We study the dynamics of the renormalisation operatorRacting on the space of C3-generalised interval exchange transformations at fixed points (which are standard periodic type IETs)

LOCAL RIGIDITY FOR PERIODIC GENERALISED INTERVAL EXCHANGE TRANSFORMATIONS

SELIM GHAZOUANI

Abstract. In this article we study local rigidity properties of generalised interval exchange maps using renormalisation methods. We study the dynamics of the renormalisation operatorRacting on the space of C³-generalised interval exchange transformations at fixed points (which are standard periodic type IETs). We show thatRis hyperbolic and that the number of unstable direction is exactly that predicted by the ergodic theory of IETs and the work of Forni and Marmi-Moussa-Yoccoz. As a consequence we prove that the local C¹-conjugacy class of a periodic interval exchange transformation, withdintervals, whose associated surface has genusgand whose Lyapounoff exponents are all non zero is a codimensiong−1 +d−1C¹-submanifold of the space ofC³-generalised interval exchange transformations. This solves a particular case of a conjecture of Marmi-Moussa-Yoccoz.

1. Introduction

The study of stability and rigidity properties of quasi-periodic and parabolic dynamical systems form a rather old class of problems in the modern theory of dynamical systems. Trying to determine whether the solar system is stable led astronomers to formulate simplified mathematical problems, one of which being the famous three-body problem. Daunted by the many difficulties arising when trying to solve it, Poincaré [22] suggested that mathematicians turn to even simpler toy models, such as the dynamics of circle diffeomorphisms.

The three-body problem was eventually solved by Kolmogorov in 1954 using a set of methods nowadays commonly referred to as KAM theory. These methods were subsequently applied by Arnol’d [1] to solve the problem of the local rigidity for analytic circle diffeomorphisms.

Later on, from the late 1970s onwards, the introduction of renormalisation as a tool in mathematics allowed mathematicians to discover a few more rigidity and universality phenomena for other classes of parabolic dynamical systems, in particular unimodal maps [2, 6, 16, 21, 23, 24], circle diffeomorphisms with critical points [4, 5] or breaks points [12, 14] and more recently circle maps with a flat interval [20]. The general question that can be asked at this point is the following

Question. What classes of dynamical systems are rigid (in some sense) and what are the mechanisms responsible for this rigidity?

The notion of rigidity is vague and can mean different things depending on the context. We give here rigidity themes that we have in mind when writing this text and which are interconnected.

• Geometric rigidity. This is when the topological structure of a dynamical system forces its geometry.

Formally, we say a smooth dynamical system is geometrically rigid if any other system which is topologically conjugate to it is differentiably conjugate to it. This is the case for certain classes of circle maps and infinitely renormalisable unimodal maps.

• Universality. We say a class of dynamical systems displays some form of universality if some universal behaviour can be observed in arbitrary parameter families. An example is parameter families of unimodal maps displaying period-doubling bifurcations and for which the structure of bifurcations asymptotically does not depend on the initial family.

• Solving cohomological equations. Analysing local geometric rigidity problems via linearising the problem often features solvingcohomological equations. Understanding the obstructions to solving cohomological equations is an important step to solving rigidity problems, and describing distributions realising these obstructions an interesting problem.

Most of known results for geometric rigidity and universality are either in dimension 1or are local results where the underlying system is a translation on a torus and KAM theory applies.

1

Ergodic theory.A lot geometric rigidity results rely partly on the fact that the combinatorial structure (and consequently the ergodic theory) of underlying dynamical systems is simple; precisely it is either a translation on a torus or an odometer. It is the case for results in KAM theory, circle diffeomorphisms, circle diffeomorphisms with critical or break points and unimodal maps. For instance, KAM theory relies on Fourier analysis to solve the cohomological equation over rotations of the n-dimensional torus. This is only made possible becausen-dimensional tori are abelian groups, translations preserve this group structure which makes for an efficient Fourier analysis. More general parabolic systems are more complicated and we do not always have ready tools for a direct analysis of their ergodic theory, study of the cohomological equation and deviations of ergodic averages.

There has been important progress in understanding the ergodic theory of parabolic dynamical systems:

[28], [15] and [9] for flows on surfaces, [7, 8] for the horocycle flow and nilflows and [17, 19] for interval exchange maps. For all these examples it is shown that deviations of ergodic average for smooth observable are governed by finitely many distributions, and for functions in the kernel of those distributions one can solve the cohomological equation. These distributions also provide finitely many obstructions to geometric rigidity.

Main result.In this article we prove a local (geometric) rigidity result for generalised interval exchange transformations. A generalised interval exchange transformation (GIET) is a bijection of the interval which is piecewise continuous, smooth and increasing on its continuity intervals (see Section 2for precise definitions).

These maps, which are obtained as first-return of smooth flows on surfaces, have vanishing entropy and are most of the time uniquely ergodic¹. Our main result is

Theorem 1. Let T0 be a periodic IET with hyperbolic Rauzy matrix. The set of generalised IETs which are C¹-conjugated to T0 by a diffeomorphism C¹-close to the identity is a C¹-submanifold of codimension d−1 +g−1. Heredis the number of intervals ofT0 and g is the genus of the associated translation surface, and we are working in the space ofC³-generalised IETs whose total non-linearity vanishes.

This result (and more) had been conjectured by Marmi, Moussa and Yoccoz in [18]. Their main conjecture (Problem 1 in [18]) is that this result is true for almost every choice of initial IETT0; here we only treat the case of periodic combinatorics. This result is nonetheless (to the best knowledge of the author) the first result describing a rigidity class of generalised interval exchange transformations.

We briefly comment on the statement of this theorem which might seem a bit technical at first glance. Consider T0 a standard interval exchange transformation. We can deform it within the Banach space ofC³-generalised interval exchange transformation and ask whether the new map is differentiably conjugate toT₀.

