Análisis y gráficos de cada una de las preguntas que integraron el Cuestionario

We apply Theorem4.10to µ^∗-potentials of suitable functions; we pass, using the map η → bη, from the convolution equation λ ∗ ρ = ρ to a Poisson type equation on V \ {0}

which involves µ^∗and bρ and we note that t^α(t⁻¹·ρ)(Hu⁺) = t^αρ(u, t ). The correspondingb convergences will play an essential role in the proof of Theorem 5.2. We denote by ρ1

the law of R − B and we consider the signed measure ρ0= ρ − ρ1, hence ρ₀(V ) = 0. Also

we show that ρ0is "small at infinity", and we define C ,C₊,C₋,σ^α. With the hypothesis of Theorem 5.2, we denote by^∗eν^αu the positive kernel on S^d−1given by Theorem4.10 and associated with µ^∗.

Proposition 5.3. One has the equations on V \ {0}, ρ =^∞Σ

0µ^k∗ (ρ − ρ1), bρ(v ) =^∞Σ

0((µ^∗)^k∗ δv)( bρ − bρ₁).

For u ∈ S^d−1, the function t → t^α−1ρb₀(u, t) is Riemann-integrable in generalized sense on ]0, ∞[ and, one has with rα(u) =R_∞

0 t^α−1ρb₀(u, t)d t, p(α) =R

|〈x, y〉|^αdν^α(x)d^αν^α(y)

t→∞limt^αρ(u, t ) =b

∗e^α(u)

L_µ(α) ^∗eν^αu(rα) = C(σ^α⊗ ℓ^α)(H_u⁺),

where C = 2_L^∗_µ^eν_(α)p(α)^α(rα)α ≥ 0 and σ^α∈ M¹(eΛ(T )) are defined by the above convergence and σ^α⊗ ℓ^αsatisfiesµ ∗ (σ^α⊗ ℓ^α) = σ^α⊗ ℓ^α.

Furthermore suppρ is unbounded and,

• In case I:σ^α= eν^α,

• In case II:^∗eν^αu(rα)σ^α=¹₂(^∗ν^α₊(rα)ν^α₊+^∗ν^α₋(rα)ν^α₋) where^∗ν^α₊(rα) ≥ 0,^∗ν^α₋(rα) ≥ 0.

The proof will follow from a series of lemmas.

We start with the following simple Tauberian lemma.

Lemma 5.4. For a non negative and non increasing function f on R^∗₊ and s ≥ 0, we denote, f^s(t) =¹_t R_t

0x^sf (x)d x. Then the condition lim

t→∞f^s(t) = c implies lim

t→∞t^sf (t ) = c.

Proof. Let b be a positive real number with b > 1 and let us observe that, since f is non increasing

1 t

Zbt t

x^sf (x)d x ≤ f (t)1 t

Zbt t

x^sd x = t^s

s + 1(b^s+1− 1)f (t).

It follows

b^s+1− 1

s + 1 t^sf (t ) ≥ b f^s(bt) − f^s(t).

Then the hypothesis gives :

liminf

t→∞

b^s+1− 1

s + 1 t^sf (t ) ≥ (b − 1)c.

Using the relation lim

We will use below the multiplicative structure of the group R^∗₊=]0,∞[, and we recall that Haar measure ℓ on the multiplicative group R^∗₊is given by ^{d t}_t .

Lemma 5.5. Assume that the V -valued random variable R satisfies equation (S), and E(|B|^α+δ) < ∞, with δ > 0. For u ∈ S^d−1and t , x > 0 we write r^α(u, t) =¹_tR_t

0x^αρb₀(u, x)d x.

Then |r^α(u, t)| ≤ 2_α+1^tα . Forδ^′small there exists C (δ^′) > 0 such that if t ≥ 1, |r^α(u, t)| ≤ C (δ^′)t^−δ′. In particular the function r^α(u, t) is b.R.i on S^d−1× R^∗₊.

Proof. Sinceρ₀ is the difference of the laws of R and R − B we have by definition of b

Hence the first term I₁^ε(t) in the above inequality satisfies

I₁^ε(t) ≤ E(|B|^α+δ)1

In the second integral above we use the change of variables x → x − x^εand we get

I₂^ε(t) ≤1 t

Z_t

2 [x^α− (x − x^ε)^α(1 − ε x^ε−1)]bP{< R − B,u >> x − x^ε}d x +k₀(ε) t

with 0 < k0(ε) < ∞.

We observe that there exists k1(ε) < ∞ such that for any x ≥ 2, x^α− (x − x^ε)^α(1 − ε x^ε−1) ≤ k1(ε)x^α+ε−1.

