A smooth Gaussian-Kronecker diffeomorphism

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A smooth Gaussian-Kronecker diffeomorphism

Mostapha Benhenda

To cite this version:

Mostapha Benhenda. A smooth Gaussian-Kronecker diffeomorphism. 2013. �hal-00869084v2�

A smooth Gaussian-Kronecker di ff eomorphism

Mostapha Benhenda October 9, 2013

Abstract

We construct a smooth Gaussian-Kronecker diffeomorphismT, on”×[0,1]^Ž, where [0,1]^Žis the Hilbert cube. To obtain this diffeomorphism, we adapt a con- struction by De La Rue [6], which uses transformations of the planar Brownian motion.

1 Introduction

We define Gaussian dynamical systems. We rely on [8]. Let γ a finite symmetric measure on the circle T. A dynamical system (Ω,A,,U) is Gaussian of spectral measureγif there exists a real centred Gaussian process (Xp)p∈Zsatisfying:

1. for any integerp,Xp=X0◦U^p, 2. B(Xp,p∈Z)=A

3. for any integersp,q,…[XpXq]=R1

0 e^2iπ(q−p)tdγ(t)

Givingγdetermines all the properties of this system. In particular, Uis ergodic if and only ifγis non-atomic, and in this case,U is even weak mixing (see [5]). The entropy ofUis zero or infinite, depending on whether or notγis singular with respect to Lebesgue measure (see [7]).

We recall thatK ⊂Tis aKronecker setif any continuous function f : K → Tis the uniform limit onKof a sequence of functions f_p:x7→ j_px, where j_pis an integer.

In the case whereγis non-atomic and concentrated onK∪(¹₂ −K), whereKis a Kronecker set of [0,1/2[, we say that the Gaussian dynamical system of spectral mea- sureγis aGaussian-Kroneckertransformation [10]. A Kronecker set always has zero Lebesgue measure, so a Gaussian-Kronecker transformation always has zero entropy.

Gaussian-Kronecker transformations have remarkable properties: they have simple spectrum L^p, for any p ≥ 1 (see [5, 14]). Moreover, they satisfy theWeak Closure Theorem[17]: any-preserving transformationScommuting withT is a weak limit of powers ofT. We can find a sequence of integers jpsuch that for anyA∈ A,

(S⁻¹A∆T^−jpA)→_p→+∞0

Lastly, they satisfy the spectral stability property: any system that is spectrally isomorphic to a Gaussian-Kronecker is actually metrically isomorphic to it.

In [15, 16], Katok raised the problem of the construction of a Gaussian-Kronecker transformation that is smooth. In this paper, we give a construction of this kind:

Theorem 1.1. There exists T ∈Diff^∞(”×[0,1]^Ž,Leb)that is Gaussian-Kronecker.

Remark1.2. Moreover, as in [6], we can getT non-loosely Bernoulli. This gives an- other example of a smooth non-loosely Bernoulli diffeomorphism, besides the example we constructed in [4]. However, contrary to [4], our construction here does not easily generalize to an uncountable family of pairwise non-Kakutani equivalent diffeomor- phisms.

To show this theorem, we provide a smooth version of a transformation constructed by Thierry De La Rue [6]. He constructed a Gaussian-Kronecker and non-loosely Bernoulli transformation using a method of approximation by periodic transformations of the trajectory of the planar Brownian motion.

This adaptation to the smooth case is made using the method of approximation by successive conjugacies, introduced by Anosov and Katok in [1]. More precisely, we generalize our construction of a smooth ergodic diffeomorphism of the annulus

”×[0,1] that is equal to a rotationSα on the boundary, but that is metrically iso- morphic to a rotation Rβ of the circle, such thatα , ±β. In our generalization, we construct a sequence of smooth periodic diffeomorphisms of”×[0,1]^Ž, where [0,1]^Ž is the Hilbert cube, such that each one is metrically isomorphic to an increasingly large family of mutually independent periodic rotations of the circle with different angles.

The limit angles of these rotations will provide the Kronecker spectrum of the limit diffeomorphism.

We need to carry the construction on”×[0,1]^Ž, instead of”×[0,1], because in order to ensure the smooth convergence of the sequence of smooth periodic dif- feomorphisms, we need a space of infinite dimensions (see page 30 for details). This framework is new with respect to known Anosov-Katok constructions.

1.1 Definitions

Let Diff^∞(M, µ) be the class of smooth diffeomorphisms ofMpreserving the Lebesgue measureµ. ForB∈Diff^∞(M, µ) and j∈Ž^∗, letD^jBbe the j^thderivative ofBif j>0, and the−j^thderivative ofB⁻¹if j<0. Forx∈M, let|D^jB(x)|be the norm ofD^jB(x) atx. We denotekBk_k=max0<|j|≤kmaxx∈M|D^jB(x)|.

Afinite measurable partitionξ¯ of a measured manifold (N, ν) is the equivalence class of a finite setξof disjoint measurable subsets ofN whose union isN, modulo sets ofν-measure zero. In most of this paper, we do not distinguish a partitionξwith its equivalent class ¯ξmodulo sets ofν-measure zero. In these cases, both are denoted ξ. Moreover, all partitions considered in this paper are representatives of a finite mea- surable partition.

