Germs of analytic families of diffeomorphisms unfolding a parabolic - - PowerPoint PPT Presentation

Germs of analytic families of diffeomorphisms unfolding a parabolic point (III)

Christiane Rousseau

Work done with C. Christopher, P. Mardeˇ si´ c, R. Roussarie and L. Teyssier

1 Minicourse 3, Toulouse, November 2010

Structure of the mini-course

◮ Statement of the problem (first lecture) ◮ The preparation of the family (first lecture) ◮ Construction of a modulus of analytic classification in

the codimension 1 case (second lecture)

◮ The realization problem in the codimension

1 case (third lecture)

2 Minicourse 3, Toulouse, November 2010

The classification theorem

• Theorem. [MRR] Two germs of generic families

unfolding a codimension 1 parabolic point are analytically conjugate if and only if they have same formal invariant a(ǫ) and same modulus

• Ψ0

^ ǫ,Ψ∞ ^ ǫ

• ^

ǫ∈Vδ

• / ∼

3 Minicourse 3, Toulouse, November 2010

The realization problem

Which a(ǫ) and modulus

• Ψ0

^ ǫ,Ψ∞ ^ ǫ

• ^

ǫ∈Vδ

• / ∼ are realizable?

4 Minicourse 3, Toulouse, November 2010

The strategy

• 1. Any a(ǫ) and
• Ψ0

^ ǫ,Ψ∞ ^ ǫ

• can be realized as the

modulus of a diffeomorphism f^

ǫ. This is the

local realization.

5 The strategy Minicourse 3, Toulouse, November 2010

The strategy

• 1. Any a(ǫ) and
• Ψ0

^ ǫ,Ψ∞ ^ ǫ

• can be realized as the

modulus of a diffeomorphism f^

ǫ. This is the

local realization.

• 2. If a(ǫ) is analytic and
• Ψ0

^ ǫ,Ψ∞ ^ ǫ

• depend

analytically on ^ ǫ, then the realization f^

ǫ can be

made depending analytically on ^ ǫ ∈ Vδ with uniform limit for ^ ǫ = 0.

6 The strategy Minicourse 3, Toulouse, November 2010

• 3. On the auto-intersection of Vδ we let
• ǫ = ^

ǫ ˜ ǫ = ^ ǫe2πi A necessary condition for the realization by a uniform family is that fǫ and f˜

ǫ be conjugate.

7 The strategy Minicourse 3, Toulouse, November 2010

• 3. On the auto-intersection of Vδ we let
• ǫ = ^

ǫ ˜ ǫ = ^ ǫe2πi A necessary condition for the realization by a uniform family is that fǫ and f˜

ǫ be conjugate.

• 4. This necessary condition, called the

compatibility condition, is also sufficient and allows to “correct” f^

ǫ to a uniform family. This

is the global realization.

8 The strategy Minicourse 3, Toulouse, November 2010

The local realization for a fixed ^ ǫ

The technique is standard: we realize on an abstract 1-dimensional complex manifold, which we recognize to be holomorphically equivalent to an open set of C.

9 The local realization Minicourse 3, Toulouse, November 2010

The local realization for a fixed ^ ǫ

The technique is standard: we realize on an abstract 1-dimensional complex manifold, which we recognize to be holomorphically equivalent to an open set of C. Indeed, we consider the two sectors U±

^ ǫ , each endowed

with the model diffeomorphism f ±

ǫ , i.e. the time-1 map of the

vector field vǫ = z2 −ǫ 1+a(ǫ)z ∂ ∂z

8 8

U+ U+ U+ U+ U− U− U− U−

8

10 The local realization Minicourse 3, Toulouse, November 2010

The gluing on U+

^ ǫ ∩U− ^ ǫ

This gluing must be compatible with f ±

ǫ on the three parts

• f the intersection, U0

^ ǫ, U∞ ^ ǫ and UC ^ ǫ.

Ξ Ξ Ξ Ξ

Ξ Ξ Ξ Ξ

− − − −

11 The local realization Minicourse 3, Toulouse, November 2010

The gluing on U+

^ ǫ ∩U− ^ ǫ

This gluing must be compatible with f ±

ǫ on the three parts

• f the intersection, U0

^ ǫ, U∞ ^ ǫ and UC ^ ǫ.