For any β ∈]0,α[, Proposition 5.1 implies that E(|R|^β) < ∞. Also R satisfies equation (S) and A, R ◦ bθ are independent. Hence Markov’s inequality gives

b

P{〈R − B,u〉x} ≤ x^−βE(|A|^β)E(|R|^β) ≤ k2(β)x^−βwith k₂(β) < ∞.

It follows that for any t with t − t^ε> 2

I₂^ε(t) ≤k₀(ε)

t + k1(ε)k₂(β)1 t

Z_t

2

x^α+ε−1

(x − x^ε)^βd x ≤ k3(ε,β)t^{α−β+ε−1}.

It remains to choose δ,ε,β in order to obtain α + ε − 1 − β < 0. We take δ so small that δ(α+δ)⁻¹< α and ε = (α+δ/p)(α+δ)⁻¹with p ∈ N, p ≥ 2, hence ε ∈]α(α+δ)⁻¹,1[. Also α + ε − 1 = α − δ(1 − 1/p)(α + δ)⁻¹> 0.

We take γ ∈]0,1[ and β = α − γδ(1 − 1/p)(α + δ)⁻¹so that

α + ε − 1 − β = (γ − 1)δ(1 − 1/p)(α + δ)⁻¹.

With p = 2, γ = 1/2 we get α+ε−1−β = −δ/4(α+δ)⁻¹. We write δ^′=inf (δ/4(α+δ)⁻¹,δ/2).

Hence, there exists k3< ∞ and δ^′> 0 such that for t ≥ 1, r₁^α(u, t) ≤ k3t^−δ′.

The same argument is valid for r₂^α, hence for some δ^′> 0 and t ≥ 1, we have r^α(u, t) ≤ C (δ^′)t^−δ′, with C (δ^′) < ∞. Furthermore, for t ∈]0,1] we have |r^α(u, t)| ≤ 2_α+1^tα , hence the function r^α(u, t) is b.R.i. on S^d−1× R^∗₊.

Lemma 5.6. We denote by r the finite measure on R^∗₊ defined by r (d x) = 1^]0,1[(x)x^αd x and we writeρ₀= ρ − ρ1. Then the function h_αon V \ {0} defined by

h_α(v) = |v|^−α(r ∗ bρ₀)(v) =1 t

Z_t

0 x^αρb₀( ˜v, x)d x ,

is b.R.i and one has (δ_u⊗ ℓ^α)(r ∗ bρ₀) =R_∞

0 t^α−1ρb₀(u, t)d t = rα(u) where t → t^α−1ρb₀(u, t) is Riemann-integrable on ]0, ∞[ in generalised sense.

Proof. By definition

Proof. In order to show that suppρ is unbounded, in view of Proposition 5.1, it suffices to show that there exists g ∈ T with lim

n→∞|gⁿ|^1/n> 1. If not, then the trace Tr g of g is bounded by d on T . On the other hand, condition i-p implies the irreducibility of the action of T on V ⊗C, as shown now. Let W ⊂ V ⊗C be a proper T -invariant subspace of V ⊗C and W its complex conjugate. Then W ∩W and W +W are complexified subspaces of subpaces of V which are also T -invariant. Using irreducibility of T we get W ∩W = {0}, W +W = V ⊗C hence V ⊗C = W ⊕W . Let g ∈ T^proxand v ∈ V \{0} with g v = λgv and write v = w + ¯w with w ∈ W , hence since W,W are g -invariant g w = λ^gw, g ¯w = λ^gw.¯ Since λgis a simple eigenvalue we get w = ¯w i.e W ∩W 6= {0} which gives a contradiction with condition i-p, hence T acts irreducibly on V ⊗C. Then Burnside’s density theorem implies that EndV ⊗ C) is generated as an algebra by T , i.e. there exists a base gi ∈ T

is bounded, right continuous and non increasing. Since equation (S) can be written as

R − B = AR ◦ bθ and A, R ◦ bθ are independant we have ρ₁= µ ∗ ρ, ρ − µ ∗ ρ = ρ − ρ1= ρ0. Furthermore,

ρ =Σⁿ

0µ^k∗ ρ0+ µⁿ⁺¹∗ ρ, bρ(v ) =Σⁿ

0((µ^∗)^k∗ δv)( bρ₀) + ((µ^∗)ⁿ⁺¹∗ δv)( bρ).

Also if r⁻denote the push-forward of r by x → x⁻¹then r⁻∗ρ =Σⁿ

0µ^k∗(r⁻∗ρ0)+µⁿ⁺¹∗ (r⁻∗ ρ).

Since Lµ < 0 the subadditive ergodic theorem applied to log|Sn(ω)| gives the conver-gence of Sn(ω)v to 0. In particular, for ξ ∈ M¹(V ), the sequence µⁿ∗ ξ converges in law to δ₀, hence lim

n→∞µƒⁿ∗ ξ(v) = (µ^∗)ⁿ∗ δv(bξ) = 0.

