Birkho sums of i.e.t.’s: KZ cocycle (9th lecture)indico.ictp.it/event/a11165/session/115/... ·...

Best moments so far...Avila-Viana simplicity criterium

Sketch of proof of Avila-Viana simplicity criterion

Birkhoff sums of i.e.t.’s: KZ cocycle (9th lecture)

Carlos Matheus and Jean-Christophe Yoccoz

CNRS (Paris 13) and College de France

May 31, 2012

C. Matheus and J.-C. Yoccoz Birkhoff sums of i.e.t.’s: KZ cocycle (9th lecture)

Table of contents

1 Best moments so far...I.e.t.’sSuspensions of i.e.t.’s and translation surfacesRauzy-Veech algorithmMatrix Bγ and unique ergodicity of i.e.t.’sTeich. and moduli spaces of transl. surf.

2 Avila-Viana simplicity criterium

3 Sketch of proof of Avila-Viana simplicity criterionPreliminary reductionProof of Theorem 1

I.e.t.’sSuspensions of i.e.t.’s and translation surfacesRauzy-Veech algorithmMatrix Bγ and unique ergodicity of i.e.t.’sTeich. and moduli spaces of transl. surf.

I.e.t.’s as generalization of circle rotations

I.e.t.’s and their combinatorial and length data

I.e.t.’s T are determined by

a comb. data (A, πt , πb) and

a length data λα, α ∈ A.

Main goal of the course

The idea was to study Birkhoff sums of i.e.t.’s via adequaterenorm. dyn. for them: this technique is well-known in Dyn. and itis best represented by Adrien Douady’s phrase

Figure: “to plough in parameter space and harvest in phase space”C. Matheus and J.-C. Yoccoz Birkhoff sums of i.e.t.’s: KZ cocycle (9th lecture)

Interlude: Suspensions of i.e.t.’s and translation surfaces

Using either Masur’s suspension construction or Veech’s “zipperedrectangles” construction, we saw that i.e.t.’s are first return mapsof translation flows on translation surfaces:

ζB ζC

ζBζC

In particular, it is natural to study the dynamics of i.e.t.’s andtranslation flows together!

Interlude: Suspensions of i.e.t.’s and translation surfaces

Using either Masur’s suspension construction or Veech’s “zipperedrectangles” construction, we saw that i.e.t.’s are first return mapsof translation flows on translation surfaces:

ζB ζC

ζBζC

In particular, it is natural to study the dynamics of i.e.t.’s andtranslation flows together!

Rauzy-Veech algorithm for i.e.t.’s

The basic step (of top type) of the renorm. process for i.e.t.’s is:

A B C D

Rauzy-Veech algorithm for suspensions

Its counterpart for suspensions of i.e.t.’s is:

M(π, λ, τ)

Matrix Bγ I

For a basic step γ of RV algorithm for i.e.t.’s, the matrix Bγ isId + Eβα where α is the winner and β is the loser.

In general, for a path γ = γ1 . . . γn obtained from several steps γi

of RV algorithm, we define Bγ = Bγn . . .Bγ1 .

The matrices Bγ are the (discrete time version) of the (extended)Kontsevich-Zorich cocycle.

The Bγ ’s are important because its entries are measuring “howmuch time x ∈ I ′tα spend in I t

β before returning to I ′” for i.e.t.’sand they connect to special Birkhoff sums.

Matrix Bγ I

Matrix Bγ II

For instance, in the case of an i.e.t. with combinatorial data(A BB A

(A B C DD C B A

), the matrices Bγ are

computed with the aid of the Rauzy diagrams below:

Matrix Bγ and Keane’s conjecture

Using:

a spectral gap property for Bγ (i.e., its entries are positive assoon as all letters of A win 2d − 3 times), and

the fact that the proj. action of the matrices Bγ describedconjugacy classes of i.e.t.’s and the cone of inv. meas.,

we saw that i.e.t.’s T with recurrent trajectory under RV algorithmare uniquely ergodic.

In particular, Keane’s conjecture on the unique erg. of almostevery i.e.t. would follow once we can justify the recurrence underRV alg. of (a.e.) i.e.t.’s. Keeping this in mind, we started thediscussion of Teich. and moduli spaces (as natural spaces for therecurrence to take place).

Matrix Bγ and Keane’s conjecture

Using:

a spectral gap property for Bγ (i.e., its entries are positive assoon as all letters of A win 2d − 3 times), and

the fact that the proj. action of the matrices Bγ describedconjugacy classes of i.e.t.’s and the cone of inv. meas.,

we saw that i.e.t.’s T with recurrent trajectory under RV algorithmare uniquely ergodic.