In the time coordinate W of vǫ this gluing is simply given by      Ψ0

^ ǫ

• n

U0

^ ǫ

Ψ∞

^ ǫ

• n

U∞

^ ǫ

T^

ǫ

• n

UC

^ ǫ

which commutes with T1. The map T^

ǫ is a translation:

it is the Lavaurs map.

8 Ξ

8

Ξ 0 8 Ξ

8 ~

Ξ

~ 0

Ξ0 Ξ

8

Ξ

8

Ξ U+ U+ U+ U+ U− U− U− U− 12 The local realization Minicourse 3, Toulouse, November 2010

The time W of vǫ

W = q−1

^ ǫ (z) =

• 1

2 √ ^ ǫ ln z− √ ^ ǫ z+ √ ^ ǫ + a(ǫ) 2 ln(z2 −ǫ),

^ ǫ = 0, −1

z +a(0)ln(z),

^ ǫ = 0.

Ω+ Ω− Ω0

+

Ω0

−

~ Ψ0 Ψ

8 8

~ Ψ Ψ0 Ψ

ε ^ ε ^ ε ^ ε ^ ε ^

Ψ0

ε ^

Ψ

8

Ψ0

ε ^ ε ^ ∼

α0 α0 Ω+

ε ^

~ Ω+

ε ^

_ Ω−

ε ^

~ Ω−

ε ^

_ 8 Ξ

8

Ξ 0 8 Ξ

8 ~

Ξ

~ 0

Ξ0 Ξ

8

Ξ

8

Ξ U+ U+ U+ U+ U− U− U− U− 13 The local realization Minicourse 3, Toulouse, November 2010

Why T^

ǫ is a translation?

In the time coordinate W, it is a diffeomorphism commuting with T1 on a strip of width larger then 1 going from ImW = −∞ to ImW = +∞.

Ω+ Ω− Ω0

+

Ω0

−

~ Ψ0 Ψ

8 8

~ Ψ Ψ0 Ψ

ε ^ ε ^ ε ^ ε ^ ε ^

Ψ0

ε ^

Ψ

8

Ψ0

ε ^ ε ^ ∼

α0 α0 Ω+

ε ^

~ Ω+

ε ^

_ Ω−

ε ^

~ Ω−

ε ^

_

14 The local realization Minicourse 3, Toulouse, November 2010

The gluing in z-coordinate

In the z-coordinate, the gluing is simply given by      Ξ0

^ ǫ = q^ ǫ ◦Ψ0 ^ ǫ ◦q−1 ^ ǫ

• n

U0

^ ǫ

Ξ∞

^ ǫ = q^ ǫ ◦Ψ∞ ^ ǫ ◦q−1 ^ ǫ

• n

U∞

^ ǫ

id

• n

UC

^ ǫ

8 Ξ

8

Ξ 0 8 Ξ

8 ~

Ξ

~ 0

Ξ0 Ξ

8

Ξ

8

Ξ U+ U+ U+ U+ U− U− U− U− 15 The local realization Minicourse 3, Toulouse, November 2010

Behavior of the gluing near the fixed points Ξ0,∞

^ ǫ

(z) = id+ξ0,∞

^ ǫ

(z) with     

• ξ0

^ ǫ(z)

• < C(^

ǫ)

• z+

√ ^ ǫ

• A

| √ ^ ǫ|

• ξ∞

^ ǫ (z)

• < C(^

ǫ)

• z−

√ ^ ǫ

• A

| √ ^ ǫ|

16 The local realization Minicourse 3, Toulouse, November 2010

The compatibility condition

Ω+ Ω− Ω0

+

Ω0

−

~ Ψ0 Ψ

8 8

~ Ψ Ψ0 Ψ

ε ^ ε ^ ε ^ ε ^ ε ^

Ψ0

ε ^

Ψ

8

Ψ0

ε ^ ε ^ ∼

α0 α0 Ω+

ε ^

~ Ω+

ε ^

_ Ω−

ε ^

~ Ω−

ε ^

_

For ^ ǫ in the auto- intersection of Vδ we have two descriptions

• f the modulus.

A necessary condition for realizability to a uniform family in ǫ is that they encode conjugate dynamics.

17 The compatibility condition Minicourse 3, Toulouse, November 2010

Parameter values in the auto-intersection

For these values, the fixed points are linearizable and there is an orbit from one point to the other.

