!

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

!

(g23" 27g3

2)(#) $ R

+ is a Seifert surface

« Proof » that R(A) = Linking(kA, )

is the trefoil knot

!

g2

3" 27g

3

2= 0

Jacobi proved that

!

(g23 " 27g3

2)(Z +#Z) = (2$ )12%(# )24

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

Step 1: find some template inside SL(2,R)/SL(2,Z) whichlooks 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 domainwith angle between 60 and 120 degrees.

Step 2 : deform lattices to make them approachthe 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 youmade it so clear that you can explain it to the man you meet on thestreet »

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

D. Hilbert