• A necessary condition for this to happen is that they are orbitally equivalent; this condition is realised if and only their generalised rotation number/Rauzy path(see [26]) is the same. Roughly, this generalised rotation number takes value in a (d−1)-dimensional simplex and provides a first set of combinatorial obstructions.

• Once we know that this first condition is satisfied, the two maps that we get have same ergodic theory (as it can be shown that they are semi-conjugate). In this case, the ergodic theory (via the work of Forni [9, 10] and Marmi-Moussa-Yoccoz [17]) provides us with a new set of obstructions which correspond to obstructions to solving the cohomological equation. There areg−1such obstructions, and correspond to Lyapounov exponents of the Konsevich-Zorich cocyle.

This result shows that these obstructions are the only obstructions to local rigidity. It is very much in the spirit of standard rigidity results about circle maps or unimodal maps, once the ergodic theory has been factored in.

Renormalisation.The proof of Theorem1makes use of renormalisation methods. A powerful idea to study parabolic dynamical systems is to consider a renormalisation operator acting on the moduli space of such systems. A renormalisation operator is a procedure by which one associates to a dynamical system T a suitably rescaled first-return map, that we denote byR(T), which is in the same class asT (in our case, a generalised IET with as many discontinuity intervals).

A general principle is that two maps T₁ andT₂ are differentiably conjugate if and only if their iterated renormalisationsRⁿ(T₁)andRⁿ(T₂)are getting close at an exponential rate. In this article we make use of a

1In parameter space, one expects the generic GIET to be Morse-Smale, but interesting cases are infinitely renormalisable maps; their combinatorial structure can be reduced to that of standard IETs which are know to almost always be uniquely ergodic, see [26].

renormalisation operatorRon generalised IETs (which is an extension of standard Rauzy-Veech induction).

Our main theorem is just a corollary of the following result

Theorem 2. Let T0 be a standard IET which is a fixed point of R and whose standard Rauzy matrix is hyperbolic. ThenRacting on the Banach manifold of C³-generalised IETs is hyperbolic at T0, its unstable space has dimension(d−1) + (g−1) and consequently its stable space has codimension(d−1) + (g−1).

Previous work on generalised interval exchange transformations.We discuss briefly previous work on the question on generalised interval exchange transformations of genus2. As already mentioned, pioneering work of Forni followed by Marmi-Moussa-Yoccoz and Marmi-Yoccoz on the solving of the cohomological equation set the stage for a discussion on rigidity properties of generalised interval exchange maps. They demonstrated the existence of obstructions to solving the cohomological equation and interpreted them as cohomology classes of the associated surface (obtained by suspending a generalised interval exchange map).

Subsequent work of Marmi-Moussa-Yoccoz implemented a KAM scheme to describe local smooth conjugacy classes of generalised IETs in high regularity (C^r-conjugacy classes forr≥2). They give a formula for the codimension of such conjugacy classes in terms of dandg. Their result cover almost every rotation number, but they fail to describeC¹-rigidity classes which are the generic case in parameter space. Their work was completed by Forni-Marmi-Matheus [11] to cover other rotation numbers, still in high regularity.

Strategy of the proof. We comment on the proof of Theorem1and Theorem2(we assume some knowledge of renormalisation theory). We consider the renormalisation operator acting at a fixed pointT0.

(1) We first show the existence of(d−1) + (g−1)unstable directions by lettingRact on the finite dimensional subspace of affine interval exchange transformations. This action can be related to the standard action of the Zorich-Konsevich cocycle and we can achieve our aim by standard ergodic- theoretic methods.

(2) The difficult part of the problem is to construct the stable space. Indeed the complement of the (d−1) + (g−1)unstable directions is infinite dimensional and we have a priori very little control on what happens there. In many standard cases deriving from circle maps (circle diffeomorphisms, circle maps with critical points or break points), a strong control is given by what is nothing short of an ergodic miracle, the Denjoy-Koksma inequality. It provides what specialists calla priori bounds for the renormalisation.

(3) We construct apre-stable space of codimension (d−1) + (g−1) satisfying the property that for any T in this pre-stable space, the sequence Rⁿ(T)

is bounded in the C²-topology. We see this latter statement as an a priori bound. This construction is the heart of the article. It makes use of the fact thatRis hyperbolic restricted to the subspace of affine IETs, various distortion bounds for one-dimensional dynamical systems and the choice of an appropriate norm using the non-linearity of one-dimensional maps. A key idea of this construction is to carry out corrections to shadow the sequence Rⁿ(T) by renormalisations of affine IETs. This was inspired by the proof of the main theorem of [17].

(4) Once those "a priori bounds" are constructed, we obtain uniform contraction in the pre-stable space as a reformulation of Herman’s theory for circle diffeomorphisms.

(5) We derive the main theorem using standard result for renormalisation of one-dimensional maps borrowed from [4].

(6) The regularity of the stable space is obtained by writing the equation defining it and using a result of Marmi-Yoccoz [19] on the regularity of solutions of the cohomological equation.

Acknowledgements.The author would like to thank Liviana Palmisano for sharing course notes about renormalisation, Michael Bromberg and Björn Winckler for interesting discussions, and Giovanni Forni his careful reading and precious comments on an early versions of this text. The author is greatly indebted to Corinna Ulcigrai for sparking his interest in the subject, her teaching, the many hours of conversation about interval exchange maps and her continued support.

2. Generalised interval exchange transformations 2.1. Basic definitions.

Definition 1. Let d≥2 be an integer. AC^r-generalised interval exchange transformation (GIET) is a map T from the interval[0,1]to itself such that

• there are two partitions[0,1] =Sd

i=1I_i^t=Sd

i=1I_i^b intodopen subinterval (the intervalsI_i^ts and I_i^bs are lying on[0,1]ordered from left to right);

• there exists a permutation σ∈S_n such that T restricted toI_i^tis an orientation preserving diffeomor- phism onto I_σ(i)^b of class C^r;

• T extends to the closure of I_i^tto aC^r-diffeomorphism onto the closure ofI_σ(i)^b .