From the above convergence on V ,we have ρ − δ0=^∞Σ

0µ^k∗ ρ0, bρ(v ) =^∞Σ

0((µ^∗)^k∗ δv)(cρ₀).

But, by Proposition 5.1, ρ({0}) = 0, hence the stated vague convergence of^∞Σ

0µ^k∗ ρ0. Since the sequence (µⁿ⁺¹∗(r⁻∗ρ))(ψ) converges to zero for any bounded Borel function ψ on V such that lim

v →0ψ(v ) = 0, we have on such functions : r⁻∗ ρ =^∞Σ

0(µ^∗)^k∗ (r⁻∗ ρ0).

We observe that, for any bounded measure ξ, we have (µ^k∗ bξ)(v ) = ((µ^∗)^k∗ δv)(bξ) and rƒ⁻∗ ξ = r ∗ bξ. It follows from the above equality that the potential^∞Σ

0((µ^∗)^k∗ δv)(r ∗ bρ₀) is finite and equal to (r ∗ bρ)(v ).

We have observed in Lemma 5.6 that the function v → |v|^−α(r ∗ bρ₀)(v) is b.R.i, hence the renewal Theorem4.10applied to µ^∗and to the function r ∗ bρ₀gives for u ∈ S^d−1,

t→∞limt^α(r ∗ bρ)(u, t ) =

∗e^α(u)

L_µ(α)(^∗eν^αu⊗ ℓ^α)(r ∗ bρ₀).

Since for fixed u, bρ(u, x) = bP{〈R,u〉 > x} is non increasing, Lemma 5.4 gives

t→∞limt^αρ(u, t ) =b

∗e^α(u)

Lµ(α)^∗eν^αu(rα).

In particular, we have^∗eν^αu(rα) ≥ 0. In case I this gives^∗eν^α(rα) ≥ 0 since^∗eν^αu=^∗eν^α. In case II, taking u ∈ Λ+(T^∗) and using^∗p^α₊(u) = 1, this gives^∗ν^α₊(rα) ≥ 0. Also, in the same way^∗ν^α₋(r_α) ≥ 0. Furthermore, in case II, using Theorem4.10,

∗eν^αu(r_α) =^∗p^α₊(u)^∗ν^α₊(r_α) +^∗p^α₋(u)^∗ν^α₋(r_α).

If^∗eν^α(rα) > 0, in case II we can define a probability measure σ^αon eΛ(T ) by

∗eν^α(r_α)σ^α=1

2(^∗ν^α₊(r_α)ν^α₊+^∗ν^α₋(r_α)ν^α₋),

while in case I , σ^α= eν^α. If^∗eν^α(rα) = 0, we have also^∗ν^α₊(rα) =^∗ν^α₋(rα) = 0, hence we can leave σ^αwith projection ν^αon P^d−1undefined in the above formulae. In any case σ^α⊗ ℓ^αis µ-harmonic.

We get another expression for the above limit, by using the formulae for^∗e^α(u),^∗p^α₊(u),

∗p₋^α(u), p(α) of section 2 paragraph 3, for case II as follows (see Theorem2.16), with p(α) =

Z

|〈x, y〉|^αdν^α(x)d^∗ν^α(y),

∗e^α(u)^∗p^α₊(u)p(α) = Z

〈u,u^′〉^α₊dν^α₊(u^′) = α(ν^α₊⊗ ℓ^α)(H_u⁺).

From above, we get

∗e^α(u)^∗eν^αu(rα) = α

p(α)((^∗ν^α₊(rα)ν^α₊+^∗ν^α₋(rα)ν^α₋) ⊗ ℓ^α)(H_u⁺).

Hence, with C = 2_L^∗_µ^eν_(α)p(α)^α(rα)α, σ^αas above and C₊=_L^∗_µ^eν_(α)p(α)^α+(rα)α, C₋=_L^∗_µ^eν_(α)p(α)^α−(rα)α,

t→∞limt^αρ(u, t ) = C (σb ^α⊗ ℓ^α)(H_u⁺),C σ^α= C₊ν^α₊+C₋ν^α₋.

In case I we get the corresponding formula. The fact that suppρ is unbounded follows

from Lemma 5.7, since k^′(α) > 0 and k(α) = 1. ä

Corollary 5.8. For any v ∈ V \ {0}, we have

t→∞limt^αPb{|〈R,v〉| > t} = Cp(α)

α (^∗e^α⊗ h^α)(v), with p(α) =R

|〈x, y〉|^αdν^α(x)d^∗ν^α(y) and C = 2_L^∗_µ^eν_(α)p(α)^α(rα)α ≥ 0. In particular, there exists b > 0 such that bP{|R| > t} ≤ bt^−α.

