Exponential decay of correlations for a real-valued dynamical system generated by a k dimensional system

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Exponential decay of correlations for a real-valued dynamical system generated by a k dimensional system

Jager Lisette, Maes Jules, Ninet Alain

To cite this version:

Jager Lisette, Maes Jules, Ninet Alain. Exponential decay of correlations for a real-valued dynamical system generated by a k dimensional system. Acta Applicandae Mathematicae, Springer Verlag, 2018.

�hal-01881754�

Exponential decay of correlations for a real-valued dynamical system generated by a k dimensional system

JAGER Lisette^∗, MAES Jules, NINET Alain 8 novembre 2017

Abstract

As a first step towards modelling real time series, we study a class of real, bounded-variables pro- cesses {X_n, n∈N} defined by a k-term recurrence relation X_n+k =ϕ(Xn, . . . , Xn+k−1). These processes are noise-free. We immerse such a dynamical system into R^k in a slightly distorted way, which allows us to apply the multidimensional techniques introduced by Saussol [SAU] for deterministic transformations.

The hypotheses we need are, most of them, purely analytic and consist in estimates satisfied by the function ϕ and by products of its first-order partial derivatives. They ensure that the induced transformationT is dilating.

Under these conditions, T admits a greatest absolutely continuous invariant measure (ACIM).

This implies the existence of an invariant density forX_n, satisfying integral compatibility condi- tions. Moreover, ifT is mixing, one obtains the exponential decay of correlations.

2010 Mathematical Subject Classification : 37C40.

Keywords and phrases : ACIM, dynamical systems, decay of correlations, dilating transforms.

Address : Laboratoire de Math´ematiques, FR CNRS 3399, EA 4535, Universit´e de Reims Champagne-Ardenne, Moulin de la Housse, B. P. 1039, F-51687, Reims, France

1 Introduction

In this work, we are concerned with a deterministick≥2 terms induction rather than with the more classical study of a dynamical systemx0, ϕ(x0), . . . , ϕⁿ(x0), . . .. We thus write the model in a probabilistic manner, X_n+k = ϕ(X_n, . . . , Xn+k−1), where X₀, . . . , Xk−1 are real random variables. We aim at proving, for this model, the exponential decay of correlations, principally under analytic assumptions about the partial derivatives of the function ϕ.

The decay of correlations has been treated by many authors for the classical model. Among the most recents works we may refer to the works of Alves, Freitas, Luzzatto, Vaienti [AFLV], Gou¨ezel [GO], Sarig [SAR], Young [YOU] and Saussol [SAU]. We shall use, most particularly, the results of the last author.

In a first paper [JMN], we have proved the result for a two-terms induction, by studying a system imbedded into R². We adopt here the same method and imbed our system intoR^k, in

∗Corresponding author : lisette.jager@univ-reims.fr, 03 26 91 83 92

a non-canonical way and we introduce a transformation T :Z 7→ T(Z), defined on a compact subset of R^k. But the hypotheses which allowed us to conclude in the 2-terms case are not relevant for a general k. Indeed, the proof of the result requires to locate very precisely the eigenvalues of a realkdimensional matrix, which is quite easy whenk= 2 and much less so for higher orders. We only consider here ordersk greater than or equal to 3.

We first obtain (Theorem 2-4.) the existence of a greatest absolute continuous invariant measure (ACIM)µ for the transformation T. If T is mixing, one obtains (Theorem 2-7.) that, for well-chosen applications f and g, there exist constants C = C(f, h) >0 and ρ ∈]0,1[ such

that :

Z

Ω

f◦Tⁿh dµ− Z

Ω

f dµ Z

Ω

hdµ

6C ρⁿ.

As a consequence, if a given R^k-valued random variable Z0 has distribution µ (the ACIM), ifF and H are convenient real valued functions, one gets (Theorem 5 for r=s= 1) :

|Cov(F(X_n), H(X₀) )|6C ρⁿ.

The corresponding inequalities are more complicated when T is not mixing (Theorem 2-6., Lemma 4).

Let us note, too, the existence of integral identities satisfied by the invariant measure (Theo- rem 3).

We apply our results to a non linear example (Section 3).

2 Hypotheses and results

Let L ∈ R^∗+ and let us consider an application ϕ : [−L, L]^k → [−L, L], defined piecewise on [−L, L]^k.