Examples of such generalised interval exchange transformations include standard interval exchange trans- formations (IET) for which the mapT is further restricted to be a translation on each of theI_i^ts andaffine interval exchange transformations (AIET) for which T is an affine map restricted to the I_i^ts.

In what follows we make the standing assumption thatr≥2. LetT be aC^r-GIET. We define

(1) ηT = D log DT

which is called thenon-linearity ofT and is well-defined because we have assumed T isC².

Iff :I−→J is a continuous function from a bounded intervalI to anotherJ, we use the following notation

||f||=||f||0= sup

x∈I

|f(x)|.

2.2. The moduli space and coordinates. We define

X_σ^r={generalised interval exchange transformation of class C^r with associated permationσ}.

Let T be a C^r-GIET, with associated permutation σ and let (I_i^t)_1≤i≤d and (I_i^b)_1≤i≤d be the "top" and

"bottom" partitions of[0,1]associated to it. We make the two following observations.

• There is a unique affine interval exchange transformationAT mappingI_i^ttoI_σ(i)^b .

• Furthermore, for alli≤d, there is a unique elementϕⁱ_T ofDiff^r([0,1]) such that the restriction ofT to I_i^t is equal to

c_i◦ϕ_i◦b_i

whereb_i is the unique orientation preserving affine map mappingI_i^t onto[0,1]andc_i is the unique orientation preserving affine map mapping [0,1]ontoI_σ(i)^b .

This operation can be inverted and therefore the map

T 7−→(AT, ϕ¹_T,· · ·, ϕ^d_T) gives an identification betweenX_σ^r andAσ× Diff^r([0,1])^d

whereAσ the space of AIETs with permutation σ. In the sequel we denote byP the space Diff^r([0,1])^d

and so we have a canonical identification X_σ^r=Aσ× P.

Using this parametrisation we can endowX_σ^rwith the structure of a Banach space directly inherited from that ofDiff^r([0,1]). When there is no possible ambiguity, we will drop the indexesσandrand simply write

X =A × P.

2.3. Renormalisation. We introduce in this paragraph a map acting uponX_σ^r which is arenormalisation operator. A fully-fledged renormalisation theory for GIETs would require that we introduce theRauzy-Veech induction, as it is done in [18]. However, because we are only going to treat a particular combinatorial case, we can spare such machinery and define everything in more elementary terms.

In the sequelT₀is a standard IET which satisfies the following self-similarity property: there existsx₀∈]0,1[

such that the first-return map ofT₀ on[0, x₀]is equal, up to affine rescaling, toT₀. Consequently, there is a neighbourhoodW ofT₀ inX and a smooth mapX:X −→[0,1[such that the following holds.

• X(T0) =x0;

• For every T ∈ W, the first return map ofT on[0, X(T)]is a GIET with permutationσ;

• ifRT denotes this first return map rescaled to define a function from [0,1]to itself, the map R:W −→ X

is continuous;

• if we denote byR_AandR_P the projection ofRon the coordinatesAandP respectively the map R_A:W −→ A

is of classC¹;

• R(T0) =T₀;

• for allT ∈ X,DR_A(T)is a bounded operator for theC^r-norm.

The facts thatRis of classC¹andDR_A(T)is a bounded operator are a consequence of the fact thatR(T)is obtained by taking compositions of the restrictions ofT to its continuity intervals on intervals whose endpoints themselves depend smoothly on T (the proof of these facts is discussed in greater detail in Appendix A).

In the sequel we will be calling R the renormalisation operator. For a given GIET T ∈ W, we will call RT =R(T)its renormalisation and when well-defined, we call the sequenceT,RT,R²T,· · ·,RⁿT,· · · its consecutive renormalisations. When it is the case that consecutive renormalisation of T are defined for all n≥0,i.e. RⁿT ∈ W for alln≥0, we say thatT is infinitely renormalisable.

Remark 3. The reason why we care about such a renormalisation operator is the following: a GIET inW is C¹-conjugate toT₀ if and only if its consecutive renormalisations converge fast enough toT₀. This rather loose statement will be made precise in Section7.

2.4. Dynamical partitions. LetT be an element ofWand assume further thatT is infinitely renormalisable.

For anyn≥0,RⁿT is the rescaling of a first return map ofT on an interval of the form[0, xn]. The interval [0, xn]is partitioned into

[0, xn] =∪^d_j=1I_n^j

andRⁿT rescaled down to[0, x_n]is equal toT^ljn on each of theI_n^js. For1≤j≤d, we introduce P_n^j ={I_n^j, T(I_n^j), T²(I_n^j),· · · , T^ljn−1(I_n^j)}

and we call

Pn =

d

[

j=1

P_n^j

the dynamical partition of leveln. One easily verifies thatPn is a partition of[0,1]into subintervals.

3. Affine interval exchange transformations

Anaffine interval exchange transformation is simply a generalised IET which is affine restricted to its intervals of continuity. In this subsection, we aim at computing the derivative of the renormalisation operator restricted to AIETs, at the fixed point T₀. We will see that this derivative can be understood fairly simply in terms of the combinatorial structure ofT₀.

3.1. Coordinates on A. Let T be an AIET with permutationσ. Denote byλ1,· · · , λd the lengths of its continuity intervals. Because these form a partition of[0,1], they must satisfy the following equation:

λ1+· · ·+λd= 1.

Furthermore, if we denote byρ1,· · · , ρd the derivatives ofT on intervals of respective lengthsλ1,· · ·, λd, we must also have

ρ1λ1+· · ·+ρdλd= 1.

These two equations, together with the further restrictions that∀i, λi>0identifyAσ to a submanifold of R^2d of dimension2d−2. For any affine interval exchange transformationT, we denote byλ(T)its associated lengths andρ(T)its slopes.

Surface associated to an IET.To an IET can be associated a a topological surface with marked points by an operation ofsuspension. Ifsis the number of marked points of this surface andgits genus we have the following relation

d= 2g+s−1.

We make the standing assumption thatsis equal to1and that g≥2.