A partitionξ⁰issubordinateto a partitionξif any element ofξis a union of ele- ments ofξ⁰, modulo sets ofν-measure zero. In this case, ifB(ξ) denotes the completed algebra generated byξ, thenB(ξ)⊂ B(ξ⁰). The inclusion mapi: B(ξ)→ B(ξ⁰) will be denotedξ ,→ ξ⁰. This notation also means thatξ⁰issubordinatetoξ. A sequence of partitionsξnismonotonicif for anyn,ξn ,→ ξn+1. These definitions and properties are independent of the choice of the representativesξandξ⁰of the equivalence classes ξ¯and ¯ξ⁰.

A measure-preserving bijective bimeasurable mapT : (M1, µ₁,B₁)→(M2, µ₂,B₂) induces anisomorphismof measure algebras, still denotedT : (µ₁,B₁)→ (µ₂,B₂). If ξ₁, ξ₂ are partitions, and ifB₁ =B(ξ₁) andB₂ =B(ξ₂), we denoteT : ξ₁ → ξ₂this induced isomorphism of measure algebras. IfM₁ =M₂,µ₁=µ₂andB₁ =B₂, thenT

is ameasure-preserving transformation. Its induced isomorphism is anautomorphism (see [12, p.43] and [18]).

Ametric isomorphism Lof measure-preserving transformationsT1 : (M₁, µ₁,B₁)→ (M₁, µ₁,B₁),T₂: (M₂, µ₂,B₂)→(M₂, µ₂,B₂) is a measure-preserving bijective bimea- surable map L : (M₁, µ₁,B₁) → (M₂, µ₂,B₂) such that LT₁ = T₂L a.e. For con- venience, when the measure is the Lebesgue measure and the algebra is the Bore- lian algebra, we omit to mention the measures and algebras, and we simply say that L: (M1,T1)→(M2,T2) is a metric isomorphism.

Let ¯ξbe a measurable partition andξa representative of this equivalent class mod- ulo sets ofµ-measure zero. Forx∈ M, we denoteξ(x) the element of the partitionξ such thatx∈ξ(x). A sequences of partitionsξnof measurable setsgeneratesif there is a set of full measureFsuch that for anyx∈F,

{x}=F\

n≥1

ξn(x)

This property of generation is independent of the choice of the representativesξnof the equivalent class ¯ξnand therefore, we will say that the sequence of measurable par- titions ¯ξngenerates. LetM/ξdenote the equivalent class of the algebra generated byξ, modulo sets ofµ-measure zero.M/ξis independent of the choice of the representative ξof the equivalent class ¯ξ. If T : M1 → M2 is a measure-preserving map such that T(ξ1)=ξ2µ-almost everywhere, we can define a quotient map:T/ξ1:M/ξ1→M/ξ2.

LetM=”×[0,1]^Ž. We consider the periodic flowStdefined by:

St:”×[0,1]^Ž →”×[0,1]^Ž (s,x) 7→(t+s mod 1,x)

Fora,b∈”¹, let [a,b[ be the positively oriented circular sector betweenaandb, with aincluded andbexcluded.

ForI⊂”or [0,1], andi∈Ž, we denote (I)i=”×[0,1]×...×[0,1]×I×[0,1]×..., whereIis located at thei^thposition. (I⊂”ifi=0,I⊂[0,1] ifi≥1).

Fori,i⁰, we denote (I)_i×(I⁰)_i⁰=(I)_i∩(I⁰)_i⁰.

By [11, p. 157], the infinite Cartesian product of the one-dimensional Lebesgue measure defines a probability measureµonM, still called Lebesgue measure.

A sequenceT_nofµ-preserving mapsweakly convergestoT if, for any measurable setE,µ(T_nE∆E)→0, whereA∆B=(A−B)∪(B−A).

letEbe the set of bijectionsf of”×[0,1]^Ž, such thatf((x_i)i∈Ž)=(f_n(x1, ...,x_n),x_n+1, ..) with fna smooth diffeomorphism of”×[0,1]ⁿ for some integern. The smooth dis- tance on finite-dimensional smooth diffeomorphisms extends toE: d(f,g)=d(fn,gn) (we takento be the maximum of the integersn(f) andn(g) associated with f andg respectively). The completion ofEfordis denoted Diff^∞(”×[0,1]^Ž). It corresponds to the set of smooth diffeomorphisms of”×[0,1]^Ž, and extends the finite-dimensional notion of smooth diffeomorphisms.

1.2 Transformations of the planar Brownian motion

We use the representation of a Gaussian dynamical system as a geometric transforma- tion of the trajectory of the planar Brownian motion, developed in [7]. We denote by (Ω,A,) the canonical space of the planar Brownian motion issued from 0, on the time interval [0,1]:

1. Ω =C₀⁰([0,1],ƒ) is the space of continuous maps from [0,1] toC, that cancels in 0.

2. is the Wiener measure onΩ,

3. Ais the Borelian sigma-algebra, completed for.

Forω ∈Ω, and 0≤u≤1, we denote byBu(ω) the position of the trajectoryωat timeu.