Under conjugation by an affine function, similar results could be obtained for an applicationϕ defined on [a, b]^k, with values in [a, b]. Recall thatk≥3.

Suppose that all following conditions are fulfilled : 1. There existsd∈N^∗ such that

[−L, L]^k=

d

[

j=1

O_j ∪ N,

where the Oj are nonempty open subsets, N is Lebesgue negligible and the union is disjoint. The boundary of each Oj is contained in a compact,C¹, (k−1)−dimensional submanifold ofR^k.

2. There existsε1 >0 such that, for every j∈ {1, . . . , d}, there exists a map ϕj defined on B_ε1(O_j) ={(x₁, . . . , x_k)∈R^k, d((x₁, . . . , , x_k), O_j)≤ε₁} with values in Rand satisfying ϕj|_Oj =ϕ|_Oj.

3. The application ϕj is bounded and C^1,α on Bε1(Oj) for an α ∈]0,1]¹, which means that ϕ_j is C¹ and that there exists C_j >0 such that, for all (u₁, . . . , u_k),(v₁, . . . , v_k) in B_ε1(O_j), alli∈ {1, . . . , k}:

∂ϕ_j

∂xi

(u1, . . . , uk)−∂ϕ_j

∂xi

(v1, . . . , vk)

≤Cj||(u₁, . . . , uk)−(v1, . . . , vk)||^α. 4. The maximal number ofC¹ arcs ofN crossing isY ∈N^∗. Moreover, one sets

σ >1

1. IfϕjisC² onBε1(Oj), it is necessarilyC^1,αonBε1(Oj) withα= 1

and imposes that

η:=

1

√σ α

+ 4

√σ−1Yγk−1

γ_k <1 whereγ_k = π^k/2

Γ(^k₂ + 1) is the volume of the unit sphere ofR^k. 5. LetA >1 and σ >1 satisfy A^2/k> σ. Set

γ =A^−1/kandM₀(σ, A) =

−(k−1)γ^k−1+q

(k−1)²γ^2k−2+ 4(k−2)γ^2k+1(_γ¹2 −σ)

2(k−2)γ^2k+1 >0.

LetM ∈ ]0, M0(σ, A)[.

Assume that, for all 2≤i≤k, allj ≤dand all (u1, . . . , u_k)∈Bε1(Oj) :

∂ϕ_j

∂x1

(u₁, . . . , u_k)

≥A,

∂ϕ_j

∂x1

(u₁, . . . , u_k)×∂ϕ_j

∂xi

(u₁, . . . , u_k)

≤M.

(These very tight conditions are due to the loss of precision in the localization of the eigenvalues of the matrixB - see (8) - in the case whenk >2.)

6. The sets O_j satisfy the following geometrical condition :² for all (u₁, u₂, . . . , u_k) and (v1, u2, . . . , uk) in Bε1(Oj), there exists a C¹ path Γ = (Γ1, . . . ,Γk) : [0,1] → Bε1(Oj) between (u₁, u₂, . . . , u_k) and (v₁, u₂, . . . , u_k), with nonzero gradient satisfying

∀t∈]0,1[, Γ⁰₁(t)

> M A²

k

X

i=2

Γ⁰_i(t) .

For everyj∈ {1, ..., d}, one denotes byUj (resp.Wj,N⁰) the image ofOj (resp.Bε1(Oj),N) un- der the transformation which associates with (u₁, . . . , u_k)∈R^k the point (u₁, γu₂, . . . , γ^k−1u_k).

The set Ω = [−L, L]×[−γL, γL]×. . .×[−γ^k−1L, γ^k−1L], with which we shall work, is the image of [−L, L]^k under the same transformation.

For every non-negligible Borel setS of R^k, for every f ∈L¹_m(R^k,R), one sets : Osc(f, S) =Esup

S

f−Einf

S

f whereEsup

S

etEinf

S

are the essential supremum and infimum onSwith respect to the Lebesgue measure m.

One then defines the normk · k_α by

|f|_α = sup

0<ε<ε1

ε^−α Z

R^k

Osc(f, Bε(x1, . . . , x_k))dx1. . . dx_k , kfk_α=kfk_L1

m+|f|_α and the setV_α={f ∈L¹_m(R^k,R), kfk_α<+∞}.