3.2. Intersection matrix. RecallT₀ the fixed point ofRand theP_n^js the sub-partitions associated with the dynamical partitions Pn. Defineaij to be the number of elements ofP_nⁱ which intersectI₀^j. TheI₀^j are just by definition the intervals of continuity ofT₀. We will denote byA thed×d= 2g×2gmatrix whose entry in place(i, j)isa_ij. We callA theintersection matrix ofA. We have the following well-known facts aboutA(we refer to [27] for details and proofs).

(1) All coefficients ofA are positive(possibly requires passing to a power ofR).

(2) (λ⁰₁,· · · , λ⁰_d)the lengths ofT0is an eigenvector of ^tA.

(3) The associated eigenvalue is simple and is the the largest eigenvalue of^tA.

(4) Apreserves a (non-degenerate) symplectic form.

We want to understand the action ofRonAclose toT0. Note thatRstabilises the subset of standard IETs (defined in coordinates byρ1=· · ·=ρd= 1). This subset identifies with the simplex∆ ={(λ1,· · ·, λd)∈ R+ | P

λi= 1}and the action of Rrestricted to it is nothing but the projective action of(^tA)⁻¹. From all these considerations we get the following fact:

T0 is an expanding fixed point ofRrestricted to IETs.

By that we mean that(DR)_T0 the derivative ofRsatisfies for all v∈T_T0∆, ||(DR)T0v||> α||v|| for a certain norm|| · || andα >1.

Another important fact is that the action of R on the slopes ρ = (ρ1,· · · , ρd) satisfies the following: if µ(T) = logρ(T) = logρ1(T),· · · ,logρd(T)

we have

µ(RT) =A·µ(T).

irrespective of the value ofλ(T).

3.3. Derivative of Rrestricted to A at T₀. We make the following standing assumption for the rest of the article:

Ais a hyperbolic matrix.

BecauseApreserves a symplectic form, it hasg eigenvalues which are (strictly) larger than1andgwhich are (strictly) smaller than1. We briefly discuss how little restrictive this assumption in3.4.

Using coordinates (λ, µ)introduced above, we writeR= (Rλ,Rµ).

Proposition 4. The following statements hold true:

(1) (D_λRµ)_T0 = 0;

(2) there existsα >1 such that(D_λRλ)_T0 isα-expanding;

(3) (D_µR_µ)_T0 is hyperbolic and hasg−1 expanding directions.

Proof. The proof is straightforward. A neighbourhood of T0 can be parametrised using coordinates(λ, µ) has above. If λ⁰ is the coordinates associated toT0, the tangent space ofX atT0 is defined by the following equations

Xλⁱ₀= 0 and

Xµiλⁱ₀= 0.

(1) (D_λR_µ)_T0 = 0is a simple consequence of the fact that the space of linear IETs is stable byR;

(2) as said above, the restriction ofRto∆is the projective action ofA. Since the line spanned by λ⁰ is the eigenline of the (simple) largest eigenvalue of A, there exists α >1such that (DλRλ)T₀ is α-expanding;

(3) The action of(DµRµ)T₀ is that of Arestricted to the subspace defined by the equationP

iµiλⁱ₀= 0.

This space is stabilised by the action ofAand consequently the action of(DµRµ)T₀ is diagonalisable withg−1eigenvalues larger than1andg smaller than one.

This proposition in particular implies thatT0is a hyperbolic fixed point ofRand that the unstable space at T0 has dimension exactly(d−1) + (g−1).

3.4. On the standing assumption. We wanted to point out that the assumption thatAbe hyperbolic is not very restrictive. For anydand combinatorics giving rise to a surface with only one marked point, there are infinitely many periodicT0and most of them have an intersection matrix which is hyperbolic. However, we would like to point out that it is not the case for all of them: Bressaud-Bufetov-Hubert have constructed infinitely many periodic IETs violating this condition, see [3].

4. Estimates

In this section we prove estimates on the distortion, the second derivative and third derivatives of iterated renormalisations. These will be crucial for the analysis of the renormalisation operator.

4.1. Distortion bounds. We prove a standard distortion lemma and apply it to show that the "profile"

coordinate remains uniformly bounded under iteration of renormalisation. This fact will be the starting point of the correction operation carried out in Section5.

Lemma 5. Let T be a GIET. LetJ ⊂[0,1]be an interval such that J, T(J), T²(J),· · ·, Tⁿ(J)are pairwise disjoint and do not contain any singularities ofT. Then for all x, y∈J we have

D(Tⁿ)(x)

D(Tⁿ)(y) ≤exp(

Z 1

0

|ηT|dLeb).

Proof. The proof is classical. We have that

log DTⁿ(x) =

n−1

X

i=0

log DT(Tⁱ(x)) and therefore

|log DTⁿ(x)−log DTⁿ(y)| ≤

n−1

X

i=0

|log DT(Tⁱ(x))−log DT(Tⁱ(x))| ≤

n−1

X

i=0

| Z Tⁱ(x)

Tⁱ(y)

ηT|.

Since the intervals[Tⁱ(y), Tⁱ(x)]are pairwise disjoints we get

|log DTⁿ(x)−log DTⁿ(y)| ≤ Z 1

0

|ηT|dLeb and exponentiating gives the expected result.

In the sequel we use the following notation forf : [0,1]−→Rof classC^r,

||f||_C^r = max

0≤i≤d||f⁽ⁱ⁾||

wheref⁽ⁱ⁾ is thei-th derivative off. We extend this norm to(C^r([0,1],R))^d simply by taking the sum of the norms on each coordinate. From the lemma above, we derive the following

Proposition 6. Recall thatX² was defined to be the set of C²-GIETs with a given combinatorial type. Let K be a pre-compact set of X² with respect to theC²-topology. There exists a constantM(K)>0such that for any T GIET renormalisablentimes belonging toK we have

||π_P Rⁿ(T)

−(Id)^d||_C1 ≤M||π_P(T)||_C2

where (Id)^d= (Id,· · ·,Id)∈ P= Diff²₊([0,1])^d . Proof. Considerφa coordinate ofπ_P Rⁿ(T)

. It is obtained by taking finitely many restrictions ofT tok pairwise disjoint intervalsI1,· · ·, Ik, composing them and rescaling them. We can therefore apply Lemma5 to such a composition to find that for allx, y∈[0,1],

Dφ(x) Dφ(y) ≤exp(

Z 1

0

|ηT|dLeb).