Proof. By definition of^∗eν^αu and since^∗p^α₊(u) =^∗p^α₋(−u) we have,¹₂(^∗ν^α_u+^∗ν^α_−u) =^∗eν^α. Hence, using Proposition 5.3,

t→∞limt^αPb{|〈R,u〉 > t} = 2^∗e^α(u)

L_µ(α) ^∗eν^α(r_α) = Cp(α) α

∗e^α(u).

The formula in the corollary follows by α-homogeneity, since it is valid for |v| = 1.

We take a base u_i ∈ V (1 ≤ i ≤ d) and write |R| ≤ Σ^d

i =1|〈R,ui〉| . For t large: bP{|R| > t} ≤

d

i =1Σ Pb{|〈R,ui〉| > t ≤ (C^′+ε)t^−αb^′α Σ^d

i =1

∗e^α(ui), with ε > 0, C^′= C^p(α)_α , hence the result.

5.3 A dual Markov walk and the positivity of directional tails

The following proposition will play an essential role in the discussion of positivity for C₊,C₋and C as defined in Proposition 5.3. We denote by Λ^∗a(Σ) the set of elements u in S^d−1such that the projection of Λa(Σ) on the half line R+u is unbounded. Here, instead of the vector space V used in paragraph 2, duality in the context of Radon transforms lead us to consider a λ-random walk on the larger vector space V ×R and to use ideas of [36], for the analysis of corresponding measurable fibered kernels (see subsection 4.1).

However, the continuity hypotheses of [36] are not in general satisfied by these kernels.

Proposition 5.9. With the hypothesis of Theorem 5.2 if M ⊂ S^d−1 is T^∗-minimal and Λ^∗a(Σ) ⊃ M, then for any u ∈ M,

C (u) = lim

t→∞t^αPb{〈R,u〉 > t} = lim

t→∞t^αρ(u, t ) > 0.b In particular with the above notations, we have C > 0.

We observe that R_n=ⁿ⁻¹Σ

0 A₁··· AkB_k+1satisfies the relation 〈R_n+1, v〉 = 〈Rn, v〉+〈B_n+1,S^′_nv 〉 where S^′_n= (A1··· An)^∗. Also h = (g ,b) ∈ H acts on E = (V \{0}) × R according to the for-mula h(v,r ) = (g^∗v, r + 〈b, v〉), hence the pair,

(S^′_nv, r + 〈Rn, v〉) = (vn,rn),

is a random walk on the right homogeneous H-space E. Actually, V × R is a vector space and the above formula for h(v,r ) defines a right linear representation of H in V × R which leaves invariant E ⊂ V × R. In particular, using the radial R^∗₊−fibration of this vector space, we see that the radial projection (v,r ) → (u,p) with v = |v|u, r = p|v|

defines an H-equivariant projection from E to S^d−1× R, hence (S^′nv, r + < Rn, v >) is also a R^∗₊-fibered Markov chain over S^d−1× R. Then we can write

E = (S^d−1× R) × R^∗₊⊂ V × R.

The action of h = (g ,b) on S^d−1× R is given by h(u, p) = (g^∗· u,h^up) with h^up = 1

|g^∗u|(p + 〈b,u〉).

The proof of Proposition 5.9 is based on the relation 〈R,v〉 = lim

n→∞< Rn, v > and on the dynamics of the random walk (vn,rn) = (S^′nv, r + < Rⁿ, v >) on V × R. We denote by

∗P its corresponding fibered Markov kernel and we study an associated ladder processb (xτ_n,Wτ_n), defined below. In terms of this process we can give a new expression for

b

ρ(v ) = bP{〈R,u〉 > t} as a potential of a non negative function on E. Then we can use a weak renewal theorem for (xτ_n,Wτ_n) and obtain Proposition 5.9.

We will consider successively the two components un = _|v^vn_n|, pn = _|v^rn_n| and finally the fibered Markov chain (vn,rn) above S^d−1× R.

Let M be a T^∗-minimal subset of S^d−1, hence (see section 2, paragraphe 3), M = Λ+(T^∗) (or Λ₋(T^∗)) in cases II, or M = Λ(T ) in case I.

The following says that Λ^∗a(Σ) is "large".

Lemma 5.10. In cases I or II’ : Λ^∗a(Σ) = S^d−1. In case II” : Λ^∗a(Σ) ⊃ Λ₊(T^∗).

Proof. Let Λ^∞_a (Σ) = Λa(Σ) ∩ S^d−1_∞ and u ∈ S^d−1, u^′_∞∈ Λ^∞a (Σ) corresponds to u^′∈ S^d−1. If 〈u^′,u〉 > 0, then u ∈ Λ^∗a(Σ). Hence the complement of Λ^∗a(Σ) in S^d−1is contained in the set {u ∈ S^d−1; 〈u,u^′〉 ≤ 0 ∀u_∞^′ ∈ Λ^∞a (Σ)}. From the discussion at the beginning of this section we know that Λ^∞a (Σ) 6= φ is T -invariant and closed, hence contains eΛ^∞(T ) in cases I, II’ or only Λ^∞₊(T ) in case II” with Λ^∞₋(T ) ∩ Λ^∞a (Σ) = φ.