We introduce similar notions on Ω : for every 0 < ε0 < γ^k−1L, for every g ∈ L^∞_m(Ω,R), we define :

N(g, α, L) = sup

0<ε<ε0

ε^−α Z

Ω

Osc(g, Bε(x1, . . . , xk)∩Ω) dx1. . . dxk. One then sets :

||g||_α,L =N(g, α, L) + 2K(Ω)ε^1−α₀ ||g||∞+||g||_L1 m

2. In favorable cases, the geometrical hypothesis can be replaced by the following one, stronger but much sim- pler : for all points (u1, u2, . . . , uk) and (v1, u2, . . . , uk) inBε1(Oj), the segment [(u1, u2, . . . , uk),(v1, u2, . . . , uk)]

is contained inBε₁(Oj)

whereK(Ω) = 2^k+2(

k

P

i=1

2γⁱ⁻¹L)^k−1 = 2^2k+1L^k−1(^1−γ_1−γ^k)^k−1.

The functiong is said to belong to Vα(Ω) if this expression is finite. This set does not depend on the choice ofε0, butN and k.k_α,L do.

There exist relations between the setsV_α(Ω) andV_α. Indeed, one can prove the following result using Proposition 3.4 of [SAU] :

Proposition 1

1. If g ∈ V_α(Ω) and if one extends g to a function f defined on R^k, setting f(x) = 0 if x /∈Ω, then f ∈Vα and

kfk_α ≤ kgk_α,L.

2. Conversely let f ∈Vα and set g=f1Ω. Theng∈Vα(Ω) and the following holds : kgk_α,L ≤ 1 + 2K(Ω)max(1, ε^α₀)

γ_k ε^k−1+α₀

! kfk_α.

Under the hypotheses 1-5 listed above, we obtain a first result :

Theorem 2 Let T be the transformation defined on Ωby : ∀u= (u1, . . . , uk)∈Uj : T(u) =Tj(u) =

u2

γ , . . . ,uk

γ , γ^k−1ϕj(u1,u2

γ , . . . , uk

γ^k−1)

. (1)

The applicationsT_j can be defined naturally on W_j by the same formula. Then

1. The Frobenius-Perron operator P :L¹_m(Ω)→L¹_m(Ω)associated with T has a finite num- ber of eigenvalues of modulus1,λ₁, . . . , λ_r.

2. For every i∈ {1, . . . , r}, the eigenspace E_i ={f ∈L¹_m(Ω) : P f =λ_if} associated with the eigenvalue λi is finite-dimensional and contained in Vα(Ω).

3. The operator P decomposes as

P =

r

X

i=1

λ_iP_i+Q,

where thePi are projections on the spacesEi,k|P_ik|₁ ≤1andQis a linear operator defined onL¹_m(Ω), such that Q(Vα(Ω))⊂Vα(Ω), sup

n∈N^∗

k|Qⁿk|₁ <∞ and k|Qⁿk|_α,L =O(qⁿ) when n→+∞, for a given q∈]0,1[. Moreover, P_iP_j = 0 if i6=j, P_iQ=QP_i = 0 for everyi.

4. The operatorP has the eigenvalue1. Set λ₁ = 1, leth∗ =P₁1_Ω anddµ=h∗ dm. Thenµ is the greatest absolutely continuous invariant measure (ACIM) of T, which means that, if ν << mand if ν is T-invariant, then ν << µ.

5. The support ofµ can be decomposed into a finite number of mutually disjoint measurable sets, on which a power ofT is mixing. More precisely, for every j∈ {1,2, . . . ,dim(E1)}, there exist a number L_j ∈ N^∗ and L_j mutually disjoint sets W_j,l (0 ≤ l ≤ L_j −1), satisfyingT(Wj,l) =Wj,l+1 mod (Lj),T^Lj being mixing on everyWj,l. One denotes byµj,l

the normalized restriction ofµ onW_j,l, defined by µ_j,l(B) = µ(B∩W_j,l)

µ(W_j,l) , dµ_j,l= h^∗1_Wj,l µ(W_j,l)dm.

Saying that T^Lj is mixing on every Wj,l means that, for everyf ∈L¹_µj,l(Wj,l) and every h∈L^∞_µj,l(W_j,l),

n→+∞lim < T^nLjf, h >_µj,l=< f,1>_µj,l<1, h >_µj,l with indifferently used notations : < f, g >_µ⁰=µ⁰(f g) =R

f g dµ⁰.

