On the affine random walk on the torus
Jean-baptiste Boyer


























maths@jbboyer.fr









Let μ be a borelian probability measure on 𝐆:=SL_d(ℤ) ⋉𝕋^d. Define, for x∈𝕋^d, a random walk starting at x denoting for n∈ℕ,

    {[               X_0                 =                 x;             X_n+1                 = a_n+1 X_n + b_n+1 ].

where ((a_n,b_n))∈𝐆^ℕ is an iid sequence of law μ.

Then, we denote by ℙ_x the measure on (𝕋^d)^ℕ that is the image of μ^⊗ℕ by the map ((g_n) ↦ (x,g_1 x, g_2 g_1 x, … , g_n … g_1 x, …)) and for any φ∈L^1((𝕋^d)^ℕ, ℙ_x), we set 𝔼_x φ((X_n)) = ∫φ((X_n)) dℙ_x((X_n)).


Bourgain, Furmann, Lindenstrauss and Mozes studied this random walk when μ is concentrated on SL_d(ℤ) ⋉{0} and this allowed us to study, for any hölder-continuous function f on the torus, the sequence (f(X_n)) when x is not too well approximable by rational points.


In this article, we are interested in the case where μ is not concentrated on SL_d(ℤ) ⋉ℚ^d/ℤ^d and we prove that, under assumptions on the group spanned by the support of μ, the Lebesgue's measure ν on the torus is the only stationary probability measure and that for any hölder-continuous function f on the torus, 𝔼_x f(X_n) converges exponentially fast to ∫ fdν.


Then, we use this to prove the law of large numbers, a non-concentration inequality, the functional central limit theorem and it's almost-sure version for the sequence (f(X_n)).


In the appendix, we state a non-concentration inequality for products of random matrices without any irreducibility assumption.


On the affine random walk on the torus
    Jean-Baptiste Boyer
    December 30, 2023
======================================









§ INTRODUCTION AND MAIN RESULTS


Let d∈, d⩾ 2 and ^d:= ^d/^d be the torus of dimension d. Let μ be a borelian probability measure on :=SL_d() ⋉^d. Define, for any x∈^d, a random walk starting at x by denoting, for any n∈,

    {[               X_0                 =                 x;             X_n+1                 = a_n+1 X_n + b_n+1 ].

where ((a_n,b_n))∈^ is an iid sequence of law μ.

Then, we denote by _x the measure on ^ that is the image of the measure μ^⊗ on ^ by the map ((g_n) ↦ (x,g_1 x, g_2 g_1 x, … , g_n … g_1 x, …)) and by _x the operator of integration against the measure _x.

We denote by P the Markov operator associated to  μ. This is the operator defined for any borelian non-negative function f on ^d and any x∈^d by

    Pf(x) = ∫_ f(gx)μ(g)

Thus, for any n∈, we have that

    P^n f(x) = ∫_ f(gx) μ^∗ n(g) = ∫_^d f(y) μ^∗ n∗δ_x(y) = ∫_(^d)^ f(X_n) _x((X_n)) = _x f(X_n)

Where we noted μ^∗ n the n-th power of convolution of the measure μ (μ^∗ 0 is by convention the Dirac measure at (I_d,0)).

Bourgain, Furmann, Lindenstrauss and Mozes studied in <cit.> the case where μ is concentrated on SL_d() ⋉{0} and they proved that, under assumptions on the support of μ, the only P-invariant probability measures on the torus where the Lebesgue's measure ν and the uniform measures on unions of rational orbits (which are finite). their result is even more precise since they give the rate of convergence of _x f(X_n) to ∫ fν in terms of diophantine properties of x and this allowed us to study the sequence (f(X_n)) for starting points x that are not too well approximable by rational points in <cit.>.

In this article, we are interested in the case where μ is not concentrated on SL_d()⋉^d/^d. A result by Benoist-Quint (see <cit.>) shows that in this case, under assumptions on the projection on SL_d() of the subgroup spanned by the support of μ, the only P-invariant probability measure on the torus is Lebesgue's measures and this proves that for any continuous function f on ^d and any x∈^d,

    1/n∑_k=0^n-1 f(X_k) ∫ fν _x-a.e.

The aim of this article is to precise the previous convergence by proving a Central Limit Theorem, a Law of the Iterated Logarithm, etc.


To do so, we are going to make a few assumptions on the subgroup spanned by {a|(a,b)∈μ}.

In the sequel, we will say that a closed subgroup  of SL_d() is strongly irreducible if it doesn't fix any finite union of non-trivial subspaces of ^d. Moreover, we will say that  is proximal if it contains an element g such that there is v_g^+ ∈^d∖{0}, λ∈ and a g-
invariant hyperplane V_g^< in ^d such that ^d=   v_g^+ ⊕_g^<, gv_g^+ = λ v_g^+ and the spectral radius of g in V_g^< is strictly smaller that |λ|. Finally, we will say that a probability measure μ on SL_d() is strongly irreducible and proximal if the closure of the subgroup spanned by the support of μ has these properties.


These two assumptions are actually assumptions on the Zariski-closure of  and so, as an example, they are satisfied if  is Zariski-dense in SL_d().


Finally, we will say that a measure μ on SL_d() has an exponential moment if there is some ε∈_+^∗ such that

    ∫_SL_d()g^εμ(g)<+∞

We will see in the sequel that our study of the random walk on the torus requires arguments of orbit closures. This is why we give a name to the property that we will use and we will see right after examples of measures satisfying it.

Let μ be a borelian probability measure on .

We say that μ satisfies an effective shadowing lemma if for any C',t' ∈_+^∗, there are C_1,C_2,M,t,L ∈_+^∗ such that for any x,y∈^d, any r∈_+^∗ and any n∈, with r⩽ C_1e^-Ln, if

    μ^∗ n({ g∈| d(gx,y) ⩽ r }) ⩾ C_2 e^-t n

then, there are x',y'∈^d such that d(x,x'),d(y,y')⩽ re^Mn and

    μ^∗ n({ g∈| gx'=y'})⩾ C'e^-t'n









For a measure to satisfy this property means that if a lot of elements g∈μ^∗ n send x close to y, it is only because x and y are close to points of the same orbit. The name comes from the theory of hyperbolic diffeomorphisms since when μ=δ_g_0, saying that μ satisfies an effective shadowing lemma means that there is some constant M such that for any large enough K, any n∈ and any x,y∈^d with d(g^n_0 x,y) ⩽ e^-K n, there are x',y' ∈^d such that d(x,x'),d(y,y')⩽ e^-(K-M)n and g^n_0 x' = y'.



This is a technical definition but we will see in section <ref> a criterion (the proposition <ref>) that allows to tell if a measure satisfies to an effective shadowing lemma and we will deduice examples from it.


In particular, we will see in example <ref> that if b_0∈^1 is such that there are C,L ∈_+^∗ such that for any q∈^∗, d(qb_0,0) ⩾C/q^L, then, any borelian probability measure μ on  whose projection on SL_d() is strongly irreducible, proximal and has an exponential moment and such that F(μ):={coefficients of b|(a,b)∈μ}⊂{0,b_0} satisfies an effective shadowing lemma.


Moreover, in example <ref> we will prove that if a_1, … , a_N ∈SL_d() generate a strongly irreducible and proximal group, then for a.e. b_1, … , b_N∈^d, the measure μ = 1/N∑_i=1^N δ_(a_i,b_i) satisfies an effective shadowing lemma.




For α∈ ]0,1], we denote by 𝒞^0,α(^d) the space of α-hölder-continuous functions f on ^d endowed with the norm

    f_α := f_∞ + m_α(f)

where

    f_∞ := sup_x|f(x)|  and m_α(f) := sup_x≠y |f(x)-f(y)|/d(x,y)^α

where d is the distance induced by some norm on ^d.

Moreover, for any two borelian probability measures ϑ_1, ϑ_2 on ^d, we denote by 𝒲_α(ϑ_1, ϑ_2) the Kantorovich-Rubinstein's distance of ϑ_1 and ϑ_2 and this is defined by

    𝒲_α(ϑ_1, ϑ_2) := sup_f∈𝒞^0,α(^d)
    f_α⩽ 1|∫ fϑ_1 - ∫ fϑ_2 |


This will allow us to prove the


Let μ be a borelian probability measure on :=SL_d()⋉^d that is not concentrated on SL_d() ⋉^d/^d and satisfies an effective shadowing lemma. Note μ_0 the projection of μ on SL_d() and assume that μ_0 is strongly irreducible, proximal and has an exponential moment.


Denote by P the Markov operator associated to μ.

Then, the Lebesgue's measure ν on ^d is the only P-invariant borelian probability measure on the torus. Moreover, for any α∈]0,1], there are C,t∈_+^∗ such that for any f ∈ C^0,α(^d) and any n∈,

    sup_x∈^d𝒲_α(μ^∗ n∗δ_x, ν) ⩽ Ce^-tnf_α

In particular, for any α-hölder-continuous function f on the torus, there is a continuous function g such that

    f-∫ f ν = g-Pg  and g_∞⩽ C f_α




We don't know if the function g that we construct in this theorem is hölder-continuous.


This theorem will allow us to prove a few of the classical results in probability theory for the sequence (f(X_n)) in the

Under the same assumptions than in theorem <ref>.

Denote, for any continuous function f on the torus, f= f-∫ fν and, for any sequence x=(X_n) ∈ (^d)^,

    S_n f(x) = ∑_k=0^n-1f(X_k) = ∑_k=0^n-1 f(X_k) - n ∫ fν

Moreover, for any t∈ [0,1], set

    ξ_n(t) = 1/√(n)(S_if(x)+ n(t-i/n)f(X_i))  for i/n⩽ t⩽i+1/n and 0 ⩽ i ⩽ n-1

Then, for any continuous function f on the torus and any x∈^d,

    S_n f(x)/n 0   _x-a.e.

Moreover, for any α∈]0,1] there is t∈_+^∗ such that for any ε∈ ]0,1] there is a constant C such that for any α-hölder-continuous function f on the torus, any x∈^d and any n∈,

    _x ( {x∈^|  |S_n f(x)|  >nεf_α}) ⩽ Ce^-t ε^2 n

Finally, set

    σ^2(f) := ∫_^d g^2 - (Pg)^2 ν

and then,


  * If σ^2(f) ≠0 then for any bounded continuous F: 𝒞^0([0,1]) → and any x∈^d,

    _x F(ξ_n)  F(W_σ^2)  and 1/ln n ∑_k=1^n 1/k F(ξ_k)  F(W_σ^2)  _x- a.e.

Where W_σ^2 denotes Wiener's measure of variance σ^2.

And for any continuous function φ on  such that t^2 φ(t) is bounded and for any x∈^d,

    1/ln n ∑_k=1^n 1/kφ( S_k f(x)/√(k))φ(W_σ^2(1))  _x- a.e.


  *  If σ^2(f)= 0 then for any x∈^d and any n∈, S_nf ∈L^∞(_x) and

    S_nf_L^∞(_x)⩽ 2Cf_α





The two convergences of (F(ξ_n)) in point (<ref>) are respectively called functional central limit theorem (FCLT) and almost-sure functional central limit theorem (ASFCLT). There is no obvious link between the convergence in law of (F(ξ_n)) and the a.e. convergence of it's logarithmic average (see <cit.> for a criterion). However, note that we have to take a logarithmic mean because of the arc sine law.