Since eΛ^∞(T ) is symmetric and condition i-p is valid, it follows Λ^∗a(Σ) = S^d−1 in cases I, II’. In case II”, we know from the end of proof of Theorem2.16that the complement of Λ^∗a(Σ) is contained in bΛ₊(T^∗) = {u ∈ S^d−1; ∀u^′∈ Λ₊(T ),〈u,u^′〉 > 0}. Since Λ₊(T^∗) ∩

−bΛ₊(T^∗) = φ, we get Λ+(T^∗) ⊂ Λ^∗a(Σ).

The random walk (vn,rn) = (Sn^′v, r + 〈Rⁿ, v〉) has H-equivariant projection S^′nv on V \ {0}, the kernel^∗P has projectionb ^∗P already defined in section 2, hence the positive homogeneous function^∗e^α⊗ h^α, lifted to E, satisfies^∗P (b ^∗e^α⊗ h^α) =^∗e^α⊗ h^α, and we can consider the new relativized fibered Markov kernel^∗Pbαon E. If (u, p) ∈ M × R, the projection xn= (un, pn) of (vn,rn) on M × R depends on the kernel^∗Qb^αgiven by

∗Qb^αϕ(u, p) = Z

ϕ(g^∗· u,h^up)^∗q^α(u, g )dλ(h)

where^∗q^αcorresponds to q^αas in section 3. The important fact for the proof of Pro-position 5.9 is that^∗Qb^αhas a stationary probability κ with κ(M ×[t,∞[) > 0 for any t > 0, such that p is not "too large" with respect to κ.

For the analysis of^∗Qb^αwe consider on bΩ the projective limit^∗Qb^α_uof the system^∗q^α_n(u,·)λ^⊗n (n ∈ N) and, by abuse of notation, the corresponding expectation will be written E^αu. Given a^∗Qe^α-stationary measure eπ^α_M, we write^∗Qb^α=R

δ_u⊗^∗Qb^α_udπe^α_M(u) and we denote by E^αthe corresponding expectation symbol. We denote by bθ^#the map of M × H^Zinto itself defined by bθ^#(u, bω) = (g₁^∗· u, bθω) where bb θ is the bilateral shift on H^Z, and ^∗Qb^α will again denote the natural bθ^#-invariant measure on M × H^Z. Also we extend S^′_n(ω) as a G-valued Z-cocycle. If η is a probability measure on X = M × R, the associated Markov measure on^aΩb= X × bΩ, is denoted by^∗Qb^α_η, and the extended shift by^aθ whereb

aθ(x,b ω) = (hb 1x, bθω). Also if η isb ^∗Qb^α-stationary we will consider the bilateral associ-ated system (Ω^#,^aθ, ηb ^#) where Ω^#= X × H^Z,^aθ is the bilateral shift and ηb ^#is the unique

aθ-invariant measure with projectionb ^∗Qb^α_η on X × bΩ.

Lemma 5.11. Let M be a T^∗-minimal subset of S^d−1, π^α_M the unique^∗Qe^α-stationary measure on M. With the above notations, we consider the Markov chain x_n = (un, pn), on X = M × R given by

u_n+1= g_n+1^∗ .un, p_n+1=p_n+ 〈b_n+1,u_n〉

|g_n+1^∗ u_n| , p0= p, u0= u,

where (g_n,bn) is distributed according to^∗Qb^α_u. Then, for any p ∈ R, xnconverges in^∗Qb^α -law to the unique^∗Qb^α-stationary measureκ, the projection of κ on M is π^α_M, κ(M ×{p}) = 0 andR

|p|^εdκ(u, p) < ∞ for ε small. We have κ^#− a.e, limsup

n→∞ |Sn^′u||pn| = ∞, lim

n→∞|S^′_−nu||p_−n| = 0.

If Λ^∗a(Σ) ⊃ M, then κ(M×]t,∞[) > 0 for any t > 0 and limsup

n→∞ |S^′nu|pⁿ= ∞,^∗Qb^α_κ− a.e.

Proof. If will be convenient to use the functions a(g , u), b(h, u) defined by h^up = a(g ,u)p+

b(h, u), and the random variables a_k,bk (k ∈ Z) defined by ak( bω, u) = a(gk,S^′_k−1· u), b_k( bω, u) = b(hk,S^′_k−1· u). Then we can express the action of hn··· h1 ∈ H on X as, un= S^′n· u, yn^p(u) = (hn···h1)^up, where

y_n^p(u) = a(S^′n,u)p + yn^◦(u) and y_n^◦(u) = Σⁿ₁a_k+1ⁿ (u)bk(u),

with aⁿ_k(u) = a(gn··· gk,S_k−1^′ · u). The random variables ak,bk are^∗Qb^α-stationary and y_n^◦ has the same law as p^◦_n( bω, u) =ⁿ⁻¹Σ

0 a₋₁··· a−kb_−k−1.