6. Moreover, there exist real constantsC >0and 0< ρ <1 such that, for everyhin V_α(Ω) andf ∈L¹_µ(Ω), the following holds :

Z

Ω

f◦T^n×ppcm(Li)h dµ−

dim(E1)

X

j=1 Lj−1

X

l=0

µ(W_j,l)< f,1>µj,l<1, h >µj,l

≤C||h||_α,Ω||f||_L1 µ(Ω)ρⁿ. 7. If, moreover,T is mixing,³ the preceding result can be stated as : there exist real constants

C >0 and 0< ρ <1 such that, for every h in V_α(Ω) and f ∈L¹_µ(Ω), one has :

Z

Ω

f ◦Tⁿh dµ− Z

Ω

f dµ Z

Ω

hdµ

≤C||h||_α,Ω ||f||_L1 µ(Ω) ρⁿ.

Let us now come back to the initial system and let us try to deduce the invariant law as- sociated with Xn. If the sequence (Xn)n is defined by the initial terms X0, . . . , Xk−1, with values in [−L, L], and the recurrence relation X_n+k = ϕ(X_n, . . . , Xn+k−1), one sets Z_n = (γ^j−1Xn+j−1)1≤j≤k. Then (Zn)n satisfies the recurrence relation Zn+1 = T(Zn), which yields the following result :

Theorem 3 If the random variableZ₀ = (γ^j−1Xj−1)_1≤j≤khas densityh∗, then, for everyn≥0, Zn has densityh∗. Computing the marginal distributions, we get as a consequence that for every n∈N, X_n has a density h_inv which has the following expressions : for every j∈ {0, . . . , k−1}

∀u∈[−L, L], hinv(u) =γ^j Z

R^k−1

h∗(z1, . . . , γ^ju, . . . , zk) dˇzj+1

where dˇz_j+1 means that one integrates with respect to all coordinates ofz butz_j+1.

Indeed, γ^jXn is the (j+ 1)−th coordinate ofZn−j ifj = 0, . . . , k−1. Let us consider a Borel setA of R. Then, for j∈ {0, . . . , k−1},

P(X_n∈A) =P(Zn−j ∈R^j×γ^jA×R^k−j−1)

= Z

R^j×γ^jA×R^k−j−1

h∗(z1, . . . , zk) dz1. . . dzk

= Z

R^j×A×R^k−j−1

h∗(z₁, . . . , γ^ju, . . . , z_k) dˇz_j+1 γ^jdu with z_j+1 =γ^ju

= Z

A

Z

R^k−1

h∗(z₁, . . . , γ^ju, . . . , z_k) dˇz_j+1

γ^jdu,

which gives the desired result.

IfF is defined on [−L, L] and if s∈ {1, . . . , k}, let us denote byTsF the function defined on Ω by

T_sF(z) =T_sF(z₁, . . . , z_k) =F(z_sγ^1−s). (2) The following Lemma is then a direct consequence of point 6 in Theorem 2, applied toT_sF and TrH fors, r∈ {1, . . . , k} :

Lemma 4 For every Borel set B of [−L, L] and every interval I of [−L, L], if Z₀ has the invariant distribution, one has :

P X_{n×ppcm(Li)+s−1} ∈B, Xr−1 ∈I

−

dim(E1)

X

j=1 Lj−1

X

l=0

µ(W_j,l)< T_s1_B,1>_µj,l<1, T_r1_I >_µj,l

≤

(2L)^kγ^k(k−1)/2+ 4(2L)^k−1γ1−r+k(k−1)/2ε^1−α₀ + 2^2kL^k−1 1−γ^k

1−γ

k−1

ε^1−α₀

Cρⁿ.

3. Which is equivalent to : if 1 is the only modulus-1 eigenvalue ofP and if, additionnaly, it is simple

More generally, let F, defined and measurable on [−L, L], be such that T_sF belongs to L¹_µ(Ω).

LetH∈L^∞_m([−L, L])be such that sup

0<δ<ε0γ^1−r

δ^−α Z

[−L,L]

Osc(H,]x−δ, x+δ[∩[−L, L])dx <+∞.