The FCLT and the ASFCLT have many corollaries such as the cental limit theorem and the almost sure central limit theorem (taking F_φ(ξ) = φ(ξ(1)) for any continuous and bounded function φ on ), the law of the iterated logarithm (see theorem 2.4 in <cit.>), a control of max_k ∈ [0,n]S_kf(x)/√(n) (taking F(ξ) :=sup_t∈ [0,1]ξ(t)), or an estimation of σ^2(f) (taking φ(x) = x^2).


Before we continue, we give an example where there is a non-constant function f such that σ^2(f)=0.


Let

    A=([ 2 1; 1 1 ]) and B =([  0  1; -1  0 ])

Then, the subgroup spanned by A and B is strongly irreducible and proximal.

Let b_0∈^1∖/ be a diophantine number[There are C,L∈_+^∗ such that for any q∈^∗, d(qb_0,0)⩾ Cq^-L.], b=(b_0,0) and μ = 1/2δ_(A,b) + 1/2δ_(BA,Bb).

Then, according to proposition <ref>, the measure μ satisfies to the assumptions of theorem <ref>.

Let g be the function defined for any x∈^2 by g(x) = d(x,0). We made everything so that for any x∈^2, g(Bx) = g(x).

Then, for any x∈^2,

    Pg(x) = 1/2 g(Ax+b) + 1/2 g(BA x+Bb) = g(Ax+b)

And,

    ∫_ |Pg(x)|^2 ν(x) = ∫_ |g(Ax+b)|^2 ν(x) = ∫_ |g(x)|^2 ν(x)

Moreover, if we set f=g-Pg, then, we just saw that σ^2(f) = ∫ g^2 -(Pg)^2ν=0 and for any x∈, n∈ and any (g_1, … g_n) ∈{(A,b), (BA,Bb)}^n, we have that

    g(g_n+1… g_1 x) =g(A g_n … g_1x+b)

And so,

∑_k=0^n-1 f(g_k …g_1 x)    = g(x) - g(g_n …g_1 x) + ∑_k=0^n-1 g(g_k+1 …g_1 x) - g(A g_k …g_1x+b) 
   = g(x) - g(g_n …g_1 x)

This proves that for any x∈, the sequence (∑_k=0^n-1 f(g_k … g_1 x)) is bounded in L^∞ (_x).


The results in section 3 of <cit.> actually prove that this example is really general.


We will see in sub-section <ref> that theorem <ref> is a quite general corollary of theorem <ref> since we can easily study functions f on the torus that writes f=g-Pg+∫ fν with g continuous and theorem <ref> precisely says that any holder-continuous function can be written in this way.

Therefore, the main point of this article is the proof of theorem <ref>. To do so, we use the same method as Bourgain, Furmann, Lindenstrauss and Mozes. In section <ref>, we prove that the only obstacle in the equidistribution of the measure μ^∗ n∗ϑ is the lower regularity of ϑ i.e. the existence of points x such that for some r depending on n,

    ϑ(B(x,r))⩾ r^ε

In particular, if μ^∗ n+m∗ϑ is far from Lebesgue's measure then there has to be points x such that

    μ^∗ m∗ϑ(B(x,r)) ⩾ r^ε

Then, our assumptions that μ satisfies an effective shadowing lemma and that μ is not a subset of SL_d() ⋉^d/^d will allow us to prove in section <ref> that this cannot happen when r≪ e^-m≪ r^ε.

The precise proof of the theorem is in subsection <ref>.

Finally, in section <ref>, we prove proposition <ref> that is a criterion that shows that under some diophantine conditions on the translations in it's support, a measure satisfies an effective shadowing lemma and we will use this criterion to produce examples of such measures.  
 
In the appendix, we state results on the products of random matrices in the case where the action is not irreducible and that we use in section <ref>. 





 §.§ Some kind of diophantine assumption is necessary


We already said (and we will prove in section <ref>) that a way to guarantee that a measure satisfies an affective shadowing lemma is to require diophantine conditions on the coefficients of the translations of it's support. In this sub-section, we prove that this kind of assumptions is indeed necessary to get theorem <ref>.


Let a,b∈SL_d() and v∈^d. Set μ= 1/2δ_(a,0) + 1/2δ_(b,v).

Assume that for some α∈ ]0,1], there are C,t∈_+^∗ such that for any α-hölder-continuous function f on the torus and any n∈,

    sup_x∈^d|P^n f(x) - ∫ fν| ⩽ Ce^-tnf_α

Then, there are constants C_0,L∈_+^∗ such that for any rational point p/q∈^d/^d,

    d(v,p/q)⩾C_0/q^L




For q∈^∗ and x∈^d, we set X_q = 1/q^d/^d and

    f_q(x) = 1- min(1, q^2αd(x,X_q)^α)

This function is chosen so that it takes the value 1 on 1/q^d/^d, it vanishes on the complementary of the 1/q^2-neighborhood of 1/q^d/^d and it is hölder-continuous with f_q_α⩽ q^2α.

In particular, we have that, for some constant C depending only on d (and on the distance on ^d),

    ∫ f_qν⩽∑_p/q∈1/q^d/^dν(B(p/q, 1/q^2) ) ⩽C/q^d

Moreover, for μ^⊗-a.e. ((a_n,b_n)), we have that

    |f_q(∑_k=1^n a_n … a_k+1 b_k) - 1 |   ⩽f_q_α d(∑_k=1^n a_n … a_k+1 b_k, X_q )^α
       ⩽e^α Mn/(e^M-1)^α d(v,X_q)^α q^2α

where we noted e^M = max(A,B). Indeed, for any p/q∈ X_q, we have that f_q(p/q) = 1 and ∑_k=1^n a_n … a_k+1 b_k can be written Dv where D is a matrix with integer coefficients and D⩽∑_k=1^n a_n … a_k+1⩽∑_k=1^n e^M(n-k).
This proves that for any n∈,

    | P^n f_q(0) - 1 | = |∫_ f_q(b) μ^∗ n(a,b) - 1 |⩽e^α Mn/(e^M-1)^α d(v,X_q)^α q^2α

But, by assumption, we also have that

    |P^n f_q(0) - ∫ f_q ν| ⩽ Ce^-tnf_q_α⩽ Ce^-tnq^2α

So, this proves that for any n,q∈^∗,

    1 - C/q^d -Ce^-tn q^2α⩽e^α Mn/(e^M-1)^α d(v,X_q)^α q^2α

Thus, for any p∈^d, any q∈^∗ such that q^d ⩾ 4C and any n such that Ce^-tn⩽ q^-2α/4, we have that

    d(v,p/q) ⩾e^M - 1/2^1/αe^Mn q^2

In particular, for n = ⌊1/tln(4Cq^2α) ⌋+1, we find that, for some constant C' depending only on M,α,t,C,

    d( v, p/q) ⩾C'/q^2+2α M/t

And this is what we intended to prove.





 
 
 
We can prove the same kind of results for rates more general than Ce^-tn and this shows that even convergences slower than exponential require some kind of diophantine assumption.
 





 §.§ Proof of theorem <ref> given the results of sections <ref> and <ref>
 


Let α∈ ]0,1] and ε∈_+^∗.

According to theorem <ref> there are constants c_0,ε' ∈_+^∗ with ε'<ε such that for any ϑ∈ M^1(^d), any t∈ ]0,1] and any n∈ with n⩾ c_0(1+|ln t|),

    𝒲_α(μ^∗ n∗ϑ, ν) ⩾ t ⇒ϑ( { x ∈^d | ϑ(B(x,r))⩾ r^ε}) ⩾ t^c_0

where

    r=e^-(Λ_1 +ε')n( t/16)^1/α

In particular, for any m,n∈, any ϑ∈ℳ^1(^d) any t∈_+^∗ small enough and any C large enough,

    𝒲_α(μ^∗ n+m∗ϑ, ν) ⩾ Ce^-t n⇒μ^∗ m∗ϑ( { x ∈^d | μ^∗ m∗ϑ(B(x,r))⩾ r^ε}) ⩾(Ce^-t n)^c_0

for

    r=e^-(Λ_1 +ε'- δ /α)n/16 ^1/α

But, since the measure satsfies to an effective shadowing lemma, according to proposition <ref>, there are C_1,C_2,t_0,L ∈_+^∗ such that for any x,y∈^d, any m∈ and any r∈_+^∗ with r⩽ C_1e^-Lm,

    μ^∗ m({ g∈| d(gx,y) ⩽ r }) ⩽ C_2 e^-t_0 m

And so, to get a contradiction, we only need to assume that

    r=e^-(Λ_1 +ε'- δ /α)n/16 ^1/α⩽ C_1 e^-Lm and r^ε = e^-ε(Λ_1 +ε'- δ /α)n/16 ^ε/α⩾ C_2 e^-t_0m

And this is always possible for n=Km with K∈ large enough and ε small enough.



























We just proved that there are C,t∈_+^∗ and K∈^∗ such that for any borelian probability measure ϑ on ^d and any m∈,

    𝒲_α( μ^∗ (K+1)m∗ϑ,ν) ⩽ Ce^-tm

Let n∈ and let m,L∈ be such that n=(K+1)m+L and 0⩽ L< K+1. Then,

    𝒲_α( μ^∗ n∗ϑ,ν)    ⩽𝒲_α( μ^∗ (K+1)m∗μ^∗ L∗ϑ,ν) ⩽ C e^-tm = Ce^-t/K+1 (n-L)
       ⩽ C e^t e^-tn/(K+1)

And this finishes the proof of the first part of the theorem.

In particular, with ϑ= δ_x for some x∈^d, we get that for any α-hölder-continuous function f on ^d and any x∈^d,

    |P^n f(x) - ∫ fν| ⩽ Ce^-tnf_α


Let f be an α-hölder-continuous function on ^d. Set, for any n∈,

    g_n=∑_k=0^n-1 P^k f-∫ fν

Then,

    (I_d-P)g_n = f-∫ fν - (P^n f -∫ fν)

And so,

    lim_n g_n - Pg_n = f-∫ fν

Moreover, the series is normally convergent since

    ∑_n   P^nf - ∫ fν_∞⩽C/1-e^-tf_α

And so, the function g=lim_n g_n exists, is continuous and satisfies

    g-Pg=f-∫ fν and g_∞⩽C/1-e^-tf_α


Now, let ϑ be a P-invariant borelian probability measure on ^d. Then, for any hölder-continuous function f,

    ∫ fϑ = ∫ P^n f ϑ∫ fν

Where we first used the P-invariance of ϑ and then the dominated convergence theorem since for any x∈^d, lim_n P^n f(x) = ∫ fν according to the first part of the proof. And, finally, as the hölder-continuous functions are dense in the space of continuous functions on the torus, this proves that ϑ = ν and so, ν is the unique P-invariant borelian probability measure on ^d. 



 §.§ Proof of theorem <ref>
 



This result is a consequence of the uniqueness of the P-invariant borelian probability measure seen in theorem <ref>. Indeed, if we manage to prove that for any x and _x-a.e. x=(X_n) ∈ (^d)^, the accumulation points of ν_n,x:=1/n∑_k=0^n-1δ_X_k are P-invariant, we will get that they have to be the Lebesgue's measure and so, for any continuous function f on the torus,

    1/n∑_k=0^n-1 f(X_k) ∫ fν  _x-a.e.