Ifσis a probability measure on [0,1] concentrated in a finite number of points α1 < ... < αt of respective weightsm1, ...,mt, we define a transformationUσ of Ω, preserving, by the following: for anyk=1, ...,t,uk=Pk

j=1mj, andu0=0. We cut the trajectoryωintpieces, corresponding to time intervals [uk−1,uk],1 ≤k≤t, then we perform a rotation of angleαkon thek^th-piece. Uσωis the trajectory obtained by gluing the rotated pieces. Thus, foru∈[uj,uj+1],

Bu◦Uσ=

j

X

k=1

e^2iπαk(Buk −Buk−1)+e^2iπαj+1(Bu−Buj)

Suppose, moreover, thatα_i= ^p_qnⁿbn,i, withpnandbn,iintegers relatively prime with the integerqn. LetUn=Uσ. Let also

cn,i,l= (

ω∈Ω/ arg(Bu_i+1(ω)−Bu_i(ω))∈

"

l qn

,l+1 qn

")

andζn,i=cn,i,l,l=0, ...,qn−1 .ζn,iis a partition stable byUn, theζn,i,i=0, ...,tn− 1 are mutually independent, and (Un|ζn,i, ζn,i) is metrically isomorphic to (R^pn

qnbn,i, ξn), whereR^pn

qnbn,iis the circle rotation of angle ^p_qⁿ

nb_n,i, andξ_nis the partition of the circle” defined byξn ={h_l

q_n,^l+_q¹

n

h,l=0, ...,qn−1}. This metric isomorphism is the basis of our construction.

Now, ifσis a non-atomic probability measure on [0,1], we can define Uσas he limit of a sequence of transformationsUσ_n, where the measuresσ_n are concentrated on a finite number of points converging sufficiently well towardsσ. For anyu∈[0,1], and anyp∈š, we have:

Bu◦U_σ^p =Z u 0

e^2iπpψ(s)dBs

whereψ(s)=inf{x∈[0,1]/σ([0,x])≥s}.

Moreover, ifσ is concentrated on [0,1/2], then (Ω,A,,Uσ) is a Gaussian dy- namical system of spectral measureγ, whereγis the symmetric probability measure on [−1/2,1/2], defined by

γ(A)= 1

2(σ(A∩[0,1/2])+σ(−A∩[0,1/2])) The underlying real Gaussian process is given byX0=Re(B1) [7].

As a corollary, ifσis non-atomic on [0,1], thenUσ is ergodic (and even weak mixing), and if, moreover,σ is singular with respect to Lebesgue measure, thenUσ

has zero entropy.

1.3 Basic steps of the proof

The metric isomorphism is obtained as the limit of isomorphisms of finite algebras. We use the lemma [1, p.18]:

Lemma 1.3. Let M1and M2be Lebesgue spaces and letξ⁽ⁱ⁾_n (i=1,2) be monotonic and generating sequences of finite measurable partitions of Mi. Let T_n⁽ⁱ⁾be automorphisms of Misuch that T_n⁽ⁱ⁾ξ_n⁽ⁱ⁾ =ξ⁽ⁱ⁾_n and T_n⁽ⁱ⁾ →T⁽ⁱ⁾in the weak topology. Suppose there are metric isomorphisms Ln:M1/ξ_n⁽¹⁾→M2/ξ⁽²⁾_n such that

LnT_n⁽¹⁾/ξ⁽¹⁾_n =T_n⁽²⁾/ξ⁽²⁾_n Ln

and

Ln+1ξ⁽¹⁾_n =ξ⁽²⁾_n then(M₁,T1)and(M₂,T2)are metrically isomorphic.

Said otherwise, if we have generating sequences of partitions and sequences of automorphisms T_n⁽ⁱ⁾ weakly converging towards T⁽ⁱ⁾, and if, for any integer n, the fol- lowing diagram commutes:

ξ⁽¹⁾_n

T_n⁽¹⁾ 55 ^Ln //

 _

ξ_n⁽²⁾ _ ii ^Tn⁽²⁾

ξ⁽¹⁾_n+1 ^Ln+1 //ξ⁽²⁾_n+1

then(M₁,T₁)and(M₂,T₂)are metrically isomorphic.

The proof of theorem 1.1 is in two steps. In the first step (lemma 1.4), we determine sufficient conditions such that there exists sequences of finite partitions and automor- phisms satisfying the assumptions of lemma 1.3 withM1= Ω,M2=M=”×[0,1]^Ž, T_n⁽¹⁾ =Un,T_n⁽²⁾ =Tn, whereTnis a smooth diffeomorphism, and such that the limitT in the smooth topology of the sequenceTnis smooth.

In the second step (lemma 1.5), we construct sequences of integers satisfying the conditions of the first step.

Lemma 1.4. There exists an explicit family of integers R1(n) ≥ n,R2,R3, such that, if there exist increasing sequences of integers tn,pn,qn,an(i⁰/tn),bn(i⁰/tn) ∈ Ž^∗, i⁰ = 0, ...,tn−1, and sequences sn(i⁰/tn)∈š^∗, i⁰=0, ...,tn−1such that, for any integer n, any i⁰=0, ...,tn−1,

1. (temporal monotonicity) tn+1=2^R2(n,qn,tn)tn. 2. (primality) a_n(i⁰/t_n)b_n(i⁰/t_n)−s_n(i⁰/t_n)q_n =1.