Sinceφis a diffeomorphism of[0,1]there exists z∈[0,1]such thatDφ(z) = 1. T belongs to a precompact set with respect to the C²-topology so in particularDT andD(T⁻¹) are bounded by a uniform constant.

BecauseηT = ^D_DT^2T and the fact that the exponential is Lispchitz on compact sets ofRwe get the existence of a constantL >0 such that for allx

Dφ(x)

Dφ(y) ≤L||D²T||.

Comparing an arbitrary pointxto zgives the expected result.

4.2. C²-bounds. In this paragraph we prove an estimate which give some uniform bounds on the second derivative of iterated renormalisation of elements in X close toT₀. The proof builds upon Lemma5. To the best knowledge of the author, this estimate is new.

Lemma 7. Let ϕ1,· · ·, ϕn ∈ C²(R,R). For allk≤n definefk=ϕk◦ϕ_k−1◦ · · · ◦ϕ1 and set f0= Id. Then we have for all n≥2 the formula

f_n⁰⁰= (f_n−1⁰ )²·(ϕ⁰⁰_n◦f_n−1) +

n

X

k=2

(f_n−k⁰ )²·(ϕ⁰⁰_n−k+1◦f_n−k)·(ϕn◦ · · · ◦ϕ_n−k+2)⁰◦f_n−k+1 Proof. We proceed by induction onn. We check that the statement holds true forn= 2:

f₂⁰⁰= (ϕ₂◦ϕ₁)⁰⁰= (ϕ⁰₁·ϕ⁰₂◦ϕ₁)⁰= (ϕ⁰₁)²·ϕ⁰⁰₂◦ϕ₁+ϕ⁰⁰₁·ϕ⁰₂◦ϕ₁. Assume the statement holds true forn≥2. We have

f_n+1⁰⁰ = (ϕ_n+1◦f_n)⁰⁰=ϕ⁰⁰_n+1◦f_n·(f_n⁰)²+f_n⁰⁰·ϕ⁰_n+1◦f_n. Replacingf_n⁰⁰ in the formula we get

f_n+1⁰⁰ = (ϕn+1◦fn)⁰⁰=ϕ⁰⁰_n+1◦fn·(f_n⁰)²+ (ϕ⁰_n+1◦fn)·(f_n−1⁰ )²·(ϕ⁰⁰_n◦f_n−1) +

n

X

k=2

(f_n−k⁰ )²·(ϕ⁰⁰_n−k+1◦f_n−k)·(ϕ⁰_n+1◦fn)·(ϕn◦ · · · ◦ϕ_n−k+2)⁰◦f_n−k+1. By the chain rule we have

(ϕ⁰_n+1◦f_n)·(ϕ_n◦ · · · ◦ϕ_n−k+2)⁰◦f_n−k+1= (ϕ_n+1◦ϕ_n◦ · · · ◦ϕ_n−k+2)⁰◦f_n−k+1

Injecting in the formula above for f_n+1⁰⁰ gives the expected result.

Consider aC², increasing diffeomorphism f :I−→J whereI andJ are two connected intervals. We denote byN(f)thenormalisation orrescaling off, it is by definition the mapf pre-composed by the unique affine map sending[0,1]ontoI and post-composed by the unique affine map sendingJ onto[0,1]. We have the following easy lemma:

Lemma 8. Let f as above. Then we have

||N(f)⁰⁰|| ≤ ||f⁰⁻¹|| · ||f⁰⁰|| · |I|

Proof. Leta=|I|andb=|J|. By definition we have N(f) :=x7−→ 1

bf(ax).

Thus

N(f)⁰⁰(x) =a²

b f⁰⁰(ax) =aa bf⁰⁰(ax).

There existsx₀∈Isuch that _f0(x¹0)= _|J|^|I| = ^a_b. Hence the result.

Using Lemma 7and Lemma8, we prove the following

Proposition 9. Let V be a pre-compact neighbourhood ofT₀ in X² with respect to theC²-topology. There exists a constantM⁰ such that the following holds. LetT ∈ V be aC² GIET renormalisable ntimes. We use the following notation π_P Rⁿ(T)

= (ϕⁿ₁,· · ·, ϕⁿ_d)∈ Diff²₊([0,1])^d

. Then we have for alli≤dand for all n∈N

||(ϕⁿ_i)⁰⁰|| ≤M⁰||(T⁻¹)⁰|| · ||T⁰⁰||

Proof. The proof is an application of Lemma7to the composition of restrictions ofT to the dynamical partition.

Recall thatϕⁿ_i is the renormalised ofT^lin restricted to an intervalI_nⁱ such thatI_nⁱ, T(I_nⁱ), T²(I_nⁱ),· · · , T^lin−1(I_n^j) are disjoint. We denote bySk the restriction of T to T^k(I_nⁱ). We have the following properties

• ϕⁿ_i = N(S_lj

n−1)◦ · · · ◦N(S₁)◦N(S₀)

• any partial productψk= N(S_lj

n−1)◦ · · · ◦N(Sk)is such that ||log(ψk)⁰|| ≤K||T⁰⁰||(ψk is a diffeomor- phism of [0,1] and therefore there existsx0 ∈[0,1]such that logψ_k⁰(x0) = 0and the claim follows from Lemma5);

• same holds for partial productsφk= N(Sk)◦ · · · ◦N(S0);

• for anyk,||N(Sk)⁰|| ≤ ||(T⁻¹)⁰|| · ||T⁰⁰|| · |T^k(I_n^j)|.