For any continuous function f on the torus, we can compute,

    ∫ fν_n,x - ∫ Pfν_n,x   = 1/n∑_k=0^n-1 f(X_k) - 1/n∑_k=0^n-1 Pf(X_k) 
       = 1/n∑_k=0^n-1 f(X_k+1) - Pf(X_k)  + 1/n (f(X_0) - f(X_n))

But, M_n = ∑_k=0^n-1 f(X_k+1) - Pf(X_k) is a martingale with bounded increments so 1/n M_n  0 a.e. and as f is bounded, we also have that 1/n (f(X_0) - f(X_n))  0 in L^∞(_x).

Thus, we just proved that for any x∈^d and any continuous function f on ^d, there is X_f ⊂ (^d)^ such that _x(X_f) = 1 and for any x∈ X_f,

    lim_n ∫ fν_n,x - ∫ Pfν_n,x=0

Let (f_i) be a dense sequence in 𝒞^0(^d) and X_∞ = ∩_i X_f_i. Then, _x(X_∞)=1 and for any x∈ X_∞ and any i∈,

    lim_n ∫ f_iν_n,x - ∫ Pf_iν_n,x=0

So, as the sequence (f_i) is dense, we get that for any continuous function f on ^d and any x∈ X_∞,

    lim_n ∫ fν_n,x - ∫ Pfν_n,x = 0

This proves that for any x∈ X_∞, the accumulation points of (ν_n,x) are P-invariant and so they are equal to ν and this proves the law of large numbers.


To prove the remaing part of the theorem, we are going to use Gordin's method and deduce the non-concentration inequality, the FCLT and the ASFCLT from these results for martingales. Indeed, according to theorem <ref>, for any α∈ ]0,1], there is a constant C such that for any α-hölder-continuous function f on the torus there is a continuous function g such that

    f-∫ fν = g-Pg  and g_∞⩽ Cf_α

Set, for x= (X_n) ∈^,

    S_n f(x) = ∑_k=0^n-1 f(X_k) -n∫ fν and M_n = ∑_k=0^n-1 g(X_k+1) - Pg(X_k)

Then,

    S_n f(x) = M_n + g(X_0) - g(X_n)

And M_n is a martingale with bounded increments.



For any n∈, we have that

    |M_n| ⩾ |S_n f(x)| - 2 g_∞⩾ |S_n f(x)| - 2Cf_α

So, using Azuma-Hoeffding's inequality, if nε>2C, we get that

I_n(x) :   =_x ( |S_nf(x) | > nεf_α)⩽_x( | M_n | ⩾(nε-2C) f_α) 

    ⩽2exp( - (n ε- 2 C)^2f_α^2/2n ( 2Cf_α)^2 ) = 2 exp(- n ε^2/8 C^2 +  ε/4 C  - 1/2n) 

And this finishes the proof of this point.


As the function g is bounded, the sequence (S_nf(x) - M_n) is bounded in L^∞(_x) and so it is clear that to prove the FCLT and the AEFCLT, it is enough to study the martingale M_n (that has bounded increments). But, according to the functional central limit theorem for martingales (see corollary 4.1 in <cit.>) and it's almost sure extension (see <cit.>), it is enough to prove the a.e. convergence of the variance (when the limit doesn-t vanish). But, for any n∈^∗,

    1/n∑_k=0^n-1_x[| M_k+1 - M_k|^2 |X_0, … ,X_k ]    = 1/n∑_k=0^n-1_x[| g(X_k+1) - Pg(X_k)|^2 |X_0, … ,X_k ] 
        = 1/n∑_k=0^n-1 P(g^2)(X_k) - (Pg(X_k))^2

So, according to the law of large numbers that we already proved and applied to the continuous function P(g^2) - (Pg)^2,

    1/n∑_k=0^n-1_x[| M_k+1 - M_k|^2 |X_0, … ,X_k ]  σ^2(f):=∫ g^2 - (Pg)^2 ν _x-a.e.

(We used the P-invariance of ν to get that ∫ P(g^2) ν = ∫ g^2 ν).

And this proves point <ref> since we suppose in it that σ^2(f) ≠0.

To conclude, remark that, using the -invariance of ν, we can compute

    ∫_∫_| g(γ x) - P g(x)|^2ν(x)  μ(γ)    =  ∫_∫_ g(γ x)^2 + g(x)^2 -2Pg(x) g(γ x) ν(x) μ(γ) 
       = 2 ∫_ g^2 - (Pg)^2 ν = 2σ^2(f)

And so, if σ^2(f) = 0, then, as g is continuous, we get that for any γ∈μ and any x∈^d, g(γ x) = Pg(x). This proves that for any n∈, M_n=0 _x-a.e. and so, S_n f(x) = g(X_0) - g(X_n).
Thus, for any x∈^d, S_nf ∈L^∞(_x) and

    sup_x∈^dsup_n∈S_nf_L^∞(_x)⩽ 2 g_∞⩽ 2C f_α

This inequality finishes the proof of point <ref>.


















































§ THE NON-EQUIDISTRIBUTION COMES FROM THE LOWER REGULARITY OF THE MEASURE


Like Bourgain, Furmann, Lindenstrauss and Mozes did for the linear random walk on the torus, we are going to prove in this section that if the measure μ^∗ n∗ϑ is far from being equidistributed, it is only because of atoms i.e. of points x∈^d such that

    ϑ(B(x,r))⩾ r^ε

for some r∈_+^∗ depending on n.

More specifically, the aim of this section is to prove the

Let μ be a borelian probability measure on SL_d() ⋉^d. Denote by μ_0 the projection of μ on SL_d() and assume that μ_0 is strongly irreducible, proximal and has an exponential moment. Let λ_1 ∈_+^∗ be the largest Lyapunov exponent of μ_0 (see appendix <ref>).


Then for any α∈]0,1] and any ε∈_+^∗, there is c_0,ε'∈_+^∗ with ε'< ε such that for any ϑ∈ M^1(^d), any t∈]0,1] and any n∈ with n⩾ c_0(1+|ln t|),

    𝒲_α(μ^∗ n∗ϑ, ν) ⩾ t ⇒ϑ( { x ∈^d | ϑ(B(x,r))⩾ r^ε}) ⩾ t^c_0

Where we set

    r=e^-(λ_1 +ε')n( t/16)^1/α



In the case of the linear random walk, this statement is a reformulation of an intermediate result (propositions 7.1 and 7.2) of <cit.>. We could prove it for the affine random walk just like they do for the linear one that is to say, by studying, for any borelian probability measure ϑ on ^d, the set of Fourier-coefficients of μ^∗ n∗ϑ and  by remarking that for any c∈^d,

    μ^∗ n∗ϑ(c)    = ∫_^d∫_SL_d() ⋉^d e^2iπ⟨ c,ax+b ⟩μ^∗ n(a,b) ν(x) 
       = ∫_SL_d() ⋉^d e^2iπ⟨ c,b⟩ϑ(^t ac) μ^∗ n(a,b)

And so,

    |μ^∗ n∗ϑ(c) | ⩽∫_SL_d() ⋉^d|ϑ(^t ac)| μ^∗ n(a,b) = ∫_SL_d()|ϑ(^t ac)| μ_0^∗ n(a)

Where we recall that we denoted by μ_0 the projection of μ onto SL_d().

Thus, if |μ^∗ n∗ϑ(c) | ⩾ t, then for many a, we also have that |ϑ(^t ac)|⩾ t and this is the key remark in the proof of BFLM.


Instead, we are going to see that this result can also be obtained as a corollary of the one of BFLM for the linear walk : at first, we are going to prove that their result gives informations on the spectral radius of the operator P in L^p(^d,ν) (even for the affine random walk) and then, that this implies the theorem.







 §.§ Spectral gap in LpTd


Let  be a second countable locally compact group acting measurably on a standard borelian space  endowed with a -invariant probability measure ν.

Let μ be a borelian probability measure on  and P the Markov operator associated to μ. This is the operator defined for any non-negative borelian function f on  and any x∈ by

    Pf(x) = ∫_ f(gx) μ(g)

As ν is a -invariant probability measure, it is clear that for any p∈[1,+∞], 1∈L^p(,ν) and P1=1. Moreover, we can prove, using Jensen's inequality that P_p= 1. So, we note, for any p∈ ]1,+∞],

    L^p_0(,ν):= { f∈L^p(,ν)| ∫ fν=0 }

and ρ_p the spectral radius of P in L^p_0(,ν). We say that P has a spectral gap in L^p_0(,ν) (or, by abuse of notations in L^p(,ν)) if ρ_p<1.

In the sequel, we will need a more flexible tool than the spectral gap. This is why, for any P-invariant subspace  of L^p(,ν) endowed with a norm . _ such that P is continuous on (, . _) and the injection of (, . _) into (L^p,  . _p) is also continuous, we set

    κ(μ,, L^p(,ν) ) := -lnlim sup_n→ +∞sup_f∈∖{0}( P^n f_p/f_)^1/n


The sequence ( sup_f∈∖{0}P^n f_p/f_) is not sub-multiplicative in general so it may converge to 0 only at polynomial rate and in this case, we would have that κ(μ,, L^p(,ν) )=0. This is impossible if = L^p() because in this case, if it converges to 0, it has to be at exponential rate.



With this definition, if (, . _)= (L^p(,ν),  . _p), then e^-κ(μ,, L^p(,ν) ) = ρ_p and, for any (, . _), we have, since the inclusion of  into L^p(,ν) is supposed to be continuous,

    κ(μ,, L^p(,ν) ) ⩾ -lnρ_p



In particular, when  is a subset of L^∞(,ν), we can define and study the function (p↦κ(μ,, L^p(,ν) )).


Remind that, according to Hölder's inequality, for any 1⩽ p⩽ p' and any function f∈L^∞ (,ν) with f_∞⩽ 1,

    f_p'^p' = ∫_ |f|^p'ν⩽∫_ |f|^p ν =f_p^p and f_p ⩽f_p'

So, we get that the function ( p↦κ(μ,, L^p(,ν) )) is decreasing whereas the function ( p↦ pκ(μ,, L^p(,ν) )) is non-decreasing.


In the same way that we defined L^p_0(^d), we set

    𝒞^0,α_0(^d) := {f∈𝒞^0,α(^d) | ∫ fν=0}



The definition of the function κ is made to get the
 
Let μ be a strongly irreducible and proximal probability measure on SL_d() having an exponential moment.

Then, for any α∈ ]0,1] small enough,

    lim_p→+∞  p κ(μ,𝒞^0,α_0(^d), L^p(^d,ν) ) = λ_1 d









This theorem implies in particular that for any ε∈_+^∗, there are p∈ and C∈_+ such that for any n∈ and any f∈𝒞^0,α_0(^d),

    P^n f_L^p(^d)⩽ C e^-(λ_1 d - ε) n/pf_α



First of all, since μ has an exponential moment, for any α∈ ]0,1] small enough, any f∈𝒞^0,α(^d) and any x∈^d,

    |Pf(x)| ⩽f_∞

and for any y∈^d,

    |Pf(x) - Pf(y)|    ⩽∫_ |f(gx) - f(gy)| μ(g) ⩽f_α∫_ d(gx,gy)^αμ(g)
       ⩽f_α d(x,y)^α∫_g^αμ(g)

So, Pf is α-hölder-continuous and P is a continuous operator on 𝒞^0,α_0(^d).