ThenT_rH∈V_α(Ω) and

E(F(X_{n×ppcm(Li)+s−1})H(Xr−1))−

dim(E1)

X

j=1 Lj−1

X

l=0

µ(W_j,l)µ_j,l(TsF)µ_j,l(TrH)

≤C(F, H, L) ρⁿ

with

C(F, H, L) =C||T_sF||_L1 µ

(2L)^k−1γ^k(k−1)/2||H||_L1

m([−L,L])

+(2L)^k−1γ(k(k−1)/2)−α(r−1) sup

0<δ<ε0γ^1−r

δ^−α Z

[−L,L]

Osc(H,]x−δ, x+δ[∩[−L, L]) dx +2^2kL^k−1

1−γ^k 1−γ

^k−1

ε^1−α₀ ||H||_L∞ m([−L,L])

.

This last result, which gives the exponential decay of correlations, is a straightforward conse- quence of Lemma 4 and of the remark in point 7, Theorem 2.

Theorem 5 If, moreover, T is mixing, then for all r, s∈ {1, . . . k}

|Cov(F(Xn+s−1), H(Xr−1))| ≤C(F, H, L) ρⁿ.

3 A nonlinear example

We can state the result :

Theorem 6 Let σ >1 be such that η:= 1

√σ + 4(k+ 1)

√σ−1 γk−1

γk

<1. (3)

Let A > σ^k2. Set γ =A^−1k and

M0(σ, A) =

−(k−1)γ^k−1+q

(k−1)²γ^2k−2+ 4(k−2)γ^2k+1(_γ¹2 −σ)

2(k−2)γ^2k+1 .

Suppose M ∈]0, M₀(σ, A)[.

Let a1, . . . , a_k, b1 be nonnegative numbers such that

a1≥2A², (4)

b₁ ≥4LM√

k−1 + 2a₁L, (5)

√a₁a_i<2M ∀i∈ {2, . . . , k}. (6) Set

ψ(x) =

k

X

i=1

a_ix²_i

!

+b₁x₁+ b²₁ 4a₁. Thenψ is positive on[−L, L]^k. Set ϕ0 =√

ψ.

Let `∈[−L, L[. Define the transformation ϕon [−L, L]^k, piecewise, by

ϕ(x) =`+ϕ<sub>0</sub>(x)−2pL if `+ϕ₀(x)∈[2pL−L,2pL+L[. (7) The applicationϕ satisfies the hypotheses 1-6 of Theorem 2.

For example, fork= 3 and L= 1, one can take

σ= 150, A= 1900, M = 260 and

a₁= 7250000, a₂ = 0, a₃ = 0.03, b₁ = 15000000.

Remark 7 Since this nonlinear transform admits, according to Theorem 2, a stationary density, it could be used as a pseudorandom number generator (cf [LM], [LY]).

The rest of this section will be dedicated to the proof. It appears in the proof that the parametersα and Y of the hypotheses satisfyα= 1 and Y ≤k+ 1.

Proof :One sees that ψ(x) = 1

4a₁(2a1x1+b1)²+

k

X

i=2

aix²_i ≥ 1

4a₁(−2a₁L+b1)² on [−L, L]^k, since, by (5), 2a1x1+b1 ≥b1−2a1L≥ 4LM√

k−1 >0. Hence ψ is positive on the compact set [−L, L]^k and consequently on an open neighbourhood U of [−L, L]^k. The function ϕ0 =√

ψ is well defined and C^∞ on U.

Hypothesis 2 is satisfied since, whatever the open subsets Oj of [−L, L]^k are, the expression (7) forϕmakes sense on the neighbourhoodU of [−L, L]^k itself.

Since ϕ₀ is smooth, Hypothesis 3 is fullfilled with α= 1.

To prove thatϕ satisfies Hypothesis 5, we only have to prove it forϕ0 on a neighbourhood of [−L, L]^k. One checks that

∂ϕ₀

∂x₁(x) = 2a₁x₁+b₁ 2p

ψ(x) ≥ 2a₁x₁+b₁ 2p

ψ(x1, L, . . . , L).

Denoting by g = g(x₁) the function appearing in the right side above, one sees that g⁰ has the sign of a₁Pk

i=2a_iL², which means that g is an increasing function. To obtain the desired condition about ∂ϕ₀

∂x₁, it suffices that g(−L) > Aon [−L, L]^k (and hence on a neigbourhood of [−L, L]^k). Now,g(−L) = −2a₁L+b1

2p

ψ(−L, L, . . . , L) > Aif and only if (−2a₁L+b1)² >4A² 1

4a1

(−2a₁L+b1)²+

k

X

i=2

aiL²

!

⇐⇒ (−2a₁L+b₁)²(a₁−A²)>4A²a₁

k

X

i=2

a_iL².