We estimate E^α(|pn^◦|^ε) for 0 < ε < τ and ε small, where p^◦n( bω, u) =ⁿ⁻¹Σ

0 a₋₁··· a_−kb_−k−1. Since (a_k··· a1)(ω,u) = a(S^′_k(ω),u) we get E^α(|a₋₁··· a_−k|^ε) = E^α(|S^′_ku|^−ε). Hence Corol-lary 3.18 gives lim

k→∞(E^α(|a₋₁··· a_−k|^ε))^1/k< 1 since k^′(α) > 0 and E(|A|^α+τ) < ∞. Also for ε small,

E^α(|bk|^ε) = E^α(|〈B1,u〉

|g^∗u| |^ε) ≤ E^α(|B1|^εγ^ε(A)) < ∞,

using^∗Qb^α-stationarity, Hölder inequality and the condition E(|B1|^α+τ)+E(|A1|^α+τ) < ∞.

Since for 0 < ε < 1 |pn^◦( bω, u)|^ε≤ⁿ⁻¹Σ

◦ |a₋₁··· a_−k|^ε|b_−k−1|^ε, we get that E^α(|p^◦n|^ε) is bounded.

The^∗Qb^α-a.e. convergence of the partial sum p^◦_n( bω, u) to p(ω, u) =b ^∞Σ

0a₋₁··· a_−kb_−k−1and the finiteness of E^α(|p|^ε) follows. By definition, p( bω, u) satisfies the functional equation p ◦^aθ = ap +b where p and (a,b) are independent. It follows that the probability meas-b ure κ on M × R given by the formula

κ = Z

δ_u⊗ δp(ω,u)b d^∗Qb^α( bω, u) is^∗Qb^α-invariant.

As observed above, the^∗Qb^α-laws of y^◦_nand p^◦_n are the same. Since the product of eπ^α_M with the law of y_n^◦is (^∗Qb^α)ⁿ(π^α_M⊗δ0) we have in weak topology : lim

n→∞(^∗Qb^α)ⁿ(π^α_M⊗δ0) = κ.

Since |yn^p( bω, u) − yn^p′( bω, u)| = a(S^′n(ω),u)|p − p^′| and a(S^′n(ω),u) = |S^′nu|⁻¹ converges

∗cQ^α_u-a.e to zero, we get the convergence of (^∗Qb^α)ⁿ(π^α_M⊗δp) to κ, for any p. On the other hand, if η^′is a^∗Qb^α-stationary measure on M ×R, its projection on M is^∗Qe^α-stationary, hence equal to π^α_M, since M is T^∗-minimal. Then, from above (^|∗Qb^α)ⁿη^′converges to κ, hence η^′= κ. The^∗Qb^α-ergodicity of κ implies the^aθ-ergodicity ofb ^∗Qb^α_κand κ^#.

If Λ^∗a(Σ) ⊃ M assume κ (M×]t,∞[) = 0, for some t > 0, i.e. the^∗Qb^α-invariant set suppκ is contained in M×] − ∞,t]. Then, for any (u,p) ∈ suppκ we have p + 〈Rn,u〉 ≤ t|Sn^′u|

∗Qb^α_u-a.e., i.e. p + 〈Rn,u〉 ≤ t|S^′nu|^∗q^α_n(u,·)λ^⊗n-a.e. for any n ∈ N and for some p ∈ R, t , u ∈ M. It follows p + 〈Rn,u〉 ≤ |S^′nu|, λ^⊗n-a.e. and since lim

n→∞|Sn^′u| = 0, lim

n→∞R_n= R, P-a.e., we have 〈R,u〉 ≤ −p, bP-a.e. . This implies that the support of the projection of ρ on Ru is bounded in direction u; since by Proposition 5.1 we have suppρ = Λa(Σ) this contradicts the condition Λ^∗a(Σ) ⊃ M. Hence κ(M×]t,∞[) > 0 for any t > 0.