According to the result of Bourgain, Furmann, Lindenstrauss and Mozes in <cit.> (that we use as stated in proposition 4.5 in <cit.>), we have that for any α∈ ]0,1] and any ε∈_+^∗, there is a constant C such that for any n∈, any t∈ ]0,1] with n⩾ -Cln t and any f∈𝒞^0,α(^d) with ∫ fν=0,

    {x||P^n f(x)|⩾ t f_α}⊂⋃_p/q∈^d/^d
    q⩽ Ct^-C B(p/q, e^-(λ_1 - ε)n)

In particular, for any L∈^∗,

    ∫ |P^n f|^L ν   ⩽ (tf_α)^L + ν( {x||P^n f(x)|⩾ t f_α}) f_∞^L
       ⩽(t^L +(Ct^-C)^d e^-(λ_1 - ε)dn)f_α^L

And so, taking t=e^-δ n with δ∈_+^∗ small enough and L∈ large enough, we find that for some constant C,

    ∫|P^n f|^L ν⩽ Ce^-(λ_1 - 2ε)dnf_α^L

And this proves (reminding that the limit exists according to remark <ref>) that

    lim_p  p κ(μ,𝒞^0,α_0(^d), L^p(^d,ν) ) ⩾λ_1 d

We are now going to prove the other inequality. Let δ∈ ]0,1/4], f∈𝒞^∞(^d) such that f=1 on B(0,δ), f_∞⩽ 1 and ∫ f =0.

Then, for any ε∈_+^∗ and any x∈ B(0, e^-(λ_1 +ε)nδ), we have that

    P^n f(x)    = ∫__g⩽ e^(λ_1 + ε)n f(gx) μ^∗ n(g)+ ∫__g⩾ e^(λ_1 + ε)n f(gx) μ^∗ n(g) 
       = μ^∗ n({ g| g⩽ e^(λ_1 + ε)n}) +  ∫__g⩾ e^(λ_1 + ε)n f(gx) μ^∗ n(g) 
       ⩾ 1 - 2 μ^∗ n({ g| g⩾ e^(λ_1 + ε)n})

But, according to theorem <ref>, there are C,t∈_+^∗ such that

    μ^∗ n({ g| g⩾ e^(λ_1 + ε)n}) ⩽ Ce^-tn

And so, for n∈ large enough, we have that for any x∈^d,

    |P^n f(x)| ⩾(1-2Ce^-tn) _B(0,e^-(λ_1 + ε)nδ)(x)

In particular, for any L∈,

    ∫ |P^n f(x)|^L ν⩾( 1-2Ce^-tn)^L ν(B(0, e^-(λ_1 + ε)nδ)) =  ( 1-2Ce^-tn)^L e^-(λ_1 + ε)dnδ^d

And this proves that

    lim_p  p κ(μ,𝒞^0,α_0(^d), L^p(^d,ν) ) ⩽λ_1 d

And this finishes the proof of the proposition.


We are now going to extend the previous result to measures on SL_d() ⋉^d by proving the

Let μ be a borelian probability measure on :=SL_d() ⋉^d. Let μ_0 be the projection of μ onto _0:=SL_d() and assume that μ_0 is strongly irreducible, proximal and has an exponential moment.






Then, for any α∈ ]0,1] small enough,

    lim_p→+∞  p κ(μ,𝒞^0,α_0(^d), L^p(^d,ν) ) ⩾λ_1 d




Theorem <ref> actually proves that for any measure μ satisfying it's assumptions we have that for any f∈𝒞^0,α(^d) with∫ fν=0, any n∈ and any p∈^∗,

    ∫ |P^n f|^p ν⩽ C^p e^-tpnf_α^p

And so, for any p∈ [1,+∞[,

    κ( μ,𝒞^0,α_0(^d), L^p(^d,ν) ) ⩾ t

And, in particular,

    lim_p→ +∞ p κ(μ,𝒞^0,α_0(^d), L^p(^d,ν) )  = +∞



We are going to prove this result in three steps. First, we are going to prove it for trigonometric functions, then, for regular ones and last, for hölder-continuous functions.


Let μ be a borelian probability measure on  and μ_0 it's projection on _0. Denote by P the Markov operator associated to μ and by  P_0 the one associated to μ_0.

Then, for any c ∈^d, any n∈ and any L∈,

    ∫_^d |P^n e_c|^2Lν⩽∫_^d |P_0^n e_c|^2Lν

Where, for c∈^d, e_c is the function defined for x∈^d by

    e_c(x) := e^2iπ⟨ c,x⟩




Using Fubini's theorem, we can make the following computation

    ∫_   |P^n_0 e_c(x)|^2Lν (x)=∫_ (P^n_0 e_c(x))^L (P^n_0f(x))^Lν(x) 
       = ∫_∫_^2Le_c(a_1x) … e_c(a_L x)  e_c(a_L+1x) … e_c(a_2Lx)μ_0^∗ n(a_1) …μ_0^∗ n(a_2L)ν(x) 
       =  ∫_^2L∫_ e^2iπ⟨ c,(a_1+ … + a_L - (a_L+1 + … + a_2L))x⟩ν(x) μ_0^∗ n(a_1) …μ_0^∗ n(a_2L) 
       = ∫_^2L_{^t(a_1+ … + a_L - (a_L+1 + … + a_2L))c=0}μ_0^∗ n(a_1) …μ_0^∗ n(a_2L)

Doing the same kind of computations for the measure μ, and noting, to simplify notations, for (a_1,b_1), …, (a_2L, b_2L) ∈μ, a_i^j = a_i +… + a_j and b_i^j = b_i + … + b_j, we find that

    ∫_ |P^n e_c|^2Lν   = ∫_^2L e^2iπ⟨ c,b_1^L - b_L+1^2L⟩_{^t(a_1^L - a_L+1^2L)c=0}μ^∗ n(a_1,b_1) …μ^∗ n(a_2L,b_2L) 
       ⩽∫_^2L_{^t(a_1^L- a_L+1^2L)c=0}μ_0^∗ n(g_1) …μ_0^∗ n(g_2L) =∫_ |P^n_0 e_c|^2Lν

Where the last inequality comes from the first part of the proof and precisely gives what we intended to prove.


For s∈_+^∗, we denote by ℋ^s(^d) the Sobolev space of exponent s.


With the same assumptions than in corollary <ref>, for any s∈_+^∗ large enough and any ε∈_+^∗, there is L∈_+ such that for any f∈ℋ^s(^d) and any n∈,

    ∫| P^n f-∫ fν|^2Lν⩽ C e^-(λ_1d - ε)nf_ℋ^s(^d)^2L





Let f∈ℋ^s(^d). Then, by definition, we can expand f in Fourier series : f = ∑_c∈^df(c) e_c with f_ℋ^s:=(∑_c∈^d (1+c^2)^s/2 |f(c)|^2)^1/2 <+∞ and so, for any L∈^∗,

    (∫_^d |P^n f|^2Lν)^1/2L⩽∑_c∈^d |f(c)| (∫_^d  |P^n e_c|^2Lν)^1/2L

Using the previous lemma and proposition <ref>, we get that for any ε∈_+^∗, there is L∈_+ such that for any c∈^d∖{0},

    ∫_^d |P^n e_c(x)|^2Lν(x) ⩽ C e^-(λ_1d- ε)nc^2L

Combining this inequality with the previous one, we get that for any f ∈ H^s(^d) with f(0) = ∫ fν=0,

    (∫_^d |P^n f|^2Lν)^1/2L⩽ C^1/2L e^-(λ_1 d-ε)n/2L∑_c∈^d∖{0} |f(c)|c⩽ C' e^-(λ_1 d-ε)n/2Lf_ℋ^s

For some constant C' depending on d, s, μ, L but non on f.




According to Jackson-Bernstein's lemma, for any α∈ ]0,1] and any s∈_+ large enough, there is a constant C such that for any f∈ 𝒞^0,α(^d), there is a sequence (f_m)∈ℋ^s(^d)^ such that for any m∈^∗,

    ∫ fν = ∫ f_mν,   f-f_m_∞⩽C/m^αf_α and f_m_ℋ^s⩽ Cmf_α
 
























































This implies that for any x∈^d and any m,n∈^∗,

    |P^n f_m(x)| ⩾ |P^n f (x)| - C/m^αf_α











Let ε∈_+^∗, m∈^∗ and t=2C/m^α. Then, using the equality t-C/m^α = t/2 and lemma <ref>, we get that

    ∫ |P^n f|^2Lν   ⩽ (tf_α)^2L + ν({ |P^n f|⩾ tf_α}) f_∞^2L
       ⩽(t^2L + ν({ |P^n f_m| ⩾(t-C/m^α) f_α}) ) f_α^2L
       ⩽(t^2L + ( 2/tf_α)^2M∫ |P^n f_m|^2Mν)f_α^2L
       ⩽(t^2L + ( 2/tf_α)^2M C^2M m^2M e^-(λ_1 d -ε)nf_α^2M)f_α^2L

So, for m=e^δ n, we get that for some constant C',

    ∫ |P^n f|^2Lν⩽ C' (e^-δα 2L n + e^δ (1+α)2M n -(λ_1 d - ε) n)f_α^2L

And so, for δ small enough and L large enough, we get that

    ∫ |P^n f|^2Lν⩽ C e^-(λ_1 d-2ε)nf_α^2L

And this is what we intended to prove.




 §.§ Equidistribution, lower regularity and spectral gap.


In this subsection, we finish the proof of theorem <ref> by studying the link between equidistribution and the lower regularity of the measure when the spectral gap is large.

 
Under the same assumptions as in theorem <ref>, for any ε∈_+^∗, there are C,t ∈_+^∗ such that for any x,y∈^d, any f∈𝒞^0,α(^d) and any n∈,

    |P^n f(x) - P^n f(y)| ⩽( e^α(λ_1 + ε) n d(x,y)^α + Ce^-tn) f_α




Let's compute, for any x,y∈^d, n∈, f∈𝒞^0,α(^d) and ε∈_+^∗,

    |P^n f(x) - P^n f(y) |    = |∫_ f(gx) - f(gy) μ^∗ n(g) | 
       ⩽∫__g⩽ e^(λ_1 + ε)n |f(gx) - f(gy)| μ^∗ n(g) 
       + ∫__g⩾ e^(λ_1 + ε)n |f(gx) - f(gy)| μ^∗ n(g) 
       ⩽ m_α(f) ∫__g⩽ e^(λ_1 + ε)n d(gx,gy)^αμ^∗ n(g)
        + 2f_∞μ^∗ n( { g∈| g⩾ e^(λ_1 + ε)n}) 
       ⩽( d(x,y)^α e^α(λ_1 + ε)n + 2  μ^∗ n( { g∈| g⩾ e^(λ_1 + ε)n}) ) f_α

Where we used the fact that for any x,y∈^d and any g∈,

    d(gx,gy) ⩽g d(x,y)

To conclude, we use theorem <ref> and we get that there are C,t∈_+^∗ such that

    μ^∗ n( { g∈| g⩾ e^(λ_1 + ε)n}) ⩽ Ce^-tn



We are now ready to prove theorem <ref>.