The result is a consequence of Lemma7 applied toN(S_li

n−1)◦ · · · ◦N(S₁)◦N(S₀). Indeed

||ϕⁿ_i)⁰⁰|| ≤

l_nⁱ

X

k=1

||φ⁰_n−k||²· ||N⁰⁰(S_n−k+1)|| · ||ψ_n−k+2⁰ ||

and replacing in the inequality

||ϕⁿ_i)⁰⁰|| ≤e^3K||T00||· ||(T⁻¹)⁰|| · ||T⁰⁰||

lⁱ_n−1

X

k=0

|T^k(I_nⁱ)|.

TheT^k(I_n^j)s are all disjoint, theexpis bounded on bounded sets and||T⁰⁰||is bounded (becauseT belongs to a C²-precompact subset of X) thus we get the result.

4.3. Bounds onDη. We prove in this paragraph bounds on the functionD(ηT)along renormalisation when r≥3. The proofs follow the same line of thought as the previous section.

Lemma 10. Letϕ1,· · ·, ϕn ∈Diff³₊([0,1]). Setψk =ϕk◦ϕ_k−1◦ · · · ◦ϕ1 andψ0= Id. Then (1) logD(ϕn◦ϕn−1◦ · · · ◦ϕ1) =Pn

k=1logD(ϕk)◦ψk−1;

(2) η(ϕn◦ϕn−1◦ · · · ◦ϕ1) =D(logD(ϕn◦ϕn−1◦ · · · ◦ϕ1)) =Pn

k=1DlogD(ϕk)◦ψk−1·Dψk−1; (3) Dη(ϕn◦ϕ_n−1◦ · · · ◦ϕ1) =Pn

k=1D²logD(ϕk)◦ψ_k−1·(Dψ_k−1)²+DlogD(ϕk)◦ψ_k−1·D²ψ_k−1; These formulae directly derive from the definition of the non-linearity η(f) =DlogDf and their proofs are left to the reader. Let f be aC², increasing diffeomorphismI −→J whereI andJ are two connected intervals.

Lemma 11. Recall thatN(f)is the rescaling of f. Then we have

|D η(N(f))

| ≤(||f⁰⁰⁰|| · ||f⁰||+||f⁰⁰||²)· ||(f⁻¹)⁰||⁴· |I|² Proof. We have that

D η(N(f))

=D(N(f)⁰⁰

N(f)⁰) = N(f)⁰⁰⁰N(f)⁰−(N(f)⁰⁰)² (N(f)⁰)²

. By the exact same reasoning as in the proof of Lemma8we get that||N(f)⁰⁰⁰|| ≤ ||f⁻¹|| · ||f⁰⁰⁰|| · |I|². We already had||N(f)⁰⁰|| ≤ ||f⁻¹|| · ||f⁰⁰|| · |I|and becauseN(f)⁻¹=N(f⁻¹)we get the expected result.

We are now ready to prove

Proposition 12. LetV be a precompact neighbourhood of T₀ in the C³-topology. LetT ∈ V be a C³ GIET renormalisablen times. We use the following notation π_P Rⁿ(T)

= (ϕⁿ₁,· · ·, ϕⁿ_d)∈ Diff³₊([0,1])^d . Then we have for all i≤dand for all n∈N

||D(η(ϕⁿ_i))|| ≤K(sup(||T⁰⁰||,||T⁰⁰⁰||)) where K:R^∗+−→R^∗+ is a continuous function which tends to0 in0.

Proof. Again we follow the lines of the proof of Proposition9but using formulae of Lemma10. Recall thatϕⁿ_i is the renormalised of T^lin restricted to an intervalI_nⁱ such thatI_nⁱ, T(I_nⁱ), T²(I_nⁱ),· · · , T^lin−1(I_nⁱ)are disjoint.

We denote bySk the restriction ofT toT^k(I_nⁱ). We have the following properties

• ϕⁿ_i = N(S_li

n−1)◦ · · · ◦N(S1)◦N(S0)

• any partial productψ_k= N(S_li

n−1)◦ · · · ◦N(S_k)is such that ||log(ψ_k)⁰|| ≤K||T⁰⁰||(ψ_k is a diffeomor- phism of [0,1] and therefore there existsx0 ∈[0,1]such that logψ_k⁰(x0) = 0and the claim follows from Lemma5);

• same holds for partial productsφk= N(Sk)◦ · · · ◦N(S0);

• for anyk,||N(Sk)⁰⁰|| ≤ ||(T⁻¹)⁰|| · ||T⁰⁰|| · |T^k(I_nⁱ)|;

• ||(φk)⁰⁰|| ≤M⁰||(T⁻¹)⁰|| · ||T⁰⁰|| by Lemma9;

•

|D η(N(Sk))

| ≤(||T⁰⁰⁰|| · ||T⁰||+||T⁰⁰||²)· ||(T⁻¹)⁰||⁴· |T^k(I_nⁱ)|² We can now apply the third formulae of Lemma10to the productϕⁿ_i = N(S_li

n−1)◦ · · · ◦N(S₁)◦N(S₀)to get

Dη(ϕⁿ_i) =

lⁱ_n−1

X

k=0

D²logD(N(S_k))◦φ_k−1·(Dφ_k−1)²+DlogD(N(S_k))◦φ_k−1·D²φ_k−1. Recall that D²logD(N(S_k)) = Dη(N(S_k)) and η(N(S_k)) = ^N_N^(S_(S^k)00

k)⁰. Putting all the inequalities above together we get

|Dη(ϕⁿ_i)| ≤exp(K||T⁰⁰||)·X

k

(||T⁰⁰⁰|| · ||T⁰||+||T⁰⁰||²)· ||(T⁻¹)⁰||⁴· |T^k(I_nⁱ)|² +M⁰||(T⁻¹)⁰|| · ||T⁰⁰||X

k

||(N(Sk)⁻¹)⁰|| · ||(T⁻¹)⁰|| · ||T⁰⁰|| · |T^k(I_nⁱ)|.