18 The compatibility condition Minicourse 3, Toulouse, November 2010

Parameter values in the auto-intersection

For these values, the fixed points are linearizable and there is an orbit from one point to the other. This allows for a third description of the modulus studied by Glutsyuk.

19 The compatibility condition Minicourse 3, Toulouse, November 2010

Parameter values in the auto-intersection

For these values, the fixed points are linearizable and there is an orbit from one point to the other. This allows for a third description of the modulus studied by Glutsyuk. Indeed, we can bring the diffeomorphism to the model (normal form) in the neighborhood of each fixed point by means of maps ϕ±. The two normalization domains intersect.

20 The compatibility condition Minicourse 3, Toulouse, November 2010

Parameter values in the auto-intersection

For these values, the fixed points are linearizable and there is an orbit from one point to the other. This allows for a third description of the modulus studied by Glutsyuk. Indeed, we can bring the diffeomorphism to the model (normal form) in the neighborhood of each fixed point by means of maps ϕ±. The two normalization domains

• intersect. The Glutsyuk modulus is given by the

comparison of the two normalizations ϕ− ◦(ϕ+)−1

21 The compatibility condition Minicourse 3, Toulouse, November 2010

Parameter values in the auto-intersection

For these values, the fixed points are linearizable and there is an orbit from one point to the other. This allows for a third description of the modulus studied by Glutsyuk. Indeed, we can bring the diffeomorphism to the model (normal form) in the neighborhood of each fixed point by means of maps ϕ±. The two normalization domains

• intersect. The Glutsyuk modulus is given by the

comparison of the two normalizations ϕ− ◦(ϕ+)−1 The Glutsyuk modulus is unique up to composition on the left and on the right by maps of the form vt

ǫ.

22 The compatibility condition Minicourse 3, Toulouse, November 2010

Construction of the Fatou Glutsyuk coordinates

As before we construct Fatou Glutsyuk coordinates, Φl and Φr, but we use lines parallel to the line of holes

Ψε

^

Ψ0 Ψ

8

G

+

ε ^

−

ε ^

− + 23 The compatibility condition Minicourse 3, Toulouse, November 2010

Construction of the Fatou Glutsyuk coordinates

As before we construct Fatou Glutsyuk coordinates, Φl and Φr, but we use lines parallel to the line of holes

Ψε

^

Ψ0 Ψ

8

G

+

ε ^

−

ε ^

− +

The Glutsyuk modulus is ΨG = Φr ◦(Φl)−1 It is unique up to composition on the left and on the right with translations and satisfies Tαr ◦ΨG = ΨG ◦Tαl

24 The compatibility condition Minicourse 3, Toulouse, November 2010

How to recover the Fatou Glutsyuk coordinates? How to recover them from the modulus (^ ǫ,a(ǫ),Ψ0

^ ǫ,Ψ∞ ^ ǫ )?

We describe the orbit space of Fǫ with the help

• f ONE Fatou coordinate and a renormalized

return map.

25 The compatibility condition Minicourse 3, Toulouse, November 2010

The renormalized return maps

Lavaurs point of view They are given by

• T ˜

α0 ◦ ˜

Ψ0 T ˜

α0 ◦ ˜

Ψ∞

• r
• ˜

Ψ0 ◦T ˜

α0

˜ Ψ∞ ◦T ˜

α0

~ Ψ0 Ψ

8

~

ε ^ ε ^

∼

α0 S+

ε ^

~ S−

ε ^

~

26 The compatibility condition Minicourse 3, Toulouse, November 2010

The renormalized return maps

Glutsyuk point of view The Fatou Glutsyuk co-

• rdinates are the coordi-

nates in which the renor- malized return maps are given by

• T ˜

α0

T ˜

α∞

Ψε

^ G

27 The compatibility condition Minicourse 3, Toulouse, November 2010

The change of coordinates The changes from Fatou (Lavaurs) coordinates to Fatou Glutsyuk coordinates are the changes

• f coordinates transforming
• T ˜

α0 ◦ ˜

Ψ0 T ˜

α0 ◦ ˜

Ψ∞

• r

˜ Ψ0 ◦T ˜

α0

˜ Ψ∞ ◦T ˜

α0

to

• T ˜

α0

T ˜

α∞

28 The compatibility condition Minicourse 3, Toulouse, November 2010

Working in the upper region

There exists maps          ˜ H0 ◦T ˜

α0 ◦ ˜

Ψ0 = T ˜

α0 ◦ ˜

H0 ˜ H∞ ◦T ˜

α0 ◦ ˜

Ψ∞ = T ˜

α∞ ◦ ˜

H∞ H

0 ◦Ψ 0 ◦Tα0 = Tα0 ◦H

H

∞ ◦Ψ ∞ ◦Tα0 = Tα∞ ◦H ∞

The maps ˜ H0,∞ and H

0,∞ are

the changes of coordinates to Fatou Glustyuk coordinates.