The idea of the proof is that if we have some point x_0 of the torus such that |P^nf(x_0)| ⩾ t, then, on a neighborhood B(x_0,r)  for r≈ e^-λ_1 n we also have that |P^n f(x)| ≈ t. But, the control on κ(μ,𝒞^0,α, L^p) implies that ν(x| |P^n f(x)| ⩾ t) ≈ e^-λ_1 dn and so, we have that ν(x| |P^n f(x)| ⩾ t) ≈ e^-λ_1 dn≈ν(B(x_0,r)) and this proves that { x ||P^n f(x)| ⩾ t} cannot be much bigger than B(x_0,r).









































Let α,t∈ ]0,1], ϑ∈ℳ^1(^d) and n∈. As, for any 0<α'<α the inclusion of 𝒞^0,α(^d) into 𝒞^0,α'(^d) is continuous, we may assume without any loss of generality that α is small enough so that corollary <ref> holds.

Assume that 𝒲_α(μ^∗ n∗ϑ,ν)⩾ t.

By definition, there is f∈𝒞^0,α(^d) with f_α⩽ 1 and such that

    |∫ P^n fϑ - ∫ f ν| ⩾t/2

We can assume without any loss of generality that ∫ fν=0 and f_α⩽ 2.
And this proves that

    ∫_^d|P^n f(x)| ϑ(x) ⩾|∫_^dP^n f(x) ϑ(x)| ⩾t/2

We set, for any n∈ and t∈ ]0,1],

    X_n,t := {x∈^d | |P^n f(x)| ⩾ t}

Then, using that P^n f_∞⩽f_∞⩽ 2, we find that

    t/2⩽∫_ |P^n f(x)| ϑ(x) ⩽t/4 + 2 ϑ(X_n,t/4)

And so,

    ϑ( X_n,t/4) ⩾t/8

Moreover, according to lemma <ref>, for any ε_2∈_+^∗, there are C,t_0∈_+^∗ such that for any x∈ X_n,t/4 and any y∈^d, we have that

    |P^n f(y)| ⩾t/4-e^α(λ_1 + ε_2)n d(x,y)^α - Ce^-t_0 n⩾t/8 -e^α(λ_1 + ε_2)n d(x,y)^α

Since we can take c_0 so large that Ce^-t_0 n⩽t/8 for n⩾ c_0(1+|ln t|).


In particular, noting r= e^-(λ_1 + ε_2 )n(t/16)^1/α, we have that for any x∈ X_n,t/4, B(x, r)⊂ X_n,t/16.

Moreover, according to the classical covering results, there is a constant C(d), depending only on d and points x_1, …, x_N ∈^d such that

    X_n,t/4⊂⋃_i=1^N B(x_i,r) ⊂ X_n,t/16

and the union has multiplicity at most C(d). 

This implies in particular that

    ∑_i=1^N _B(x_i,r)⩽ C(d) _X_n,t/16

And so, taking the integral against the measure ν and using the equality ν(B(x,r)) = r^d, we get

    Nr^d ⩽ C(d) ν(X_n,t/16)

To sum-up, we found points x_1,…, x_N with N⩽C(d) ν(X_n,t/16)/r^d, such that

    ϑ(⋃_i=1^N B(x,r)) ⩾t/8

So, noting ℐ:={i∈ [1,N] | ϑ(B(x_i,r)) ⩾t/16N}, we get that

    ϑ(⋃_i∈ℐ B(x_i,r)) ⩾t/16

And finally, for any x∈⋃_i∈ℐ B(x_i,r), there is, by definition of ℐ, some i ∈ℐ such that

    B(x,2r) ⊃ B(x_i, r)

And so,

    ϑ(B(x,2r)) ⩾t/16N

In conclusion, we proved that

    ϑ({x∈^d | ϑ(B(x,2r)) ⩾t/16 N}) ⩾t/16

To finish, note that we have that

    N ⩽ C(d) ν(X_n,t/16)/r^d = C(d) (16/t)^d/α e^(λ_1 + ε_2)dnν(X_n,t/16)

And that, according to Markov's inequality and corollary <ref>, for any ε_1 ∈_+^∗, there are C,L∈_+ such that

    ν(X_n,t/16)    ⩽(16/t)^2L∫_ |P^n f(x)|^2Lν(x)⩽(16/t)^2L Ce^-(λ_1d-ε_1)nf_α^2L
       ⩽(32/t)^2L Ce^-(λ_1d-ε_1)n





This proves that for some constant C depending on ε_1,d,μ,

    N ⩽ C/t^C e^(ε_1 + ε_2 d ) n

And so, taking ε_1,ε_2 small enough and c_0 large enough, we get that

    t/16 N⩾ t^C+1/C e^-(ε_1+ ε_2 d) n⩾ (2r)^ε

and this is what we intended to prove.



















































































































































































§ MEASURE OF POINTS-STABILIZERS


The aim of this section is to prove the following

with the same assumptions as in theorem <ref>, there are C,t ∈_+^∗ such that for any x∈^d and any n∈,

    μ^∗ n⊗μ^∗ n({g_1,g_2 ∈| g_1 x = g_2 x}) ⩽ Ce^-tn



This proposition will be a direct corollary of lemmas <ref> and <ref> since if μ is not concentrated on SL_d()⋉^d, then the measure μ_1 of lemma <ref> is not concentrated on SL_d() ⋉{0}.

To evaluate the measure of the stabilizer of a point, we are going to lift the situation from ^d to ^d since we know better the products of random elements of SL_d() ⋉^d (through the theory of products of random matrices since this group can be identified to a subgroup of SL_d+1()) than those of elements of SL_d() ⋉^d.

To understand what we are going to do, remark that if μ = 1/2δ_(g_1,v_1) + 1/2δ_(g_2,v_2) with g_1,g_2∈SL_d(), v_1∈^d/^d and v_2∈^d such that the coefficients of v_2 and 1 are -linearly independent, then, for μ^∗ n-a.e. (g,v)∈SL_d()⋉^d, we can write

    v=M_1 v_1 + M_2 v_2

with M_i ∈ℳ_d().

So, in particular, noting v_1, v_2 some representatives of v_1,v_2 in ^d, we get that if v=0 in ^d, then there is p∈^d, such that,

    M_1 v_1 + M_2 v_2 = p

As we assumed that the coefficients of v_2 and 1 are -linearly independent and that v_1∈^d, we get that M_2 v_2 = 0 and M_1 v_1 = p.

So, we set

    μ_1 = 1/2δ_(g_1,0) + 1/2δ_(g_2, v_2)∈ℳ^1 (SL_d() ⋉^d )

and what we just proved is that

    μ^∗ n( SL_d() ⋉{0}) ⩽μ_1^∗ n( SL_d() ⋉{0})

And so we are let with a problem on probability measures on SL_d() ⋉^d.

Thus, to lift the situation from ^d to ^d, we are going to project the translation part of elements in the support of μ onto a complementary subspace of ^d in the -vector space ^d. To do so, we fix some -linear projection π_:^d →^d onto ^d and we remark that for any v∈^d/^d, v-π_ v ∈^d is well defined and the application SL_d() ⋉^d ∋ (g,v) ↦ (g,v-π_ v) ∈SL_d() ⋉^d is a group-morphism.


Now, we can prove the following
 
Let μ be a borelian probability measure on SL_d() ⋉^d.

Then, for any n∈ and any y∈^d,

    μ^∗ n⊗μ^∗ n( {(g_1,g_2) | g_2^-1 g_1 ∈Stab(y)}) ⩽μ_1^∗ n⊗μ_1^∗ n( {(g_1,g_2) | g_2^-1 g_1 ∈Stab(y-π_ y)})

where μ_1 is the measure on SL_d() ⋉^d defined by μ_1 (A) = μ(φ^-1(A)) where φ: SL_d() ⋉^d →SL_d() ⋉^d is the function defined by

    φ(g,v) = (g,v-π_ v)

and π_ is some -linear projection onto ^d.



As φ is a morphism, we only need to prove that for any g∈SL_d() ⋉^d and any y∈^d, if gy=y then φ(g) (y-π_ y) = (y-π_ y).

Write g=(a,b) with a∈SL_d() and b∈^d. Then, φ(g) = (a,b-π_ b).
So,

    φ(g) (y-π_ y) = a (y-π_ y) + b-π_ b

But, gy=y, so, noting b, y some representatives of b,y, we get that there is p∈^d such that

    a y + b = y+p

Projecting onto ^d, we also get that

    a π_y + π_b = π_y+p

And this proves that

    a (I_d-π_)y+ (I_d-π_)b = (I_d-π_)y

Finally, as (I_d-π_)y and (I_d-π_)b don't depend on the choices of the representatives of y and b, this finishes the proof of the fact that

    a(y-π_ y) + b-π_ b = y-π_ y

and this finishes the proof of the lemma.








































Let μ be a borelian probability measure on SL_d() ⋉^d that is not concentrated on SL_d() ⋉{0} and has an exponential moment. Let μ_0 be the projection of μ onto SL_d() and assume that μ_0 is strongly irreducible and proximal.


Then, there are C,t ∈_+^∗ such that for any x∈^d and any n∈,

    μ^∗ n⊗μ^∗ n( {(g_1,g_2) | g_1x= g_2 x}) ⩽ Ce^-tn




We denote by μ̃ the measure on SL_d()⋉^d defined by μ̃(A) = μ(A^-1) for any borelian subset A of SL_d()⋉^d and where A^-1 := {g^-1  | g∈ A}.


Let λ_1 ⩾…⩾λ_d be the Lyapunov exponents of μ_0 (see appendix <ref>).

Then, the largest Lyapunov exponent of μ̃ is -λ_d. Moreover, as λ_1 ⩾…⩾λ_d, λ_1>0 and λ_1 + … + λ_d = 0, we have that λ_d <0 and so, -λ_d>0.




Let n∈, ε∈_+^∗ and x∈^d. We can compute,

    I_n(x):   =μ^∗ n⊗μ^∗ n({g_1,g_2 ∈| g_1 x=g_2x }) = μ^∗ n⊗μ̃^∗ n({(g_1,g_2) ∈^2 | g_2g_1 x=x }) 
       ⩽μ^∗ n⊗μ̃^∗ n({(g_1,g_2)| 1+g_2^-1 g_1x/1+x⩽ e^(λ_1 - λ_d- ε)n})

But

    1+g_2 g_1x/1+x = 1+g_2 g_1x/1+g_1x1+g_1x/1+x

So, we obtain that

    I_n(x)    ⩽μ^∗ n({ g_1| 1+g_1x/1+x⩽ e^(λ_1 - ε/2)n}) 
       + ∫_μ̃^∗ n({ g_2| 1+g_2g_1x/1+g_1x⩽ e^(-λ_d - ε/2)n}) μ^∗ n(g_1)













And we can conclude with corollary <ref> applied to the measures μ and μ̃.














































































































§ EFFECTIVE SHADOWING LEMMAS
 


The aim of this section is to prove a criterion to produce measures satisfying an effective shadowing lemma.

First of all, we recall the

Let μ be a borelian probability measure on .

We say that μ satisfies an effective shadowing lemma if for ant C',t' ∈_+^∗, there are C_1,C_2,M,t,L ∈_+^∗ such that for any x,y∈^d, any r∈_+^∗ and any n∈ with r⩽ C_1e^-Ln, if

    μ^∗ n({ g∈| d(gx,y) ⩽ r }) ⩾ C_2 e^-t n

then there are x',y'∈^d such that d(x,x'),d(y,y')⩽ re^Mn and

    μ^∗ n({ g∈| gx'=y'})⩾ C'e^-t'n



The criterion that we are going to prove will use the diophantine properties of the translation parts of the elements in μ. More specifically, we want a condition that ensures that if (g_1,v_1), (g_2,v_2)∈μ^∗ n are such that v_1 and v_2 are close (in some sense) then v_1=v_2. This is why we make the following 


Let d∈^∗ and B⊂^d a finite subset.