Furthermore, arguments as in the proof of Proposition 5.1, using that suppλ has no fixed point in V , show κ(M × {p}) = 0 for any p ∈ R. From Theorem 3.10 we know

that lim

n→∞

1

nlog|Sn^′u| = Lµ(α) > 0,^∗Q^α_u-a.e. .Furthermore, since κ is ^∗Qb^α-ergodic and κ(M × {0}) = 0 we have limsup

n→∞ |pn| > 0, bQ^α_κ-a.e.. Then we get limsup

n→∞ |S^′nu| |pn| = ∞,

∗Qb^α_κ-a.e. If Λ^∗a(Σ) ⊃ M, from above we have κ(M×]0,∞[> 0, and again using ergodicity, limsup

n→∞

p_n > 0. Since lim

n→∞|Sn^′u| = ∞^∗Qb^α_u− a.e, it follows limsup

n→∞ |S^′nu|pn = ∞, ^∗Qb^α_κ -a.e. Using Theorem 3.2, we have also, for any u ∈ M and^∗Qb^α_u-a.e.: lim

n→∞

1

nlog|S^′_−nu| =

−Lµ(α) < 0. The conditionR

|p|^εdκ(u, p) < ∞ implies limsup

n→∞

log|pn|

|n| ≤ 0,κ^#-a.e. Then we get, lim

n→∞|S^′_−nu||p−n| = 0,κ^#-a.e.

For the analysis of^∗Pb_α, bρ we consider the optional time τ on X × bΩ given for p6= 0 by:

τ = inf{n > 0 ; p⁻¹〈Rn,u〉 > 0}, τ = ∞ if p⁻¹〈Rn,u〉 ≤ 0 for every n.

We observe that τ is independent on p as long as p > 0 or p < 0. By definition of pn: p + 〈Rn,u〉 = pn|S^′nu|, p⁻¹p_n|S^′nu| = 1 + p⁻¹〈Rn,u〉.

We note that (un, pn) is the radial projection of (vn,rn) on S^d−1× R ⊂ V × R while the projection (0,r_n) of (v_n,r_n) on {0} × R satisfies rn= r (p⁻¹p_n|S^′nu|). In (u, p,r ) coordin-ates on the set {r 6= 0}, the process (vn,rn) can be written as (xn,rn), and for any t > 0 the dilation (v,r ) → t(v,r ) reduces to (x,r ) → (x,tr ). Since κ(M ×{0}) = 0,^∗P is also a meas-ˆ urable Markov fibered kernel above S^d−1× R^∗ (see subsection 4.1). This measurable setting will be useful below.Also,

τ = inf{n > 0 ; p⁻¹p_n|S^′nu| > 1}, τ = ∞ if p⁻¹p_n|Sn^′u| ≤ 1, for every n.

In particular p⁻¹pτ> 0 where the notation pτis used if τ is finite. Also we define σn= τ ◦ (^aθ)b ^σn−1 and σ0= 0, τn =Σⁿ

1σ_k, so that τ can be seen as the first ladder index and p⁻¹pτ|S^′τu| as the first ladder height of the R^∗-valued measurable Z-cocycle ,

W_n(u, p, bω) = p⁻¹p_n|S^′nu|

over the dynamical system (Ω^#,^aθ, κb ^#), which is well defined since κ(M × {0}) = 0. The random times τncan be seen as the successive times of increase for r⁻¹r_k= p⁻¹p_k|S^′_ku|

along the random walk (vn,rn). On the other hand, by Poincaré recurrence theorem we have κ^#-a.e, limsup

n→∞

p⁻¹p_n≥ 1. Since lim

n→∞|Sn^′u| = ∞,then τ, τn are finite^∗Qb^α_κ-a.e and we have p⁻¹pτ_n> 0.

With the above notation the λ-random walk (vn,rn) on E ⊂ V × R can be written as u_n+1= g_n+1^∗ .u_n, p_n+1=p_n+ 〈b_n+1,u_n〉

|g_n+1^∗ u_n| , r_n+1= rn|g_n+1^∗ u_n|p_n+1p_n⁻¹.

We denote X = M × R and X₊= M×]0,∞[⊂ X , hence x ∈ X₊ implies xτ∈ X₊ if τ < ∞.

We consider also the submarkovian stopped operator ^∗Pb^τ on V \ {0} × R. In (u,p,r ) coordinates on the set {r 6= 0} with x = (u,p) the associated process at time n starting at (x,r ) ∈ X₊× R^∗₊is (x_τn,r |Sτ^′_nu|pτ_np⁻¹), hence the restriction of^∗Pb^τto X₊× R^∗₊is well defined and is a measurable fibered kernel on X₊×R^∗₊⊂ E . This restriction will be again denoted by^∗P^τ. Since^∗P (b^∗e^α⊗ h^α) =^∗e^α⊗ h^αand τ is finite^∗Q^α_κ-a.e. , the kernel^∗Pb_α^τ given by :

∗Pb_α^τϕ = (^∗e^α⊗ h^α)⁻¹Pb^τ(^∗e^α⊗ h^αϕ),

is also a measurable Markov fibered kernel on X ×R^∗₊which satisfies^∗Pb_α^τ1 = 1, κ⊗ℓ a.e.

The following lemma expresses the function bρ(p⁻¹v ) = bP{p⁻¹〈R,u〉 > t} = ψ(v, p) on E as a^∗Pb^τ-potential of a non negative function on E.