In particular, Keane’s conjecture on the unique erg. of almostevery i.e.t. would follow once we can justify the recurrence underRV alg. of (a.e.) i.e.t.’s. Keeping this in mind, we started thediscussion of Teich. and moduli spaces (as natural spaces for therecurrence to take place).

Teich. and moduli spaces

We considered Q(M,Σ, κ) and M(M,Σ, κ) the Teich. and modulispaces of transl. surf.: in the first case we identify transl. struct.isomorphic under homeos isotopic to id. while in the second casethere is no restriction to the isotopy class of the homeos.

Also, we have Q(1)(M,Σ, κ) and M(1)(M,Σ, κ) the Teich. andmoduli spaces of unit area of transl. surf.

For example, below we illustrate two unit area torii representingthe same point in moduli space but not in Teich. space:

“Less concrete but more conceptual”

Paraphrasing J.-C. Yoccoz’s lecture, a big advantage of Teich. andmoduli spaces is the fact that they allow us to be “less concretebut more conceptual” (an important philosophical step inDynamics since the works of H. Poincare).

More precisely, instead of sticking to the (concrete) RV algorithmand KZ cocycle Bγ , we can think (more conceptually) about theTeichmuller flow and the (continuous time version of) KZ cocycleas follows.

Nice structures on Teich. and moduli spaces

Teich. and moduli spaces (of unit area transl. surf.) comeequipped with:

a SL(2,R)-action and hence a Teich. flow gt = diag(et , e−t);

a complex manifold/orbifold structure;

a natural “Lebesgue” (Masur-Veech) meas. µMV ;

Remark

Masur-Veech measure is finite on M(1)(M,Σ, κ), so that modulispaces are the nice places to get recurrence. However, we considertogether Teich. and moduli spaces because Teich. spaces work asuniversal covers of moduli spaces...

Teich. and moduli spaces, and Teich. flow in genus 1

A nice way to appreciate the difference between Teich. and modulispace is to look at the pictures below illustrating the genus 1 case:

−1/2 1/2

KZ cocycle

In this language, the cont. time version of (restricted) KZ cocycleover Teich. flow gt was the quotient of the trivial cocycle onQ(1)(M,Σ, κ)× H1(M,R),

(p, v) 7→ (gt(p), v)),

by the mapping class group Diff+(M,Σ)/Diff+0 (M,Σ).

In genus 1, a schematic representation of KZ cocycle is:

In general, KZ cocycle “is”:

In genus 1, a schematic representation of KZ cocycle is:

In general, KZ cocycle “is”:

Deviations of ergodic averages for i.e.t.’s I

Before yesterday, J.-C. Yoccoz showed that Veech boxes in modulispaces allow to connect the RV algorithm and the Teich. flow, andthe matrices Bγ and KZ cocycle.

Then, yesterday, using this connection, he showed that Zorichphenomenon for deviations of Birkhoff sums SNT (x) of a.e. i.e.t.T

lim supN→∞

log dist(SNT (x),Dj(T ))

log N= λj

is naturally explained by the (non-negative) Lyapunov exponents1 = λ1 > λ2 ≥ · · · ≥ λg (≥ 0) of KZ cocycle w.r.t. Masur-Veechmeasures.

Deviations of ergodic averages for i.e.t.’s I

Before yesterday, J.-C. Yoccoz showed that Veech boxes in modulispaces allow to connect the RV algorithm and the Teich. flow, andthe matrices Bγ and KZ cocycle.

Then, yesterday, using this connection, he showed that Zorichphenomenon for deviations of Birkhoff sums SNT (x) of a.e. i.e.t.T

lim supN→∞

log dist(SNT (x),Dj(T ))

log N= λj

is naturally explained by the (non-negative) Lyapunov exponents1 = λ1 > λ2 ≥ · · · ≥ λg (≥ 0) of KZ cocycle w.r.t. Masur-Veechmeasures.

Deviations of ergodic averages for i.e.t.’s II

Actually, the picture for Zorich phenomenon was only completeunder the so-called Kontsevich-Zorich conjecture that theLyapunov exponents λi of Masur-Veech measures are simple (i.e.,they have multiplicity 1).

After an important partial result of G. Forni, we know that thisconjecture is true due the works of A. Avila and M. Viana.

Deviations of ergodic averages for i.e.t.’s II

Actually, the picture for Zorich phenomenon was only completeunder the so-called Kontsevich-Zorich conjecture that theLyapunov exponents λi of Masur-Veech measures are simple (i.e.,they have multiplicity 1).