Ω+ Ω− Ω0

+

Ω0

−

~ Ψ0 Ψ

8 8

~ Ψ Ψ0 Ψ

8 ε ^ ε ^ ε ^ ε ^ ε ^

Ψ0

ε ^

Ψ

8

Ψ0

ε ^ ε ^

∼

α0 α0 Ω+

ε ^

~ Ω+

ε ^

_ Ω−

ε ^

~ Ω−

ε ^

_ 29 The compatibility condition Minicourse 3, Toulouse, November 2010

The compatibility condition

It is given by:

˜ H∞ ◦( ˜ H0)−1 = TDǫ ◦H0 ◦(H∞)−1 ◦TD′

ǫ

It is possible to normalize the coordinates so that Dǫ ≡ −2πia.

Corollary: The functions Ψ0,∞

^ ǫ

are 1-summable in

√ ^ ǫ.

The directions of non-summability are the Glutsyuk directions (real multipliers). Theorem: The family

{(ψ0

^ ǫ,ψ∞ ^ ǫ )}^ ǫ∈V

is realizable if and only if the compatibility condition is satisfied.

30 The compatibility condition Minicourse 3, Toulouse, November 2010

Proof of the Corollary In upper region

         ˜ H0 ◦T ˜

α0 ◦ ˜

Ψ0 = T ˜

α0 ◦ ˜

H0 ˜ H∞ ◦T ˜

α0 ◦ ˜

Ψ∞ = T ˜

α∞ ◦ ˜

H∞ H

0 ◦Ψ 0 ◦Tα0 = Tα0 ◦H

H

∞ ◦Ψ ∞ ◦Tα0 = Tα∞ ◦H ∞

This implies            ˜ H0 = id+O(C

0)

˜ H∞ = T2πia ◦ ˜ Ψ∞ +O(C

0)

H

0 = id+O(C 0)

(H

∞)−1 = Ψ ∞ ◦T2πia +O(C 0)

Ω+ Ω− Ω0

+

Ω0

−

~ Ψ0 Ψ

8 8

~ Ψ Ψ0 Ψ

8 ε ^ ε ^ ε ^ ε ^ ε ^

Ψ0

ε ^

Ψ

8

Ψ0

ε ^ ε ^

∼

α0 α0 Ω+

ε ^

~ Ω+

ε ^

_ Ω−

ε ^

~ Ω−

ε ^

_ 31 The compatibility condition Minicourse 3, Toulouse, November 2010

In lower region

         ˜ K0 ◦ ˜ Ψ0 ◦T ˜

α0 = T ˜ α0 ◦ ˜

K0 ˜ K∞ ◦ ˜ Ψ∞ ◦T ˜

α0 = T ˜ α∞ ◦ ˜

K∞ K

0 ◦Tα0 ◦Ψ 0 = Tα0 ◦K

K

∞ ◦Tα0 ◦Ψ ∞ = Tα∞ ◦K ∞

The functions K are given by:          ˜ K0 = T− ˜

α0 ◦ ˜

H0 ◦T ˜

α0

˜ K∞ = T− ˜

α0 ◦ ˜

H∞ ◦T ˜

α0

K

0 = Tα0 ◦H 0 ◦T−α0

K

∞ = Tα0 ◦H ∞ ◦T−α0.