Finally, because ||(N(Sk)⁻¹)⁰|| is uniformly controlled by _||(T^||T−1^0||)⁰||, that P

|T^k(I_nⁱ)| and P

|T^k(I_nⁱ)|² are smaller than 1and that T belongs to a bounded C³ neighbourhood ofT₀, we get the expected result.

5. Construction of a pre-stable space

This section is the heart of the article. We construct what we call a "pre-stable" space which is a submanifold of X of codimensiond−1 +g−1, satisfyinga priori bounds for the geometry of the dynamical partitions.

We now make the standing assumptionr= 3.

5.1. Notations and preliminaries. In the sequel, we place ourselves in a neighbourhoodW ofT0 for the C³-topology. Up to restricting this neighbourhood further, we can identify it with an open neighbourhood of 0in the Banach space upon whichX =A × P is modelled. In these coordinates, we will use the notation T0= (0_A,0_P)where0_P represents the point(Id,Id,· · ·,Id)∈Diff^r₊([0,1])and 0_Arepresents T0seen as an element of A. Note that

ηT₀ ≡0 therefore by restrictingW further we can assume that

∀T ∈ W, ||ηT|| ≤ for any choice of a positive (this is possible sincer= 3).

Some more notation.We then write a neighbourhood ofT₀inAas a productU × SwhereU is the subspace of unstable directions of R atT₀ andS is the subspace of stable directions. Consequently, we identify a neighbourhood ofT₀in X to a product S × U × P whereP abusively denotes (a neighbourhood of 0in) the Banach space upon which(Diff^r₊([0,1]))^d is modelled. In these coordinates, we write

R= (RA,RP) = (RS,RU,RP).

Finally, we denote byπ_A, π_S, π_U andπ_P the projection fromX ontoA,S,U andP respectively.

5.2. Action of R. Recall from Section3 thatT0 is a hyperbolic fixed point ofRrestricted toA. We collect in this paragraph important properties ofR.

(1) R(0) = 0;

(2) R(A) =A;

(3) Ris continuous;

(4) R_Ais of classC¹;

(5) 0Ais a hyperbolic fixed point ofRrestricted toA;

(6) DR_A is a bounded operator.

A difficulty that we face is that R is not smooth, it is not derivable in the P direction. It is a simple consequence of the fact that the map(ϕ, ψ)7→ϕ◦ψ

Diff^r₊([0,1])×Diff^r₊([0,1])−→Diff^r₊([0,1])

is not differentiable. To be able to perform the construction to come, we nonetheless need some control on this map.

An appropriate choice of a distance. Recall thatDiff^r₊([0,1])is Banach manifold whose tangent space at any point identifies with the Banach spaceC₀^r([0,1],R)ofC^r real-valued functions which vanish at0and1.

We endowDiff²₊([0,1])with the following distance dη(f, g) =

Z 1

0

|ηf−ηg|.

Because the tangent space at any point ofDiff²₊([0,1])identifies withC₀²([0,1],R)we can also use the formula R1

0 |η_f−η_g| to define a distance onC₀²([0,1],R). We refer to it as theη-distance. It is more refined than the C¹-norm but less than theC²-norm. We then endow P = (Diff³₊([0,1]))^d with theη-distance: precisely, if ϕ= (ϕ₁,· · ·, ϕ_d)∈ Pandψ= (ψ₁,· · ·, ψ_d)∈ P then

dη(ϕ, ψ) =

d

X

i=1

dη(ϕi, ψi).

Proposition 13. For anyδ >0 there exists a neighbourhood (for theC³-norm) ofT0 such that the restriction of RP to the P coordinates is (1 +δ)-Lipschitz, with respect todη, restricted to this neighbourhood.

The key to the proof of this proposition are the following facts Lemma 14. For any two functionf, g∈ C²(R,R), we have

(1) for any real numbera,η(a·f) =η(f);

(2) for any real numbera,η(f◦ma) =a·η(f)◦ma wherema:=x7→ax;

(3) η(f◦g) =g⁰·ηf◦g+η(g).

We leave the proof of these elementary statements to the reader. We are now ready to give the proof of Proposition 13.

Proof. Fix > 0. Let T1 and T2 be two GIETs close to T0 such that π_A(T1) =π_A(T2) . Let π_P(T1) = (ϕ¹₁,· · · , ϕ¹_d)andπ_P(T2) = (ϕ²₁,· · ·, ϕ²_d). We want to show that

dη(RP(T1),RP(T2))≤(1 +)dη(πP(T1), πP(T2))

provided T₁ andT₂ are in a sufficiently smallC²-neighbourhood ofT₀. We have the following facts (1) for alli,||(ϕ¹_i)⁰−(ϕ²_i)⁰||0≤K1dη(ϕ¹_i, ϕ²_i)for a certain constantK1;

(2) for alli,||ϕ¹_i −ϕ²_i||0≤K2dη(ϕ¹_i, ϕ²_i)for a certain constant K2;

(3) the symmetric difference of the dynamical partition associated with T1 and T2 is less than K0· sup_i||ϕ¹_i −ϕ²_i||0 where K is a uniform constant depending on the combinatorics of the dynamical partition only.

The first two facts derive from the facts thatηf= ^f_f⁰⁰0 and that we are in aC²-neighbourhood ofT0. Let us give a proof of the first fact. First we show that there existsx0such that (ϕ¹_i)⁰(x0) = (ϕ²_i)⁰(x0). This derives from the fact that if it were never the case we would have(ϕ¹_i)⁰(x)>(ϕ²_i)⁰(x)(or the opposite inequality) for allx, contradicting that the range of bothϕ1andϕ2 is[0,1]. Now for allx∈[0,1]

|logDϕ1(x)−logDϕ2(x) =| Z x

x0

η(ϕ1)−η(ϕ2)| ≤dη(ϕ¹_i, ϕ²_i)

and we get (1) because the exponential map is Lipschitz on bounded sets. The second point is proved in a similar fashion. The third fact is a consequence of the first two facts together with the hypothesis π_A(T₁) =π_A(T₂).