We that that B is (C,L)-diophantine if for any non zero (M_b)∈^B,

    d(∑_b∈ B M_b b,0)⩽ C/max_b |M_b|^L⇒∑_b M_b b=0







More generally, we say that B is diophantine if it is (C,L)-diophantine for some C,L∈_+^∗.



With this definition, a diophantine subset can contain rational points.



Let b∈^1 ∖/. Asking {b} to be diophantine is asking for C,L∈_+^∗ such that for any q∈^∗,

    d(qb,0) > C/q^L

The name comes from this property.



Let d⩾ 2 and B a subset of ^d. It is not the same thing to say that B is diophantine and that {coefficients of b| b∈ B} is a diophantine subset of ^1 (consider B={(b_1,b_2)} with b_1 diophantine and b_2 not). This last property is stronger but this is the one that we will need in the sequel and we refer to lemma <ref> for more details.



 
Let N∈^∗. Then, for a.e. b_1, … b_N∈^1, the set {b_1, … b_N} is diophantine.


We are now ready to state the main result of this section
 
Let μ be a borelian probability measure on SL_d() ⋉^d and let μ_0 be it's projection on SL_d(). Assume that μ_0 is strongly irreducible, proximal and that it has an exponential moment and that {coefficients of b| (a,b)∈μ} is a diophantine subset of ^1.

Then, μ satisfies an effective shadowing lemma.


To prove this proposition, we first come back to the difference for a subset B of ^d between being diophantine and having elements whose coefficients form a diophantine subset of ^1.

 
Let B be a finite subset of ^d and

    F:= {coefficients of B}⊂^1

Then, F is (C,L)-diophantine if and only if there is C'∈_+^∗ such that for any non zero (M_b) ∈ℳ_d()^B,

    d(∑_b∈ B M_b b,0) ⩽C'/(max_b M_b)^L⇒∑_b M_bb=0




First, assume that F is (C,L)-diophantine and set C'= C/(d|B|)^L.

Let 0≠(M_b) ∈ℳ_d()^B be such that

    d(∑_b∈ B M_b b,0) ⩽C'/(max_b M_b)^L

Each coefficient of ∑_b M_bb is a sum of elements of F multiplied by integers that are smaller than d |B| max_b M_b. In other words, for any coefficient of ∑_b M_bb, we get a sum ∑_f∈ F L_ff with |L_f| ⩽ d|B| max_b M_b and

    d( ∑_f L_ff,0) ⩽C'/max_b M_b^L⩽C'(d|B|)^L/max_f|L_f|^L = C/(max_f |L_f|)^L

And as F is (C,L)-diophantine, this implies that ∑_f L_ff=0 and so, as this is true for any coefficient of ∑_b M_bb, we get that ∑_b M_bb=0.


Reciprocally, if there is C' ∈_+^∗ such that for any 0≠(M_b) ∈ℳ_d()^B 

    d(∑_b∈ B M_bb,0) ⩽C'/(max_b M_b)^L⇒∑_b M_bb=0

Then, we set C= C'/d^L and let 0≠(L_f) ∈^F be such that

    d(∑_f∈ FL_ff,0) ⩽ C/(max_f |L_f|)^L

For any element f of F, choose an element (b(f),i(f)) in 

    {(b,i) ∈ B× [1,d] | f is the i-th coefficient of b }

Now, for any b∈ B, we denote by M_b the matrix where we set L_f in the i-th column if there is f∈ F such that (b(f),i(f)) = (b,i) and 0 otherwise.

Thus, by definition,

    ∑_b∈ B M_bb = ([ ∑_f L_ff;        ⋮; ∑_f L_ff ])

And max_b M_b⩽ d max_f |L_f| so

    d(∑_b M_bb,0) ⩽ C/(max_f|L_f|)^L⩽ C'/(max_bM_b)^L

And this proves, as B is (C,L)-diophantine, that ∑_b M_bb=0 and so we also get that ∑_f L_f f=0 and this finishes the proof of the lemma.



From now on, we set

    B(μ):={b| (a,b)∈μ} and F(μ):={coefficients of b|b∈ B(μ)}





To prove proposition <ref>, we are going to use some control of the translation parts of elements of μ^∗ n. To do so, for any Q∈^∗ and any finite subset B of ^d, we set

    X_Q(B) = {p + ∑_b∈ B M_bb/q| p∈^d, q∈, |q|⩽ Q, (M_b)∈ℳ_d()^B, max_b∈ BM_b⩽ Q}




Thus some x∈^d belongs to X_Q(B) if there is q∈^∗ with q⩽ Q such that each coefficient of qx can be obtained from translations by coefficients of elements of B with multiplicities smaller than Q.

This definition is made so that for any M large enough we have that for any n∈^∗, with large probability, any element (g,v)∈μ^∗ n is such that v∈ X_e^Mn(B(μ)).

To make this idea more precise, we set, for any M∈_+^∗ and n∈,

    _n^M := {( a_i) ∈_0^n |max_k∈ [1,n]max_1⩽ i_1<…<i_k ⩽ na_i_k… a_ i_1⩽ e^Mn}

The aim of the following lemma is to prove that elements of _n^M are generic.
 
Let μ be a borelian probability measure on  and let μ_0 be the image of μ on SL_d(). Assume that μ_0 has an exponential moment.

Then, for any M∈_+^∗ large enough, there is t∈_+^∗ such that for any n∈,

    μ^⊗ n_0(_n^M) ⩾ 1 - e^-tn

Moreover, if B(μ) is finite (see equation (<ref>)), then, for any (a_1,b_1), … ,(a_n,b_n) ∈μ such that (a_i) ∈_n^M,

    ∑_k=1^n a_n … a_k+1 b_k ∈ X_e^(M+1)n(B(μ))




First, remark that as μ_0 is concentrated on SL_d(), for μ_0-a.e. g∈_0, g⩾ 1 and so, noting δ∈_+^∗ such that ∫__0g^δμ_0(g)<+∞, we have that for any M∈_+^∗ and any n∈,

    μ_0^⊗ n (_n^M):   =μ_0^⊗ n({ (g_i) ∈_0^| max_k∈ [1,n]max_1⩽ i_1<…<i_k ⩽ ng_i_k… g_ i_1⩾ e^Mn}) 
       ⩽μ_0^⊗ n( { (g_i) ∈_0^|  g_n…g_1⩾ e^Mn}) 
       ⩽ e^-δ Mn( ∫__0g^δμ_0(g))^n

And this finishes the proof of the first part of the lemma.

To prove the second one, take (a_1,b_1), … ,(a_n,b_n) ∈μ such that (a_i) ∈_n^M.

Then, we can write

    ∑_k=1^n a_n … a_k+1 b_k = ∑_b'∈ B(μ)(∑_k=1^n _b_k=b' a_n … a_k )b'

and, for any b'∈ B(μ),

    ∑_k=1^n _b_k=b' a_n … a_k ⩽ ne^Mn

This proves that ∑_k=1^n a_n … a_k+1 b_k ∈ X_ne^Mn(B(μ)).


In the sequel, we will have to control sums of the form ∑ c_i a_i where the a_i are μ^∗ n-generic and c_i ∈. To do so, we will use the
 
Let μ_0 be a strongly irreducible and proximal borelian probability measure on SL_d() having an exponential moment.

Set

    _0^1 := { a∈_0 |  |1/nlna- λ_1|⩽ε}

and

    _0^d := { (a_i)∈_0^d | ∀ i∈ [1,d], a_i∈_0^1 and (∑_i c_i a_i)≠0 }

where we put c_1 = 1-d and c_i=1 for i∈ [2,d].

Then, for any ε∈_+^∗, there are C,t∈_+^∗ such that for any n∈,

    μ^∗ n(_0^1) ⩾ 1-Ce^-tn et μ^∗ n⊗…⊗μ^∗ n(_0^d) ⩾  1 - Ce^-tn




In <cit.>, this lemma is not stated as we do it here but it corresponds to the lemmas 4.3, 4.6 and 7.9 and to some part of the proof of proposition 7.3.


We are now ready to prove proposition <ref>. The proof consists in two lemmas. In lemma <ref>, we study points that are far from points of X_e^n(B(μ)) and in lemma <ref> we study points that are close to it.


With the assumptions of proposition <ref>, for any M∈_+^∗ large enough there are C,t∈_+^∗ such that for any r∈ ]0,1], any n∈, any x,y∈^d, if

    μ^∗ n({g∈| d(gx,y)⩽ r}) ⩾ Ce^-tn

then there is x'∈ X_e^Mn(B(μ)) such that

    d(x,x') ⩽ re^Mn





The idea that makes the demonstration work is that if a_1 x + b_1, … , a_d x+b_d are close to each-others, then setting c_1=1-d and c_i=1 for i∈ [2,d], we get that ∑_i c_i a_i x is close to ∑_i c_i b_i. But, according to lemma <ref>, with large probability, the matrix ∑_i c_i a_i is invertible and, according to lemma <ref>, the b_i belong to X_e^Mn(B(μ)), and so x∈ X_e^Mn(B(μ)).



We keep the notations _0^1 and _0^d from lemma <ref> and the notation _n^M from lemma <ref>. Then, for any ε∈_+^∗ and any large enough M we denote by C,t the constants given by lemma <ref> and by t_0 the one given by lemma <ref>.

Then, we set

    _∗ := { ((a_1,b_1), …, (a_d,b_d))∈^d |(a_i)∈_0^d  and for any i,  b_i ∈ X_e^(M+1)n(B(μ)) }

By definition, we have that

    μ^∗ n⊗…⊗μ^∗ n(_∗) ⩽ Ce^-tn + d e^-t_0n

So, we can compute, for any x,y∈^d,

    I_n(x,y):   =(μ^∗ n({g∈| d(gx,y)⩽ r}))^d 
       = ∫_^d_B(y,r)(a_1 x+b_1) …_B(y,r) (a_dx+b_d) μ^∗ n(a_1,b_1) …μ^∗ n(a_d,b_d)  
       ⩽ Ce^-tn + de^-t_0n + ∫__∗∏_i=1^d_B(y,r)(a_i x+b_i) μ^∗ n(a_1,b_1) …μ^∗ n(a_d,b_d)

Thus, if M is large enough, we can find C,t ∈_+^∗ such that for any r∈ ]0,1], any n∈, if

    μ^∗ n({g∈| d(gx,y)⩽ r}) ⩾ Ce^-tn

then

    ∫__∗∏_i=1^d_B(y,r)(a_i x+b_i) μ^∗ n(a_1,b_1) …μ^∗ n(a_d,b_d) >0

In particular, there is ((a_1,b_1), …, (a_d,b_d))∈^∗ such that for any i,

    d(a_i x+b_i, y) ⩽ r

Now, we let x, y, b_i be representatives of x,y,b_i in ^d. We have that for any i∈ [1,d] there is p_i ∈^d such that

    a_i x + b_i - y - p_i⩽ r

So, noting c_1 = 1-d and c_i = 1 for i≠1, we get that

    (∑_i c_i a_i)x + ∑_i c_i b_i - ∑_i c_ip_i⩽ 2d r

But, by definition of _0^d, (∑_i c_i a_i )∈^∗.