The compatibility condition becomes

˜ K∞◦( ˜ K0)−1 = K

0◦(K ∞)−1◦T2πia+D′

ǫ

Ω+ Ω− Ω0

+

Ω0

−

~ Ψ0 Ψ

8 8

~ Ψ Ψ0 Ψ

8 ε ^ ε ^ ε ^ ε ^ ε ^

Ψ0

ε ^

Ψ

8

Ψ0

ε ^ ε ^

∼

α0 α0 Ω+

ε ^

~ Ω+

ε ^

_ Ω−

ε ^

~ Ω−

ε ^

_ 32 The compatibility condition Minicourse 3, Toulouse, November 2010

The 1-summability follows

In upper region:            ˜ H0 = id+O(C

0)

˜ H∞ = T2πia ◦ ˜ Ψ∞ +O(C

0)

H

0 = id+O(C 0)

(H

∞)−1 = Ψ ∞ ◦T2πia +O(C 0)

In lower region:            ( ˜ K0)−1 = ˜ Ψ0 +O(C

0)

˜ K∞ = id+2πia+O(C

0)

K

0 = Ψ 0 +O(C 0)

(K

∞)−1 = id+2πia+O(C 0)

Substituting in the compatibility condition:

• ˜

H∞ ◦( ˜ H0)−1 = T2πia ◦H

0 ◦(H ∞)−1 ◦TD′

ǫ

˜ K∞ ◦( ˜ K0)−1 = K

0 ◦(K ∞)−1 ◦T2πia+D′

ǫ

yields the existence of a constant A such that:

| ˜ Ψ∞ −Ψ

∞| < AC

| ˜ Ψ0 −Ψ

0| < AC

The 1-summability in √ǫ follows from Ramis-Sibuya’s theorem since |C

0| ∼ exp

• − 2π

2| √ ǫ|

• 33

The compatibility condition Minicourse 3, Toulouse, November 2010

The global realization

How to correct? Newlander-Nirenberg’s theorem.

We construct a family over an abstract manifold by gluing

(˜ z, ˜ ǫ) =

• (gǫ(z),ǫ)
• n the right

(z,ǫ)

• n the left

where

gǫ ◦f ◦g−1

ǫ = ˜

f

Adding ǫ = 0 yields a C∞ manifold. Why?

◮ |f − ˜

f| = O(exp(−

A

√

|ǫ|))

◮ Hence gǫ = id+O(exp(−

A

√

|ǫ|))

34 The global realization Minicourse 3, Toulouse, November 2010

End of the proof The abstract manifold has an almost complex structure which is integrable and is a product. Hence it is a neighborhood of the origin in C2 with coordinates (Z,ǫ).

35 The global realization Minicourse 3, Toulouse, November 2010

The Riccati case

We rather consider

• ψ0

^ ǫ = E◦Ψ0 ^ ǫ ◦E−1

ψ∞

^ ǫ = E◦Ψ∞ ^ ǫ ◦E−1

where E(W) = exp(−2πiW)

36 The global realization Minicourse 3, Toulouse, November 2010

The Riccati case

We rather consider

• ψ0

^ ǫ = E◦Ψ0 ^ ǫ ◦E−1

ψ∞

^ ǫ = E◦Ψ∞ ^ ǫ ◦E−1

where E(W) = exp(−2πiW) The Riccati case corresponds to

• ψ0

^ ǫ(w) = w 1+A(^ ǫ)w

ψ∞

^ ǫ (w) = exp(−4π2a(ǫ))(w+B(^

ǫ))

37 The global realization Minicourse 3, Toulouse, November 2010

The Riccati case

We rather consider

• ψ0

^ ǫ = E◦Ψ0 ^ ǫ ◦E−1

ψ∞

^ ǫ = E◦Ψ∞ ^ ǫ ◦E−1

where E(W) = exp(−2πiW) The Riccati case corresponds to

• ψ0

^ ǫ(w) = w 1+A(^ ǫ)w

ψ∞

^ ǫ (w) = exp(−4π2a(ǫ))(w+B(^

ǫ)) Then the compatibility condition is equivalent to say that there exists a presentation of the modulus with A(ǫ) and B(ǫ) analytic in ǫ.

38 The global realization Minicourse 3, Toulouse, November 2010

Conjecture If ψ0

^ ǫ and ψ∞ ^ ǫ are both nonlinear, then the only

case where ψ0

^ ǫ and ψ∞ ^ ǫ can be taken depending

analytically in ǫ is the Riccati case.

39 The global realization Minicourse 3, Toulouse, November 2010

Conjecture If ψ0

^ ǫ and ψ∞ ^ ǫ are both nonlinear, then the only

case where ψ0

^ ǫ and ψ∞ ^ ǫ can be taken depending

analytically in ǫ is the Riccati case. Otherwise, the compatibility condition is so violent that it forces non analyticity.

40 The global realization Minicourse 3, Toulouse, November 2010

The end

41 The global realization Minicourse 3, Toulouse, November 2010