We now want to find an estimate of

dη(ψ1, ψ2) =X

i

Z 1

0

|η_ψ1 i −η_ψ2

i|

whereπP(R(T1)) = (ψ₁¹,· · · , ψ_d¹)and πP(R(T2)) = (ψ²₁,· · ·, ψ_d²). The strategy is to decompose this sum in order to rewrite it as a new sum of integral of difference of the form|η_φ¹

i −η_φ²

i|over the dynamical partition, neglecting the subset of[0,1] for which the the dynamical partition ofT1 differs from that ofT2. First, let us point out that because of Lemma 14, all the quantities we are dealing with are invariant by rescaling of the ϕ_i at the source and/or at the target by affine maps (ηscales by a factorawhen the source is scaled byabut the Lebesgue measure scales by _a¹ which makesR

η globally invariant). We can therefore think of theϕ^=1,2_i s as the (non-rescaled) restrictions ofT=1,2 to its branches.

If I₁¹,· · ·I_d¹ andI₁²,· · ·I_d² are the base intervals of the respective partitions associated withT1 and T2, let J_i = I_i¹∩I_i² for all i. By the facts stated above we have that the iterated images(up to times defining R) of the J_is cover all of [0,1] up to a set of measure at most K₀·d_η(ϕ¹, ϕ₂). Therefore we have that d_η(ψ¹, ψ₂)≤P

i

R

∪Ji|η_ψ¹

i −η_ψ2 i|+R

Q|ηT1|+R

Q|ηT2| whereQ= [0,1]\ ∪Ji. Eachψ¹_i (and respectively ψ_i²) is a composition of restrictions ofT₁ (respectivelyT₂) to elements of dynamical partitions. Recall that by Lemma 14we have for any two functionsf, g

η(f◦g) =g⁰·ηf◦g+η(g).

Assume for the sake of simplicity thatψ_i¹andψ²_i are obtained by composition only two restrictions ofT1 and T2. We would then have

Z

I_j

|η_ψ1 i −η_ψ2

i|= Z

I_j

|η(T₁)²−η(T₂)²|.

Injecting using the composition formula gives Z

Ij

|η_ψ1 i −η_ψ2

i|= Z

Ij

|ηT₁+DT1·ηT₁◦T1−(ηT₂+DT2·ηT₂◦T2)|

and we get Z

I_j

|η_ψ¹

i −η_ψ²

i| ≤ Z

I_j

|ηT₁−ηT₂|+

Z

I_j

DT1|ηT₁◦T1−ηT₂◦T1|+

Z

I_j

DT1|ηT₂◦T1−ηT₂◦T2|+

Z

I_j

|DT1−DT2| · |ηT₂◦T2|.

To control each term of this sum we use the following facts

• a simple change of variable givesR

IjDT₁|ηT1◦T₁−η_T2◦T₁|=R

T1(Ij)|ηT1−η_T2|;

• R

IjDT₁|η_T2◦T₁−η_T2◦T₂| ≤ ||DT₁||R

Ij||Dη_T2|| · |T₁−T₂| ≤ |I_j|K₂||DT₁||||Dη_T2||d_η(ϕ¹_j, ϕ²_j) since T₁ restricted toI_j is equal toϕ¹_j up to rescaling;

• FinallyR

I_j|DT1−DT2| · |ηT₁◦T1| ≤ ||ηT₁|| · ||DT1−DT2|| ≤ ||ηT₁||K1dη(ϕ¹_j, ϕ²_j).

Putting everything together and by taking a sufficiently smallC³-neighbourhood we get Z

Ij

|η_ψ1 i −η_ψ2

i| ≤ Z

Ij∪T1(Ij)

|η_T1−η_T2|+

dd_η(ϕ¹_j, ϕ²_j).

This reasoning directly carries over to the case whereψ_i¹ andψ²_i are obtained by a fixed but arbitrarily larger number of iterations ofT₁ andT₂. We thus obtain that

X

i

Z 1

0

|η_ψ¹

i −η_ψ2

i| ≤(1 +)X

i

Z 1

0

|η_ϕ¹

i −η_ϕ2 i| which is the expected result.

5.3. Invariant cones. We now construct a continuous family of cones in a neighbourhood ofT₀ which are invariant for the action of Ron X. Recall that we are using the distance d_η on the coordinate P. This distance induces a distance ofX²=A × P (the space of twice continuously differentiable GIETs).

• In the sequel, we restrict our attention to a neighbourhood of T₀ for theC³-topology.

• The Banach space structure ofX is induced by an identification ofX with an open subset of an affine space modelled onR^2d−2×(C₀²([0,1]))^d. Consider any norm|| · ||on A 'R^2d−2 which derives from a scalar product and makes the stable and unstable spaces inAofRatT0orthogonal and make the product withdtimes theη-distance to get a distance onR^2d−2×(C₀²([0,1]))^d.

• The neighbourhood W of T0 in X identifies canonically with a neighbourhood of 0 in R^2d−2× (C₀³([0,1]))^d. In this section we make this identification; theη-distance is thus the distance induced by theη-distance onR^2d−2×(C₀²([0,1]))^d.

• We will use coordinates (s, u, h) ∈ S × U ×(C₀²([0,1]))^d in this identification. In particular h = (h1,· · · , hd)corresponds to diffeomorphisms(Id +h1,· · ·,Id +hn)∈ P.

For anyx∈ W and anyδ >0we define the following cone

C_x^δ:={x+u+ (s+h)|u∈ U, s∈ S, h∈(C₀²([0,1]))^d and||s|| ≤δ||u||, dη(π_P(x), π_P(x) +h)≤δ||u||}.

Lemma 15 (Invariant cones). There existsλ1>1,δ >0,1>0 andα1>0 such that, up to restrictingW further we have that∀x∈ W

(1) R(C_x^δ∩Bx(1))⊂Int(C_R(x)^δ );