3. (monotonicity) qndivides qn+1.

4. (isomorphism) qndivides an+1(i/tn+1)−an(i⁰/tn), for i=i^0tn+1_t

n , ...,(i⁰+1)^tn+1_t

n −1.

5. (convergence of the diffeomorphism, generation, Kronecker) 0<

pn+1

q_n+1 − pn

q_n

≤1/R1









 n,qn,

t_n+1−1

Y

i=0

bn+1(i/tn+1)











then there exists a smooth ergodic measure-preserving diffeomorphism T of M that is a Gaussian transformation. Moreover, if

6. (Kronecker) the translation of”^tnof vector ^p_qⁿ

n(b_n(0/t_n), ...,b_n((t_n−1)/t_n))has a fundamental domain of diameter smaller than1/n,

then T is a Gaussian-Kronecker.

Lemma 1.5. For any n≥1, there exists increasing sequences of integers

tn,pn,qn,an(i⁰/tn),bn(i⁰/tn) ∈ Ž^∗, i⁰ = 0, ...,tn−1, and sequences sn(i⁰/tn) ∈ š^∗, i⁰=0, ...,tn−1satisfying the assumptions of lemma 1.4.

We divide the proof of lemma 1.4 in three main parts. In the first part of the proof, we construct a sequence of monotonic and generating partitions ofΩ, calledζ_n^∞, which is stabilized by the transformationUn. To that end, we use assumptions 1, 2, 3, 4 and 5.

In the second part of the proof, we elaborate sufficient conditions onBn∈Diff^∞(M, µ), so that ifTn = B⁻¹_n S^pn

qnBn weakly converges towards an automorphismT, then there exists a metric isomorphism between (Ω,U,) and (M,T, µ). To that end, we apply lemma 1.3: we construct a monotonous and generating sequence of partitionsξ_n^∞of M and a sequence of isomorphisms ¯K_n^∞ : Ω/ζ_n^∞ → M/ξ^∞_n, such that ¯K_n^∞Un =TnK¯_n^∞ and ¯K_n^∞_+1|ζ∞

n =K¯_n^∞. In the construction of this isomorphism, assumption 4 is important.

Moreover, elements ofξ_n^∞must be chosen in a way that ensures the monotonicity of the sequence ¯K_n^∞. This condition of monotonicity induces combinatorial constraints on the elements of the partitionξ_n^∞.

In the third part of the proof, we construct diffeomorphismsTn=B⁻¹_n S^pn

qnBnonM

stabilizingξ^∞_n, obtained by successive conjugations from the rotationS^pn

qn. In particular, in this part, we use smooth quasi-permutations, introduced in [4].

1.4 Construction of suitable sequences of integers: proof of lemma 1.5

Lemma 1.6. There exist increasing sequences of integers tn,pn,qn,an(i⁰/t_n),bn(i⁰/t_n)∈

Ž^∗, i⁰ = 0, ...,tn −1, and sequences sn(i⁰/t_n) ∈ š^∗, i⁰ = 0, ...,tn−1, satisfying the assumptions of lemma 1.4.

Proof. We construct these sequences by induction. Lett₀ = p₀ =q₀ =1,b₀(0/t₀)= s₀(0/t₀)=1,a₀(0/t₀)=2. Suppose we have definedt_k,p_k,q_k,a_k(i⁰/t_k),b_k(i⁰/t_k),s_k(i⁰/t_k), satisfying the assumptions of lemma 1.4, up to the rank k = n, and let us define tn+1,pn+1,qn+1,an+1(i/tn+1),bn+1(i/tn+1),sn+1(i/tn+1).

We can definetn+1=2^R2(n,qn,tn)tn. Let

b(n)=(bn(0/tn), ...,bn((tn−1)/tn))

b(n)˜ =(bn(0/tn), ...,bn(0/tn),bn(1/tn), ...,bn(1/tn), ...,bn((tn−1)/tn), ...,bn((tn−1)/tn))

=(˜bn(0/t_n+1), ...,b˜n((tn+1−1)/t_n+1)) where eachbn(i/t_n) is repeatedtn+1/t_ntimes.

To get conditions 5, 6, as in [1, 3], we seekb(n+1) of the form:

b(n+1)=qnv(n+1)+(e_n+1qn+1)˜b(n) wherev(n+1)∈š^tn+1ande_n+1∈Ž.

Letb_n=Qtn−1

i⁰=0b_n(i⁰/t_n). Since gcd({b˜_n(0/t_n+1), ...,b˜_n((t_n+1−1)/t_n+1)}=gcd({b_n(0/t_n), ...,b_n((t_n− 1)/t_n)})=1, then we can apply a result in [3, section 1.3.2]: there existsv(n+1)∈š^tn+1

ande_n+1∈Ž, there existsR4(n,b(n)) such that:

kv(n+1)k ≤R4(n,bn) en+1≤R4(n,bn)

and such that the translation flow of”^tn+1 of vector b(n+1) has a fundamental domain B(n) ⊂ ”^tn+1−1 × {0} of diameter smaller than 1/(2n). Therefore, for any qn+1 ≥ R5(n,b(n)) for someR5(n,b(n)), and any pn+1 such that gcd(pn+1,qn+1) = 1, the translation of”^tn+1 of vector ^p_qⁿ⁺¹

n+1b(n+1) has a fundamental domain of diameter smaller than 1/n. Hence condition 6.