After an important partial result of G. Forni, we know that thisconjecture is true due the works of A. Avila and M. Viana.

Reduction to a countable shift

As J.-C. Yoccoz mentioned yesterday, one of the ideas of Avila andViana is to think of KZ cocycle (wrt Masur-Veech measures) as alocally constant symplectic cocycle over a countable shift bylooking at Bγ ’s over loops in Rauzy diagrams:

Setting for the simplicity criterium I

Λ be a countable alphabet, Σ = ΛN, Σ = ΛZ := Σ− × Σ;

p+ : Σ→ Σ and p− : Σ→ Σ− are the canonical projections;

f : Σ→ Σ unilateral shift, f : Σ→ Σ bilateral shift;

Ω =⋃

n∈NΛn is the set of words of Λ;

given a word ` ∈ Ω, we form the cylinders

Σ(`) := x ∈ Σ : x starts by `,

Σ−(`) := x ∈ Σ : x ends by `

for µ a f -inv. prob., µ is the unique f -inv. prob. s.t.p∗+(µ) = µ and µ− := p∗−(µ).

Setting for the simplicity criterium II

Let µ a f -inv. prob. with bounded distortion, i.e., ∃C (µ) > 0 s.t.

C (µ)−1µ(Σ(`1))µ(Σ(`2)) ≤ µ(Σ(`1`2)) ≤ C (µ)µ(Σ(`1))µ(Σ(`2))

In other words, µ is not very far from being a Bernoulli measure.

Exercise

By mimicking the proof of ergodicity of Bernoulli measure, showthat any µ with bounded distortion is ergodic.

Setting for the simplicity criterium II

Let µ a f -inv. prob. with bounded distortion, i.e., ∃C (µ) > 0 s.t.

C (µ)−1µ(Σ(`1))µ(Σ(`2)) ≤ µ(Σ(`1`2)) ≤ C (µ)µ(Σ(`1))µ(Σ(`2))

In other words, µ is not very far from being a Bernoulli measure.

Exercise

By mimicking the proof of ergodicity of Bernoulli measure, showthat any µ with bounded distortion is ergodic.

Setting for the simplicity criterium III

Let A : Σ→ Sp(d ,R) be a locally constant log-integrablesymplectic cocycle, that is,

A(x) = Ax0 for any x = (x0, x1, . . . ) ∈ Σ

and ∫Σ

log ‖A±1(x)‖dµ(x) =∑`∈Λ

µ(Σ(`)) log ‖A±` ‖ <∞

Setting for the simplicity criterium IV

Definition

A cocycle A as above is pinching if there exists a word `∗ ∈ Ω s.t.the eigenv. of the matrix A`

∗are real and distinct in modulus.

Notation: In the sympl. space Rd , d even, let G (k) be theGrassm. of isotropic, resp. coisotropic, k-planes of Rd for1 ≤ k ≤ d/2, resp. d/2 ≤ k ≤ d .

Definition

A pinching cocycle A is twisting if ∀ 1 ≤ k ≤ d/2 ∃ `(k) ∈ Ω words.t. the matrix A`(k) is twisting wrt A`

∗, i.e.,

A`(k)(F ) ∩ F ′ = 0

for any A`∗-inv. F ∈ G (k), F ′ ∈ G (d − k).

Setting for the simplicity criterium IV

Definition

A cocycle A as above is pinching if there exists a word `∗ ∈ Ω s.t.the eigenv. of the matrix A`

∗are real and distinct in modulus.

Notation: In the sympl. space Rd , d even, let G (k) be theGrassm. of isotropic, resp. coisotropic, k-planes of Rd for1 ≤ k ≤ d/2, resp. d/2 ≤ k ≤ d .

Definition

A pinching cocycle A is twisting if ∀ 1 ≤ k ≤ d/2 ∃ `(k) ∈ Ω words.t. the matrix A`(k) is twisting wrt A`

∗, i.e.,

A`(k)(F ) ∩ F ′ = 0

for any A`∗-inv. F ∈ G (k), F ′ ∈ G (d − k).

Statement of a version of Avila-Viana simplicity criterion

Theorem (Avila-Viana)

Let A be a cocycle over (f , µ) or (f , µ) as above. If A is pinchingand twisting, its Lyapunov spectrum (wrt µ or µ) is simple.

Remark

In fact, this simplicity criterion works for other groups of matrices(e.g. U(p, q)), but this is another history...

A comment on the twisting hypothesis I

There are several versions of this criterium e.g. by Avila and Vianathemselves!