So,

    1 ⩽|( ∑_i c_i a_i)| ⩽∑_i c_i a_i^d ⩽ d^2d e^(λ_1 + ε)dn

Let U = ∑_i c_i a_i. Then U is invertible and we can write,

    U^-1 = 1/ U V

with V ∈ℳ_d() and for some constant C(d) depending only on d, V⩽ C(d)U^d-1.

Thus, we get that

    x+U^-1∑_i c_i b_i - U^-1∑_i c_ip_i ⩽ 2drU

To conclude, we only need to remark that we can write

    b_i = ∑_b ∈ B(μ) M_g^i b  with max_b M_b^i⩽ e^(M+1)n

And this proves that

    -U^-1∑_i c_i b_i +U^-1∑_i c_i p_i =∑_i c_i V p_i/(U) + ∑_b∈ B( μ)(∑_i c_i VM_b^i)b /(U)

And that

    max_b∈ B(μ)∑_i c_i V M_b^i   ⩽ 2d Vmax_bM_b^i⩽ 2d C(d) e^(M+1)n e^(λ_1 + ε)(d-1)n

So, maybe for some M'⩾ M we get that

    x':=-U^-1∑_i c_i b_i +U^-1∑_i c_i p_i  ∈ X_e^M'n(μ)

and d(x,x') ⩽ re^M'n.







With the same assumptions as in proposition <ref>, there are C_0,L such that for any M∈_+ large enough, there are C,t∈_+^∗ such that for any n∈^∗, any r∈ ]0,1] with

    r ⩽C_0/e^MLn

we have that for any y∈^d, any x'∈ X_e^(M+1)n(B(μ)) and any x∈^d with d(x,x')⩽ r,

    (μ^∗ n({ g∈| d(gx,y) ⩽ r}) )^2⩽ Ce^-tn + μ^∗ n⊗μ^∗ n({g_1,g_2 | g_1x'=g_2 x'})



To simplify our notations, we set B= B(μ) and F=F(μ). 
let x,x',y∈^d as in the proposition. By definition of X_Q(B), there is (M_v)∈ℳ_d() ^B with maxM_v⩽ Q, there is p∈^d and q∈^∗ with |q|⩽ Q such that

    x' = p+ ∑_b∈ B M_b b/q



Let, for any ε∈_+^∗ and M∈,

    _∗:= { (a,b) ∈| a⩽ e^(λ_1 + ε)n et b∈ X_e^(M+1)n(B)}

Lemmas <ref> and <ref> prove that for any ε∈_+^∗ and any M∈_+ large enough, there are C,t∈_+^∗ such that for any n∈,

    μ^∗ n(_∗) ⩾ 1-Ce^-tn

Let's compute

    I_n(x,y):   =(μ^∗ n({ g∈| d(gx,y) ⩽ r}) )^2 
       = ∫_^2_B(y,r)(g_1 x) _B(y,r) (g_2 x) μ^∗ n(g_1) μ^∗ n(g_2) 
       ⩽ 2μ^∗ n(_∗) + ∫_^2__∗(g_1)__∗(g_2) _d(g_1x,g_2x)⩽ 2rμ^∗ n(g_1) μ^∗ n(g_2) 
       ⩽ 2Ce^-tn + ∫_^2__∗(g_1)__∗(g_2) _d(g_1x,g_2x)⩽ 2rμ^∗ n(g_1) μ^∗ n(g_2)

Moreover, for g_1 = (a_1,b_1) and g_2=(a_2,b_2), we have that

    d(g_1 x',g_2 x')⩽ d(g_1 x',g_1 x) + d(g_1x,g_2 x) + d(g_2 x,g_2 x') ⩽a_1 r + d(g_1 x,g_2 x) + a_2r

And this proves that

    I_n(x,y) ⩽ 2 Ce^-tn + ∫_^2__∗(g_1)__∗(g_2) _d(g_1x',g_2x')⩽ 3re^(λ_1 + ε)nμ^∗ n(g_1) μ^∗ n(g_2)

To conclude, we only need to prove that, under the diophantine condition, if x'∈ X_e^(M+1)n, g_1,g_2∈_∗ are such that d(g_1 x',g_2 x') ⩽ 3re^(λ_1+ ε)n then g_1 x'=g_2 x'.

To do so, remark that if x'∈ X_e^(M+1)n(B) and g=(a,b)∈_∗, alors, a x'+b ∈ X_2 e^(M+1 +λ_1 + ε)n(B)

Thus, we have that (g_1 - g_2) x' ∈ X_4e^(M+1+λ_1 +ε)n(B) and that d((g_1 - g_2)x',0) ⩽ 3re^(λ_1 + ε)n.

In particular, there is q∈^∗ with q⩽ 4e^(M+1+λ_1 +ε)n such that q(g_1-g_2)x' is a sum of elements of b multiplied on the left by matrices of norm smaller than (4e^(M+1+λ_1 +ε)n)^2 and

    d(q(g_1-g_2)x',0) ⩽ |q|3r e^(λ_1 + ε)n⩽ 12re^(M+1+2λ_1 +2ε)n

So, as F is (C,L)-diophantine, according to lemma <ref>, there are constants C',L such that if

    12re^(M+1+2λ_1 +2ε)n⩽C'/(4e^(M+1+λ_1 +ε)n)^2L

then we have that qg_1 x'=qg_2 x'. But, this proves that g_1 x' = g_2 x' + p/q for some p∈^d. And

    d(p/q,0)=d(g_1x',g_2 x') ⩽ 3r e^(λ_1 + ε)n

So, if 1/|q| > 3r e^(λ_1 +ε)n, we have that p/q=0 and so, g_1 x'=g_2 x'.




According to lemma <ref>, we have that for any M∈_+^∗ large enough, there are C,t∈_+^∗ such that for any r∈ ]0,1], any n∈ and any x,y∈^d, if

    μ^∗ n({g∈| d(gx,y)⩽ r}) ⩾ Ce^-tn

then there is x'∈ X_e^Mn(B(μ)) such that

    d(x,x') ⩽ re^Mn

But, in this case, we have, according to lemma <ref>, that if

    re^Mn⩽C_0/e^LMn,

then

    (μ^∗ n({ g∈| d(gx,y) ⩽ r}) )^2⩽ Ce^-tn + μ^∗ n⊗μ^∗ n({g_1,g_2 | g_1x'=g_2 x'})

And this is what we intended to prove since

    μ^∗ n⊗μ^∗ n({g_1,g_2 | g_1x'=g_2 x'}) = ∫_μ^∗ n( { g_1 | g_1 x' = g_2 x' }) μ^∗ n(g_2)








§ PRODUCTS OF RANDOM MATRICES
 

In this section, we are going to recall some of the properties of products of random matrices.

To do so, we fix some finite dimensional -vector space  that we endow with an euclidian norm.

Let μ be a borelian probability measure on :=GL().

We set, for any g∈, N(g) = max(g,g^-1) and we say that μ has a moment of order 1 if

    ∫_ln N(g) μ(g)<+∞

and that it has an exponential moment if there is some ε∈_+^∗ such that

    ∫_ N(g)^εμ(g) <+∞

Remark that there is some constant C depending only on () such that for any g∈SL_d(), g^-1⩽ C g^ and so, if μ is a measure on SL(), it is enough to ask that for some ε∈_+^∗,

    ∫_g^εμ(g)<+∞




We would like to study the product g_n … g_1 where (g_i) is an iid sequence of law μ.

The first result in this direction is Oseledec's theorem :

Let μ be a borelian probability measure on :=GL() having a moment of order 1.


Then, there are m_1 , … , m_r ∈^∗ with ∑_i m_i=, there is Λ_1>…>Λ_r ∈ and some measurable function from ^ into the space of flags :=_1 ⊃…⊃_r+1:={0} of  such that _i =  ∑_j=i^r m_j and such that for μ^⊗-a.e. ω =(g_n) ∈^,


  * g_1 _i^ω = _i^ϑω where ϑ is the shift on ^.

  * For any x∈_i^ω∖_i+1^ω,

    lim_n 1/nlng_n … g_1 x=Λ_i




We call Lyapunov exponents the paramaters Λ_1, … ,Λ_r and we note

    λ_1 = … = λ_m_1 = Λ_1,   λ_m_1+1 = … = λ_m_1+m_2 = Λ_2,    etc.

However, if x∈, this theorem doesn't say anything on the behavior of g_n … g_1 x for some generic (g_n)∈^ because we have no information on the sequences (g_n) such that x∈_i^(g_n). To avoid this problem, we usually assume that the subgroup of  spanned by the support of μ acts irreducibly on  (it doesn't fix any non-trivial subspace of ). And in this case, Furstenberg proved the

Let μ be a borelian probability measure on  having a momen tof order 1 and whose support generates a group acting irreducibly on .

Then, for any x∈∖{0},

    1/nlng_n … g_1 xλ_1  a.e.



This irreducibility assumption is not good enough for us since we will identify SL_d() ⋉^d with the subgroup

    ([ SL_d()     ^d;      0      1 ])
 
of SL_d+1() whose action on ^d+1 is not irreducible.

A first important case of reducible actions on  is when the support of μ generates a group

    {( [ Γ_1   ∗;   0 Γ_2 ])}

where Γ_i < SL(_i) with _1 ⊕_2 = and Γ_i acts irreducibly on _i. Indeed, in this case, we can study the action on _1 and on _2 to get the one on . This motivates the following


Let μ be a borelian probability measure on GL().

We say that some subspace  of  is adapted to μ if it is proper, invariant by the subgroup of GL() spanned by the support of μ and if there are Δ_1>Δ_2 ∈ such that


  * For any x∈/∖{0},

    1/nlng_n … g_1 xΔ_1 a.e.


  * For any x∈∖{0},

    lim sup1/nlng_n … g_1 x⩽Δ_2 a.e.




This definition is only useful since there is always an adapted subspace.

 
Let μ be a borelian probability measure on GL() having a moment of order 1.

Then there is some subspace  of  that is adapted to μ.


This theorem proves that we can always block-triangularize the group _μ spanned by the support of μ. Indeed, we can find by induction Δ_1>… >Δ_s ∈ and a flag =_1 ⊃…⊃_s+1:=0 adapted to μ : _i is _μ-invariant and for any i and any x∈_i / _i+1∖{0},

    1/nlng_n … g_1 xΔ_i  a.e.

The flag that we obtain in this way is included in the flag given by Oseledec's theorem (it means that {_i}⊂{_i} and that {Δ_i}⊂{Λ_i} with Δ_1 = Λ_1) but it has the advantage of being invariant and the convergence in equation (<ref>) gives us Furstenberg's law of large numbers without any irreducibility assumption.


We are going to precise the convergence in equation (<ref>) through a non-concentration inequality that we state in next
 
Let μ be a borelian probability measure on GL() having an exponential moment.