We write:

v(n+1)=(v_n+1(0/t_n+1), ...,v_n+1((t_n+1−1)/t_n+1)) Let

s_n+1(i/t_n+1)=s_n(i⁰/t_n)+a_n(i⁰/t_n)[v_n+1(i/t_n+1)+e_n+1b_n(i⁰/t_n)]

Let

µn+1(i/tn+1)=bn(i⁰/tn)+qn[vn+1(i/tn+1)+en+1bn(i⁰/tn)]

Letdn+1≥R1(n,qn,bn+1) be an integer, let cn+1(i/t_n+1)=dn+1sn+1(i/t_n+1)

t_n+1−1

Y

k=0,k,i

µ_n+1(k/t_n+1) and let

qn+1=qn









 1+dn+1

t_n+1−1

Y

k=0

µn+1(k/tn+1)











Notice thatqn+1is independent ofi(it is important to obtain condition 2). Let an+1(i/tn+1)=an(i⁰/tn)+qncn+1(i/tn+1)

Thus, assumption 4 is also satisfied.

We show assumption 2 at rankn+1. We have:

an+1(i/t_n+1)b_n+1(i/t_n+1)= an(i⁰/t_n)+qncn+1(i/t_n+1)

qnvn+1(i/t_n+1)+(e_n+1qn+1)b_n(i⁰/t_n)

=1+q_ns_n(i⁰/t_n)+e_n+1(1+s_n(i⁰/t_n)q_n)+a_n(i⁰/t_n)v_n+1(i/t_n+1) +c_n+1(i/t_n+1) q_nv_n+1(i/t_n+1)+(e_n+1q_n+1)b_n(i⁰/t_n)

=1+qnsn(i⁰/tn)+an(i⁰/tn)(en+1bn(i⁰/tn)+vn+1(i/tn+1))

+dn+1sn+1(i/tn+1)

t_n+1−1

Y

k=0,k,i

µn+1(k/tn+1) qnvn+1(i/tn+1)+(en+1qn+1)bn(i⁰/tn)











=1+qn











sn+1(i/tn+1)









 1+dn+1

t_n+1−1

Y

k=0,k,i

µn+1(k/tn+1)





















=1+s_n+1(i/t_n+1)q_n+1

Hence assumption 2 at rankn+1.

1.4.1 Proof thatTis Kronecker First, we show:

Lemma 1.7. Let Ln

i⁰ tn

!

="

pn

qn

bn(i⁰/t_n),pn

qn

bn(i⁰/t_n)+ 1 nqn

#

and

L=\

n≥1 t_n−1

[

i⁰=0

Ln(i⁰/tn) If the translation ^p_qⁿ

nb(n)of the ”^tn-torus has a fundamental domain of diameter smaller than1/n, then L is a Kronecker set.

Proof. We adapt the proof of [6, p.6]. We must show that for anyf :L→”continu- ous, for any >0, there existsk∈šsuch that

sup

x∈L

|f(x)−kx| ≤

We fixn ≥1 and first, we suppose that f is constant on each intervalL_ni0

tn

, and we denotezn(i⁰/t_n) = f

Ln

_i0

tn

. Let zn(i⁰/t_n) =(zn(0), ...,zn((tn−1)/t_n)). Since the translation ^p_qⁿ

nb(n) of the”^tn-torus has a fundamental domain of diameter smaller than 1/n, then there exists 0≤k≤qn−1 such that

|kp_n qn

b(n)−u| ≤ 1 n Therefore, for any 0≤i⁰≤qn−1,

|kpn

qn

b_n(i⁰/t_n)−z_n(i⁰/t_n)| ≤ 1 n Letx∈Land 0≤i⁰≤tn−1 such thatx∈Ln

_i0

t_n

. We have:

x− p_n qn

b_n(i⁰/t_n)

≤ 1 nqn

Therefore,

|kx−f(x)|=|kx−zn(i⁰/tn)| ≤

kx−kpn

qn

bn(i⁰/tn) +

kpn

qn

bn(i⁰/tn)−zn(i⁰/tn)

≤ k nqn +1

n

|kx−f(x)| ≤2

n (1)

Now, we suppose f : L→ ”is any continuous map. Let >0. Lis compact, so by Heine’s theorem, f is uniformly continuous. There existsη > 0 such that for any x,y∈L, if|x−y| ≤η, we have:

|f(x)−f(y)| ≤/2 (2)

We fixn ≥max (4/,1/η). Let fn :L →”such that for anyi⁰=0, ...,tn−1, any y∈Ln(i⁰/tn), fn(y)= f_pn

q_nbn(i⁰/tn)

. By relation (1), there exists 0≤k≤qnsuch that for anyx∈L,

|kx−fn(x)| ≤2 n

Letx ∈ Land leti⁰ such thatx∈ Ln(i⁰/tn). Since fn is constant onLn(i⁰/tn), we have:

|kx− f(x)| ≤ |kx−fn(x)|+|fn(x)−f(x)|=|kx−fn(x)|+|fn(x)−f(x)|

|kx−f(x)| ≤2 n +

f p_n qn

bn(i⁰/t_n)

!