For instance, in one of their works, they required a stronger(“infinitary”) version of twisting property asking that for everyF ∈ G (k), F ′ ∈ G (d − k), one can find a word ` s.t.A`(F ) ∩ F ′ = 0.

However, as it turns out, it suffices to ask the weaker (“finitary”)version above of twisting wrt to a pinching matrix.

A comment on the twisting hypothesis II

On the other hand, in the weaker version of twisting, it isimportant to ask for a word `(k) “twisting” all A`

∗-inv. isotropic

subspaces at once!

Indeed, simplicity of Lyap. spect. may fail if this is not the case:for example

Σ = 0, 1N, µ = 1/2− 1/2 Bernoulli measure

(2 00 1/2

), A1 =

(0 −11 0

)leads to a cocycle with two vanishing exponents.

A few words on reducing KZ cocycle to the shift case I

As J.-C. Yoccoz mentioned yesterday, the verification of pinchingand twisting in the case of the cocycle over a countable shiftderived from KZ cocycle depends on a combinatorial study ofRauzy diagrams and the idea is to proceed by induction.

In particular, as announced yesterday, we’ll not enter on thisverification here. Instead, we’ll propose again the following exerciseshowing that even the pinching property may be not automatic ingeneral.

A few words on reducing KZ cocycle to the shift case II

Exercise

Compute Bγ of the path D → B2 → D → C → D → A3 on our“preferred” genus 2 Rauzy diagram (see below), calculate itscharacteristic polynomial and determine the moduli of itseigenvalues. Is this Bγ a pinching matrix?

Preliminary reductionProof of Theorem 1

Key Theorem

The key result towards Avila-Viana simplicity criterion is:

Theorem 1 (Avila-Viana)

For every 1 ≤ k ≤ d/2, ∃ a map Σ− → G (k), x 7→ ξ(x) s.t.

the map ξ := ξ p− is invariant: A(x)ξ(x) = ξ(f (x));

for µ−-a.e. x ∈ Σ−, σk (A`(x,n))

σk+1(A`(x,n))→∞ and ξ`(x ,n) → ξ(x)

∀F ′ ∈ G (d − k), ξ(x) ∩ F ′ = 0 w/ positive µ−-prob. on x

`(x , n) is the terminal word of x of length n;

σ1 ≥ · · · ≥ σd are singular values of a matrix (see next slide);

ξ` is the subspace generated by the k largest semi-axis of theellipse A`(‖v‖ = 1) (see next slide).

Key Theorem

The key result towards Avila-Viana simplicity criterion is:

Theorem 1 (Avila-Viana)

For every 1 ≤ k ≤ d/2, ∃ a map Σ− → G (k), x 7→ ξ(x) s.t.

the map ξ := ξ p− is invariant: A(x)ξ(x) = ξ(f (x));

for µ−-a.e. x ∈ Σ−, σk (A`(x,n))

σk+1(A`(x,n))→∞ and ξ`(x ,n) → ξ(x)

∀F ′ ∈ G (d − k), ξ(x) ∩ F ′ = 0 w/ positive µ−-prob. on x

`(x , n) is the terminal word of x of length n;

σ1 ≥ · · · ≥ σd are singular values of a matrix (see next slide);

ξ` is the subspace generated by the k largest semi-axis of theellipse A`(‖v‖ = 1) (see next slide).

Semi-axes of ellipses and singular values

σ2σ1

Reduction of simplicity criterion to Theorem 1

Intuitively, the simplicity of Lyap. spect. of a cocycle A as abovefollows from Theorem 1 because:

(a) starting with ξ(x) ∈ G (k) for x ∈ Σ− and “reversing time”,we get analogous objects ξ∗(y) ∈ G (d − k) for y ∈ Σ;

(b) by the 3rd item of Thm 1, one can show that the transversalityproperty ξ(x) ∩ ξ∗(y) = 0 for µ-a.e. (x , y) ∈ Σ;

(c) by the 1st and 2nd items of Thm 1, ξ(x) ↔ k largestexponents of A, and ξ∗(y) ↔ d − k smallest exponents of A;

(d) by (c) and (b), one can “separate” the kth exponent of Afrom the (k + 1)th exponent of A.

u-states

Definition

A u-state (u = unstable) is a prob. m on Σ× G (k) s.t.

q∗m = µ, where q : Σ× G (k)→ Σ is the can. proj., and

for some constant C (m) <∞,

m(Σ−(`0)× Σ(`)× X )

µ(Σ(`))≤ C (m)

m(Σ−(`0)× Σ(`′)× X )

µ(Σ(`′))

∀ `0, `, `′ ∈ Ω and X ⊂ G (k).