Then, for any ε∈_+^∗, there are C,t∈_+^∗ such that for any n∈,

    μ^∗ n( { g ∈GL()| | 1/nlng - Λ_1 | ⩾ε}) ⩽ Ce^-tn

Moreover, if  is adapted to μ then, for any x∈ and any n∈,

    μ^∗ n({ g∈GL()|  e^(Λ_1 - ε)n d(x,) ⩽gx⩽ e^(Λ_1 + ε)nx}) ⩾ 1- Ce^-tn

where we noted

    d(x,) = inf_y∈x-y



To prove this theorem, we first prove the following
 
Let μ be a borelian probability measure on GL() having an exponential moment.

Let Δ_1>…>Δ_s∈ and :=_1 ⊃…⊃_s+1:={0} be the flag adapted to μ and given by induction by theorem <ref>.

Then for any ε∈_+^∗, there are C,t∈_+^∗ such that for any n∈ and any x∈∖{0},

    μ^∗ n( { g ∈GL()| Δ_s - ε⩽1/nlngx/x⩽Δ_1 + ε}) ⩾ 1 - Ce^-tn




The proof of this lemma is an adaptation of the proof of proposition 3.2 in <cit.> where Benoist and Quint only have polynomial moments.

First, for g∈ and X= x∈(^d), we set

    σ(g,X) = lngx/x, φ(X) = ∫_σ(g,X) μ(g)   et σ'(g,X) = σ(g,X) - φ(X)

Then, for any X= x ∈(^d) and any sequence (g_n) ∈GL()^, noting X_k = g_k … g_1 X, we have that

    lng_n … g_1 x/x = ∑_k=0^n-1σ(g_k+1, X_k) = ∑_k=1^n σ'(g_k+1, X_k) + ∑_k=0^n-1φ(X_k)

Now, let M_n = ∑_k=0^n-1σ'(g_k+1,X_k).

We can compute

    [ M_n+1 - M_n | X_0, …,X_n ] = [σ'(g_n+1, X_n) | X_0, … ,X_n ] = ∫_σ'(g,X_n) μ(g) = 0

This proves that M_n is a martingale.
Moreover, for any ε∈_+^∗,

    [ e^ε|M_n+1 - M_n|| X_0, …, X_n ] = ∫_ e^ε|σ'(g,X_n)|μ(g)

And the inequality

    -lng^-1⩽σ(g,X) = lngx/x⩽lng

shows that

    [ e^ε|M_n+1 - M_n|| X_0, …, X_n ] ⩽∫_ e^2εlnmax(g, g^-1)μ(g)

And so, as μ has an exponential moment, there is ε∈_+^∗ and some constant C_0 such that for any X∈() and any n∈,

    _X [ e^ε|M_n+1 - M_n|| X_0, …, X_n ] ⩽ C_0  a.e.

Thus, according to the non-concentration inequality for martingales (see theorem 1.1 in <cit.>), for any ε∈_+^∗, there is C,t∈_+^∗ such that for any X∈() and any n∈,

    _X( |M_n| ⩾ε n)⩽ Ce^-tn

(it is important to remark that the constants C,t don't depend on X but only on ε and C_0).


To conclude, we only need to study ∑_k=0^n-1φ(X_k). To do so, we just proved (using Borel-Cantelli-s theorem) that

    1/n M_n  0  a.e.

Moreover, by definition of Δ_1, … , Δ_s and _1, … ,_s, we have that for any X∈(),

    Δ_s ⩽lim inf_n 1/n∑_k=0^n-1σ(g_k+1, X_k) ⩽lim sup_n 1/n∑_k=0^n-1σ(g_k+1, X_k) ⩽Δ_1  a.e.

This proves that

    Δ_s ⩽lim inf_n 1/n∑_k=0^n-1φ(X_k) ⩽lim sup_n 1/n∑_k=0^n-1φ(X_k) ⩽Δ_1  a.e.

And so, for any stationary probability measure ν on (),

    Δ_s ⩽∫φν = lim_n ∫_x 1/n∑_k=0^n-1φ(X_k) ν(x) ⩽Δ_1

Finally, using proposition 3.1 of <cit.>, we get that for any ε∈_+^∗, there are C,t∈_+^∗ such that for any n∈ and any X∈(^d),

    _X( Δ_s - ε⩽1/n∑_k=0^n-1φ(X_k) ⩽Δ_1+ ε)⩽ Ce^-tn

And this finishes the proof of the lemma.




We refer to the proof of proposition 4.1 in <cit.>.


First, remark that for any basis (v_1 , … , v_d) of  there is a constant C such that for any g∈GL(),

    1/Cgv_1⩽g⩽ C max_i∈ [1,d]gv_i

And so, the non-concentration inequality for lngv/v for any v∈∖{0} implies the one for lng.

Then, remark that according to lemma <ref>, for any ε∈_+^∗, there are C,t∈_+^∗ such that for any x∈ and any n∈,

    μ^∗ n({ g∈GL() | gx > e^(Λ_1 + ε)nx}) ⩽ Ce^-tn


We endow / with the norm

    x_/ := inf_y∈x-y

Then, we have that for any x∈,

    x⩾ d(x,) = π x_/

where π is the projection onto / and, by definition, since  is adapted to μ, for any x∈/∖{0},

    1/nlng_n … g_1 x_/Λ_1  a.e.

This proves that {0} is adapted to the image of μ in GL(/). And so, according to lemma <ref>, for any ε∈_+^∗, there are C,t∈_+^∗ such that for any n∈ and any x∈/∖{0},

    μ^∗ n ({ g| | 1/nlngx/x - Λ_1| ⩾ε}) ⩽ Ce^-tn

And this proves that for any n∈ and any x∈,

    μ^∗ n({ g| gx⩽ e^(Λ_1 - ε)n d(x,) }) ⩽ Ce^-tn

And this finishes the proof of the theorem.


We end this section with the study, for a borelian probability measure μ on SL_d() ⋉^d, of the translation part b of the μ^∗ n-generic elements g=(a,b). The aim is to prove that if we make some assumptions on μ, for any n, with large μ^∗ n-probability, an element g=(a,b)∈SL_d() ⋉^d is such that b≈ e^Λ_1 n.

To do so, we wirst compare the Lyapunov exponents of μ and of it's projection on SL_d() in next


Let μ be a borelian probability measure on SL_d() ⋉^d having an exponential moment and let μ_0 be the projection of μ onto SL_d().


See μ as a probability measure on SL_d+1() and define Λ_1(μ) this way.

Then,

    Λ_1(μ) = Λ_1(μ_0)




First for any n∈^∗ and any ε∈_+^∗,

    μ^⊗({| 1/nlng_2n… g_n+1 - Λ_1 (μ)| ⩾ε}) = μ^⊗({|1/nlng_n … g_1 - Λ_1 (μ)| ⩾ε})

So, according to lemma <ref>,

    1/nln g_2n… g_n+1Λ_1(μ)  μ^⊗-a.e.


We can write, for any g∈SL_d() ⋉^d,

    g =( [ a b; 0 1 ])

and if g has law μ, then a has law μ_0.

Thus,

    g_2n… g_1 = ([                                                       a_2n… a_1 a_2n… a_n+1∑_k=1^n a_n … a_k+1 b_k + ∑_k=n+1^2n a_2n… a_k+1 b_k;                                                               0                                                               1 ])

And so,,

    g_n … g_1 ⩾a_n … a_1

This proves that Λ_1(μ) ⩾Λ_1(μ_0).

Let now Ω∈^ be such that μ^⊗ (Ω)=1 and for any (g_n) ∈Ω,

    lim_n 1/nlng_n… g_1  =lim_n 1/nlng_2n… g_n+1 = Λ_1(μ)

and

    lim_n1/nlna_n… a_1  = lim_n 1/nlna_2n… a_n+1 = Λ_1(μ_0)

This way, for any ε∈_+^∗, there is N_ε such that for any n∈ with n⩾ N_ε,

    e^(Λ_1(μ) - ε)n⩽g_n … g_1  , g_2n… g_n+1⩽ e^(Λ_1(μ) + ε)n

and,

    a_n … a_1 , a_2n… a_n+1⩽ e^(Λ_1(μ_0) + ε)n

Thus, we have that for any ε∈_+^∗ and any large enough n,

    e^2 (Λ_1(μ)-ε)n⩽g_2n… g_1 ⩽max( e^2(Λ_1(μ_0) + ε)n, e^(Λ_1(μ) + Λ_1(μ_0) + 2ε)n + e^(Λ_1(μ) + ε)n)

This proves that for any ε∈_+^∗,

    2(Λ_1(μ) - ε) ⩽max (2Λ_1(μ_0) + ε, Λ_1(μ) + Λ_1(μ_0) + 2ε)

And so, we get that

    2Λ_1(μ) ⩽max( 2Λ_1(μ_0), Λ_1(μ) + Λ_1(μ_0))

Finally, as we already proved that Λ_1(μ_0) ⩽Λ_1(μ), the previous inequality actually is an equality and we get the expected result.



Until now, we didn't say anything on the positivity of Λ_1.

If μ is a measure on SL(), then, as for μ-a.e. g∈, (g)=1, we have that λ_1 + … + λ_() = 0 and so, λ_1=0 if and only if for any i, λ_i=0.

To get conditions that ensure that λ_1>0, we will say that a subgroup  of SL_d() is strongly irreducible if it doesn't fix any non trivial finite union of subspaces of ^d.


We remind the following result
 
Let μ be a borelian probability measure on SL() having a moment of order 1 and such that the subgroup spanned by the support of μ is strongly irreducible and non-compact.

    ∫_SL_d()lngμ(g) <+∞


Then, Λ_1 >0.  





Let μ be a borelian probability measure on SL_d() ⋉^d having an exponetial moment and that is not concentrated on SL_d() ⋉{0}. Assume that the projection onto SL_d() of the subgroup spanned by the support of μ is strongly irreducible and non-compact.


Then, for any ε∈_+^∗, there are C,t∈_+^∗ such that for any n∈ and any x∈^d,

    μ^∗ n({ g ∈SL_d() ⋉^d |  | 1/nln1+gx/1+x - Λ_1 | ⩾ε}) ⩽ Ce^-tn




We see μ as a probability measure on SL_d+1() and we are going to prove that the subspace  adapted to μ and given by theorem <ref> is {0}.

First, since the projection of μ on SL_d() spans a group _μ that acts strongly irreducibly and non-compactly on ^d, the only subspaces of ^d+1 that can be invariant by μ are ^d+1, Vect(e_1, … e_d), Vect(e_d+1) and {0}. But, assuming that μ is not a subset of SL_d() ⋉{0} implies that Vect(e_d+1) is not invariant by the group spans by the support of μ. Moreover, according to theorem <ref>, there is Λ_1∈_+^∗ such that for any x∈Vect(e_1,…, e_d) ∖{0},

    1/nlng_n … g_1 xΛ_1  a.e.

Then, for any x∈^d+1∖Vect(e_1, …, e_d+1), we have, according to lemma <ref> that

    lim sup_n 1/nlng_n … g_1 x⩽Λ_1

So we also have that ≠Vect(e_1, … ,e_d) and, as  is proper, we have that ={0}.

Thus, according to lemma <ref>, for any ε∈_+^∗, there are C,t∈_+^∗ such that for any x∈^d+1∖{0},

    μ^∗ n( {g ∈SL_d() ⋉^d | | 1/nlngx/x- Λ_1 | ⩾ε}) ⩽ Ce^-tn

And in particular, with x=x_0+e_d+1 for x_0∈Vect(e_1, …, e_d) we get the expected result.




amsalpha