−f(x) Since

|x− p_n qn

bn(i⁰/t_n)| ≤ 1 nqn

≤ 1 n ≤η By estimation (2), we conclude:

|kx−f(x)| ≤ 2

n+/2≤

Corollary 1.8. For a suitable choice of R₁in lemma 1.4, T is Gaussian-Kronecker.

Proof. We must show that the limitσofσnin the weak topology is non-atomic, and that supp(σ) ⊂ L. Let > 0. Sincetn → +∞, then fornsufficiently large, for any x∈”, sup_x∈”σn(x)≤. Therefore,σis non-atomic.n To show that the support ofσ is included inL, let:

f : {0, ...,tn+1−1} → {0, ...,tn−1}

i 7→ i⁰s.t. _tⁱ⁰

n ≤ ⁱ

t_n+1 <ⁱ⁰_t⁺¹

n

Letn∈Ž. For anyi∈ {0, ...,tn+1−1}, we have:

pn+1

qn+1

b_n+1(i/t_n+1)−pn

qn

b_n(f(i)/t_n)= pn+1

qn+1

− pn

qn

!

b_n+1(i/t_n+1)

Therefore, for anym≥0, anyi∈ {0, ...,tn+m−1}, and for a suitable choice ofR1in lemma 1.4,

pn+m

qn+m

bn+m(i/tn+m)− pn

qn

bn(f^m(i)/tn)=

m−1

X

k=0

pn+k+1

qn+k+1

−pn+k

qn+k

!

bn+k+1(f^k(i)/tn+k+1)

≤

+∞

X

k=n

p_k+1 qk+1

− p_k qk

!

bk+1(f^k(i)/t_n+1)≤ 1 nqn

Therefore, for any integersn,m,_q^pn+m

n+mbn+m(i/t_n+m)∈Ln=Stn−1

i⁰=0Ln(i⁰/t_n).

Therefore,supp(σ_n+m)⊂Ln. SinceLnis closed, thensupp(σ)⊂Lnfor anyn, and sosupp(σ)⊂L.

2 Partitions of Ω = C

([0, 1], ƒ)

The aim of this section is to show the following proposition:

Proposition 2.1. If assumptions 1, 2, 3, 4, 5, of lemma 1.4 hold, there exists measurable partitions(ζ_n^m)n≥0,n<mofΩ, such thatζ_n^mis stable by Un, and such that at m fixed, for n <m,ζ_n^m ,→ ζ_n^m₊₁. Moreover, there exists an isomorphism Q^m_n :ζn → ζ_n^mcommuting with Un.

Moreover, at n fixed,ζ_n^mconverges as m→ +∞towards a partitionζ_n^∞, stable by Un. Moreover, the sequence(ζ^∞_n)n≥0is monotonous and generates.

A natural partition stabilized byUnis given byζ_n=∨^tn−1

i=0 ζ_n,i, where:

ζ_n,i=c_n,i,l,l=0, ...,q_n−1 with

cn,i,l=(

ω∈Ω/arg(Bu_i+1(ω)−Bu_i(ω))∈

"

l qn

,l+1 qn

")

In order to apply lemma 1.3, we need a monotonous and generating sequence of partitions. In lemma 2.13, we show thatζ_ngenerates. However,ζ_nis not monotonous, because for any integersk,i,l,l⁰,

ω∈Ω/ arg

Bk

tn+_tnⁱ⁺¹₊₁(ω)−Bk

tn+_tn+1ⁱ (ω)

∈I l qn+1

1

ω∈Ω/ arg

Bk+1

tn (ω)−Bk tn(ω)

∈Il0 qn

Therefore, as in [1, 3], we "monotonize"ζn. This "monotonization" is performed in two steps: in the first step, we construct a partition ζ_nⁿ⁺¹ stable byUn and such thatζ_nⁿ⁺¹ ,→ ζn+1 (lemma 2.2). We also need that most elements ofζ_nⁿ⁺¹ have a size controlled independently ofqn+1, to ensure the smooth convergence ofTntowardsT.

In the second step, we iterate this procedure, so as to obtain a partitionζ_n^m, such thatζ_n^m,→ ζm. Atmfixed, forn<m,ζ_n^m,→ζ^m_n+1. We take the limitm→+∞. This gives a monotonous partitionζ_n^∞stable byUn(corollary 2.12). This procedure is the same as in [1, 3]. Sinceζ_ngenerates (lemma 2.13), thenζ_n^∞too (corollary 2.15).

To show proposition 2.1, we need some notations and auxiliary transformations.

We slightly modify those introduced by De La Rue [8, p. 395].

Letθ=argB1, and letG,Dtransformations ofΩdefined by Bu◦G= √

2B^u

2

B_u◦D= √ 2(Bu+1

2

−B1 2)

Since, by scaling, Bu ◦G andBu ◦Dare Brownian motions, thenG andD are

-preserving. Moreover,G⁻¹(A) andD⁻¹(A) are independent.