Example

Given ν prob. on G (k), m = µ× ν is a u-state w/ C (m) = C (µ)2.

u-states

Definition

A u-state (u = unstable) is a prob. m on Σ× G (k) s.t.

q∗m = µ, where q : Σ× G (k)→ Σ is the can. proj., and

for some constant C (m) <∞,

m(Σ−(`0)× Σ(`)× X )

µ(Σ(`))≤ C (m)

m(Σ−(`0)× Σ(`′)× X )

µ(Σ(`′))

∀ `0, `, `′ ∈ Ω and X ⊂ G (k).

Example

Given ν prob. on G (k), m = µ× ν is a u-state w/ C (m) = C (µ)2.

Existence of invariant u-states

Proposition 1

There are (f ,A)-inv. u-states.

This result follows from the “usual” Krylov-Bogolyubov arg.: takem0 any u-state, let m(n) = (f ,A)n

∗m0 and extract an accum. pt mfrom Cesaro averages. Of course, m is (f ,A)-inv.

So, it remains to show that m is a u-state. Here, this follows from:

Exercise

Show that C (m(n)) ≤ C (m0) · C (µ)2 for all n.

Existence of invariant u-states

Proposition 1

There are (f ,A)-inv. u-states.

This result follows from the “usual” Krylov-Bogolyubov arg.: takem0 any u-state, let m(n) = (f ,A)n

∗m0 and extract an accum. pt mfrom Cesaro averages. Of course, m is (f ,A)-inv.

So, it remains to show that m is a u-state. Here, this follows from:

Exercise

Show that C (m(n)) ≤ C (m0) · C (µ)2 for all n.

What is the current goal?

Our goal is to find the section x 7→ ξ(x) ∈ G (k) (cf. Thm 1) viaan (f ,A)-inv. u-state because such meas. should capture thedynamical behavior of (f ,A) and hence it should “see” x 7→ ξ(x).

In order to formalize this, we’ll need a martingale convergencetheorem.

Proposition 2

Let m s.t. q∗m = µ and let

mn(x)(X ) :=m(Σ−(`(x , n))× Σ× X )

m(Σ− × Σ× X )

for any x ∈ Σ− and X ⊂ G (k). Then, for µ−-a.e. x ∈ Σ−,

mn(x)→ m(x) prob. on G (k).

Assume now that m is a (f ,A)-inv. u-state and let’s apply themartingale convergence theorem.

We get a family of m(x) of prob. on G (k) and we claim that if wecan show that m(x) is a Dirac mass m(x) = δξ(x) for a.e. x , thenx 7→ ξ(x) satisfies Theorem 1.

For ex., A(x)(ξ p−(x)) = ξ p−(f (x)) (1st [“inv.”] item of Thm1) would follow from the (f ,A) invariance of m, while the othertwo items depend on the pinching and twisting properties of (f ,A).

Why are m(x) Dirac masses? I

By definition,

mn(x)(X ) =m(Σ−(`(x , n))× Σ× X )

m(Σ−(`(x , n))× Σ× G (k))

Thus, by (f ,A)-inv. of m, we get

mn(x)(X ) =m(Σ− × Σ(`(x , n))× A−`(x ,n)(X ))

m(Σ− × Σ(`(x , n))× G (k))

Since m is a u-state, it follows that

mn(x)(X ) ≤ C (m)2 m(Σ− × Σ× A−`(x ,n)(X ))

m(Σ− × Σ× G (k)),

that is,

mn(x)(X ) ≤ C (m)2m(Σ− × Σ× A−`(x ,n)(X )) := νn(x)(X )

Why are m(x) Dirac masses? II

Actually, the previous inequality is “symmetric” (because the def.of u-state is “symmetric”) and one has

C (m)−2 ≤ mn(x)(X )/νn(x)(X ) ≤ C (m)2,

that is, mn(x)(X ) is equivalent to νn(x)(X ).

On the other hand, we can rewrite

νn(x)(X ) := m(Σ× A−`(x ,n)(X )) = A`(x ,n)∗ ν(X )

where ν = r∗m and r : Σ× G (k)→ G (k) is the can. proj.

Thus, m(x) = lim mn(x) is equivalent to any accum. pt of νn(x).

C (m)−2 ≤ mn(x)(X )/νn(x)(X ) ≤ C (m)2,

Why are m(x) Dirac masses? III

In view of the previous slide, our task is reduced to show that

Proposition 3

νn(x) = A`(x ,n)∗ ν accum. some Dirac mass for µ−-a.e. x ∈ Σ−.

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