Using successively (4), (5) and (6), one gets

(−2a₁L+b1)²(a1−A²)≥(−2a₁L+b1)²(A²)≥4A²(4M²)(k−1)L²>4A²a1 k

X

i=2

aiL²,

which shows thatg(−L)> Aand ∂ϕ₀

∂x1

(x)> Aon [−L, L]^k and on a neigbourhood of [−L, L]^k. One has

∂ϕ₀

∂x1

(x)∂ϕ₀

∂xi

(x)

= a_i|x_i|(2a₁x₁+b₁)

2ψ(x) ≤ a_i|x_i|(2a₁x₁+b₁) 2ψ(x1,0, . . . ,0, xi,0, . . . ,0).

This can be written as

∂ϕ₀

∂x1

(x)∂ϕ₀

∂xi

(x)

≤√ aia1

(√

ai|x_i|)(√

a1x1+₂^√b1_a

1) (√

a1x1+₂^√b1_a

1)²+ (√

a1x1)², and it is easy to see that it is smaller than

√aia1

2 , hence strictly smaller than M according to (6) on [−L, L]^k and on a neigbourhood. This achieves the proof that Hypothesis 5 is verified.

To verify Hypothesis 1, we must explicit the open sets. For p∈ Z, define the open setsO_p by :

O_p ={x∈]−L, L[^k:`+ϕ0(x)∈]2pL−L,2pL+L[}.

One sees that, forp≤ −1,O_p is empty and that, otherwise, O₀ ={x∈]−L, L[^k:ψ(x)<(L−`)²},

O_p ={x∈]−L, L[^k: ((2p−1)L−`)<sup>2</sup> < ψ(x)<((2p+ 1)L−`)²}, p≥1.

The setsO_p are open and may be empty.

Put S_p ={x ∈R^k :ψ(x) = ((2p−1)L−`)²}. IfS_p∩[−L, L]^k is not empty, ∂ψ

∂x1

(x) >0 is valid for every point ofS_p∩[−L, L]^k (because of (5)), sox₁ can be considered, locally, as aC^∞ function of the other x_i and S_p∩[−L, L]^k is a finite union of C^∞ submanifolds. The edges of [−L, L]^k are parts of hyperplanes, hence are C^∞ too. This gives Hypothesis 1.

A submanifold S_p crosses at most k hyperplanes, which implies that the maximal crossing number,Y, is smaller thank+ 1. This, together with (3), gives Hypothesis 4.

Hypothesis 6 is satisfied under its simple form. Indeed, let U = (u₁, u₂, . . . , u_k) and V = (v1, u2, . . . , uk) be two points of the same set O_p ⊂ [−L, L]^k. On [−L, L]^k, _∂x^∂ψ

1(x) >0. Hence, fort∈[0,1], if one assumes that−L < u₁ < v₁< L,

ψ(U)≤ψ(tU+ (1−t)V)≤ψ(V),

since the only coordinate that changes is the first one. Thereforeψ(tU+ (1−t)V) is in the same interval asψ(U) andψ(V). Consequently,tU+ (1−t)V is in O_p.

This achieves the proof of the theorem.

4 Proofs

Theorem 2 is a consequence of Theorems 5.1 and 6.1 of [SAU], which rely on [ITM], as well as [HK] in the case when d= 1, where the use of bounded-variation functions is possible. The difficulty lies in verifying thatT satisfies Hypotheses (PE1) to (PE5).

To prove that (PE2) is satisfied, we shall first establish thatT_j is a C¹ diffeomorphism on W_j onto T_j(W_j). Hypothesis 3 about ∂ϕ_j

∂x₁ assures that T_j is a local diffeomorphism. To check that it is injective, let us consider two different pointsu and vof Wj, such thatTj(u) =Tj(v). Then u_i =v_i for every 2≤i≤k and

ϕ_j

u₁,u₂

γ , . . . , u_k γ^k−1

=ϕ_j

v₁,u₂

γ , . . . , u_k γ^k−1

.

Using the geometrical hypothesis 6 and applying the fundamental theorem of calculus to t 7→

ϕj(Γ(t)) leads to a contradiction. The regularity hypotheses on the ϕj (and hence on the Tj) allow to prove that det(DT_j⁻¹) isα-H¨older, provided the domain is conveniently restricted. One can see that there exist, for each βj > 0, an open and relatively compact set V_j and a real constantcj such that the following holds