LetΓ =G,D, and for anyn∈Ž,Γⁿis the set of words of lengthnon the alphabet Γ. LetθG=θ◦G,θD=θ◦D,BG=B1◦G,BD=B1◦D, and having defined the random variablesθWandBWforW ∈Γⁿ, we letθGW=θW◦G,θDW =θW◦D,BGW=BW◦G, BDW =BW◦D. Thus, we obtain families of random variablesθW andBW defined by induction.

Forn∈Ž, we denoteA_n=B(θW,W ∈Γⁿ) andB_n =∨ⁿ_k=0A_k,B_∞=∨⁺_k=^∞₀A_k. For any finite measurable partitionN ={N1, ...,Nq},q∈Ž^∗, let

cn

i 2ⁿ,Nl

=n

ω∈Ω/arg Bi+1

2n(ω)−Bi 2n(ω)

∈Nl

o

and letA_n(N) = {cn

_i

2ⁿ,Nl

,i = 0, ...,2ⁿ−1,l = 0, ...,q−1}. In particular, if Nl=[l/q,(l+1)/q[, let

cn

i 2ⁿ, l

q

!

= (

ω∈Ω/arg Bi+1

2n(ω)−Bi 2n(ω)

∈

"

l q,l+1

q

")

and letA_n(q)={cn

_i

2ⁿ,_q^l

,i=0, ...,2ⁿ−1,l=0, ...,q−1},B_n(q)=∨ⁿ

k=0A_k(q).

Lett_n⁰such thattn=2^t0n. Let also φ: Γ^t0n → n _k

2^t0n,k=0, ...,2^t0n−1o a1...at⁰_n 7→ P^tn⁰

i=1 1_ai=D 2^t0n−(i+1)

Let

A_W,p(q)=B arg

B_φ(W)+ j+1 2p+|W|

−B_φ(W)+ j 2p+|W|

∈

"

l q,l+1

q

"

,l=0, ...,q−1,j=0, ...,2^p−1

!

B_W,p(q)=∨ⁿ_k=0A_W,k(q) In particular, we have:

A_p+1(q)=A_G,p(q)∨ A_D,p(q) The main step in the proof is the following proposition:

Proposition 2.2. If assumptions1,2,3,4,5, of lemma 1.4 hold, then for any n≥0, there exists a measurable partitionζ_nⁿ⁺¹,→ζ_n+1ofΩ, stable by U_n, there exists Qⁿ_n⁺¹ : ζn →ζ_nⁿ⁺¹isomorphism such that:

1. For any c∈ζn,(c∆Qⁿ_n⁺¹c)≤ ¹

2ⁿq^tnn. 2. Qⁿ_n⁺¹Un=UnQⁿ_n⁺¹.

3. Elements ofζ_nⁿ⁺¹are given by relation (5).

Proof of proposition 2.2. First, we need the lemma:

Lemma 2.3. For any p≥0,q>0,-almost surely, …h

BW|A_W,p(q)i

=…h

BW|B_W,p(q)i Proof. The wordWis fixed. Foru∈[0,1], letB⁰_u= √

2^|W|(B ^u

2|W|+φ(W)−Bφ(W)). By scal- ing,B⁰_uis another Brownian motion issued from 0, and we have: B⁰₁ = √

2^|W|B_W. Let θ⁰=argB⁰₁, and forn∈Ž, we denoteA⁰_n(q)=B

θ⁰_W0 ∈[l/q,(l+1)/q[,l=0, ...,q−1,W⁰∈Γⁿ andB⁰_n(q)=∨ⁿ_k=0A⁰_k(q). We haveA_W,p =A⁰_p,B_W,p =B⁰_p. Therefore, it suffices to

show,-almost surely:

…h

B⁰₁|A⁰_p(q)i

=…h

B⁰₁|B⁰_p(q)i We show that…h

B1|A_p(q)i

=…h

B1|B_p(q)i

-almost surely. Let up=…

…[B1|B_p(q)]−…[B1|A_p(q)]

2

We show that for any integerp ≥0,up =0. Since=A₀(q)=B₀(q), thenu0=0.

It suffices to show thatup+1 =up. We follow the method of the proof of relation (15) in [8, p.397]. We have:

up+1=…

…h B¹

2

|B_p+1(q)i

−…h B¹

2

|A_p+1(q)i +…h

B1−B¹

2

|B_p+1(q)i

−…h B1−B¹

2

|A_p+1(q)i

2

up+1=…

"

1 2 …h

B1◦G|B_p+1(q)i

−…h

B1◦G|A_p+1(q)i +…h

B1◦D|B_p+1(q)i

−…h

B1◦D|A_p+1(q)i

2#

For any complex numbersa,b,|a+b|² =|a|²+b²+ab¯+ab, where ¯¯ zdenotes the conjugate of the complex numberz. Therefore,

up+1= 1 2…

…h

B1◦G|Bp+1(q)i

−…h

B1◦G|Ap+1(q)i

2+ …h

B1◦D|Bp+1(q)i

−…h

B1◦D|Ap+1(q)i

2

+1 2…

…h

B₁◦G|B_p+1(q)i

−…h

B₁◦G|A_p+1(q)i …h

B₁◦D|B_p+1(q)i

−…h

B₁◦D|A_p+1(q)i

+1 2…

…h

B1◦G|B_p+1(q)i

−…h

B1◦G|A_p+1(q)i …h

B1◦D|B_p+1(q)i

−…h

B1◦D|A_p+1(q)i Now, we apply the claim:

Claim 2.4. Let X a random variable on(Ω,A,), andG,Hsigma-algebras such that B(X,H)andGare independent. We have, a.e.:

…X|G ∨ H=…X|H

Proof. We reproduce the proof of [13]. We need to show that …[X| B(G ∪ H)]=…[X| H],

that is, we need to show that…[X | H] can serve as the conditional expectation ofX givenB(G ∪ H), i.e. show that

• …[X| H] isB(G ∪ H)-measurable,

• …[X| H] is integrable,

• R

A…[X| H] d=R

AXdfor allA∈ B(G ∪ H).

The first two are obvious. For the third, let us note that (note that by linearity, we can assume thatXis non-negative)

B(G ∪ H)3A7→

Z

A

…[X| H] d

and

B(G ∪ H)3A7→

Z

A

Xd

are two measures defined onB(G ∪ H) with equal total mass being…[X]. Hence, it is enough to show that the two measures are identical on some∩-stable generator of B(G ∪ H). Here, we use that

{A∩B|A∈ G,B∈ H }

is indeed a∩-stable generator ofB(G ∪ H). Therefore, it suffices to show that Z

A∩B

…[X| H] d=Z

A∩B

Xd, A∈ G,B∈ H This is true, because sinceGandH are independent,

…1A∩B…[X| H]=…1A1B…[X| H]=…[1A]…[1B…[X| H]]

By the defining property of conditional expectation,…[1B…[X | H]] = …[1BX].

Moreover, 1BX∈ B(X,H), which is independent ofGby assumption. Therefore, …1A∩B…[X| H]=…[1_A]…[1_BX]=…[1_A1_BX]=Z

A∩B

Xd

We have thatA_p+1(q) = A_G,p(q)∨ A_D,p(q) and B_p+1(q) = B_G,p(q)∨ B_D,p(q).

Moreover, the sigma-algebraB(B_G,B_G,p(q)) is independent ofB_D,p(q).

Indeed,B(B_G,B_G,p(q))⊂ B(Bi+1 2p −Bi

2p,i=0, ...,2^p−1−1) and B_D,p(q)⊂ B(Bi+1

2p −Bi

2p,i=2^p−1, ...,2^p−1), and these two sigma-algebras are in- dependent, because the Brownian motion has mutually independent increments. Like- wise,B(BD,B_D,p(q)) is independent ofB_G,p(q). Therefore, by claim 2.4, we get:

u_p+1= 1 2…

…h

B₁◦G|B_G,p(q)i

−…h

B₁◦G|A_G,p(q)i

2

+ …h

B1◦D|BD,p(q)i

−…h

B1◦D|AD,p(q)i

2

+1 2…

…h

B₁◦G|B_G,p(q)i

−…h

B₁◦G|A_G,p(q)i …h

B₁◦D|B_D,p(q)i

−…h

B₁◦D|A_D,p(q)i

+1 2…

…h

B1◦G|BG,p(q)i

−…h

B1◦G|AG,p(q)i …h

B1◦D|BG,p(q)i

−…h

B1◦D|AD,p(q)i SinceB_G,p(q) andB_D,p(q) are independent, we get:

up+1= 1 2…

…h

B1◦G|BG,p(q)i

−…h

B1◦G|AG,p(q)i

2

+ …h

B1◦D|BD,p(q)i

−…h

B1◦D|AD,p(q)i

2

SinceA_G,p(q) = G⁻¹(A_p(q)) andB_G,p = G⁻¹(B_p(q)), and since G is measure- preserving, we obtain:

… …h

B₁◦G|B_G,p(q)i

−…h

B₁◦G|A_G,p(q)i

2

=…

…h

B1◦G|G⁻¹(B_p(q))i

−…h

B1◦G|G⁻¹(A_p(q))i

2

=… …h

B₁|B_p(q)i

−…h

B₁|A_p(q)i

2

◦G =…

…h

B₁|B_p(q)i

−…h

B₁|A_p(q)i

2

=u_p SinceDis also measure-preserving, we can do the same for the right-hand side of the equation and therefore,u_p+1=u_p.

Lemma 2.5. Let j_k→+∞and r_k→+∞two sequences of positive integers. We have:

+∞

_

k=0

B_jk(rk)=B_∞

Proof. Sincerk→+∞, then for anyt≥0,u≥0,A_t=W₊∞ k=uA_t(rk).

Since jk→ +∞, there existsk0(t) such that for anyk≥k0, jk ≥t. Moreover, for anyt≤ jk,A_t(rk)⊂ B_jk(rk). Therefore,

A_t= ⁺

∞

_

k=k₀

A_t(rk)⊂

+∞

_

k=k₀

B_jk(rk)⊂

+∞

_

k=0

B_jk(rk) Therefore,

B_∞=⁺

∞

_

t=0

A_t⊂

+∞

_

k=0

B_jk(r_k)