Lemma 5.12. With t v = u ∈ S^d−1, t > 0 and p 6= 0, we write

τ = inf{n > 0; p⁻¹〈Rn,u〉 > 0},τ = ∞ if p⁻¹〈R,u〉 ≤ 0 for any n ∈ N,

ψ(v, p) = bP{p⁻¹〈R,u〉 > t}, ψτ(v, p) = bP{t < p⁻¹〈R,u〉 ≤ t + p⁻¹〈Rτ,u〉;τ < ∞}, where Rτ=^τ−1Σ

0 A₁··· AkB_k+1,ψ^α= (^∗e^α⊗ h^α)⁻¹ψ, ψ^α_τ = (^∗e^α⊗ h^α)⁻¹ψτ. Then, ψ =X^∞

0

(^∗Pb^τ)^kψ_τ, ψ^α= X∞

0

(^∗Pb_α^τ)^kψ^α_τ.

Proof. We write 〈R − Rn, v〉 = 〈Rⁿ,S^′_nv 〉, where Rⁿ =^∞Σ

nA_n··· AkB_k+1 hence if τ < ∞ :

〈R −Rτ, v〉 = 〈R^τ,S^′_τv 〉. By definition of τ, since p⁻¹〈R,u〉 > t > 0 and the convergence of R_nto R imply τ < ∞, we have

ψ_τ(v, p) = ψ(v,p) − bP{〈R − Rτ,u〉p⁻¹> t ; τ < ∞}.

On the other hand, since p⁻¹p_τ> 0, b

P{〈R − Rτ,u〉p⁻¹> t ; τ < ∞} = bP{〈R^τ,u_τ〉p⁻¹> t

|Sτ^′u|; τ < ∞} =^∗Pb^τψ(v, p).

It follows ψ_τ= ψ −^∗Pb^τψ, ψ =ⁿ⁻¹Σ

0 (^∗Pb^τ)^kψ_τ+^∗Pb^τnψ with

∗Pb^τnψ(v, p) = bP{t|Sτ^′_nu|⁻¹< p⁻¹〈R^τn,uτ_n〉 ; τn< ∞}.

For (x, bω) ∈ M × bΩ we have either τn(x, bω) = ∞ for some n hence^∗P^τnψ = (^∗P^τ)^kψ = 0 or lim

n→∞τ_n(x, bω) = ∞. In the second case, lim

n→∞|S^′τ_nu|⁻¹ = ∞, bP − a.e. and, since Rτ_n

converges bP−a.e to R, we have lim

n→∞

∗Pb^τnψ = 0, ψ =^∞Σ

0(^∗Pb^τ)^kψ_τ. The last relation follows from the definitions of ψ^α_τ and^∗Pb_α^τ, since ψ^α_τ is non negative.

Remark. The functionψ satisfies the basic property of potentials for^∗Pb^τ,

∗Pb^τψ ≤ ψ, lim

n→∞(^∗Pb^τ)ⁿψ = 0.

This property is a key for understanding non triviality of the Cramer-type estimate for ψ. It is also valid for other natural functions such as ψ defined by

ψ(v, p) = bP{sup

n≥1|Wn| > t}.

Let τ be as above and Λ^∗a(Σ) ⊃ M, hence using Lemma 5.11 we have κ(X+) > 0. Let

∗Qb^α,τ(x,.) be the law of xτunder^∗Qb^α_x. Since τ < ∞,^∗Qb^α_κ-a.e. , this kernel is a measurable Markov kernel with respect to κ on X₊, hence^∗Qb^α,τis not κ-ergodic in general and it is natural to consider the first return time to X₊as well as the corresponding induced Markov operator^∗Qb^α₊ on X₊. Following the idea of ([36], Lemma 2) we interpret the ladder index τ as a first return time to a subset of Ω^#, we construct a stationary measure for^∗Qb^α,τon X₊, and we show the finiteness of the corresponding expectation of τ.

Lemma 5.13. Assume Λ^∗a(Σ) ⊃ M and write

τ = inf{n > 0 ; p⁻¹p_n|S^′nu| > 1} = inf{n > 0 ; p⁻¹〈Rn,u〉 > 0}, τ = ∞ if p⁻¹p_n|S^′nu| ≤ 1 for any n ∈ N.

Then the stopped operator^∗Qb^α,τpreserves X₊and admits a stationary ergodic probab-ilityκ^τ on X₊ which is absolutely continuous with respect to κ. The integral E^α₀(τ) = RE^α_u(τ)dκ^τ(u, p) is finite.

Withγ^α_τ = Lµ(α)E^α₀(τ) we have lim

n→∞

1

nlog(|S^′τ_nu|pτ_n

p ) = γ^ατ∈]0,∞[,^∗Qb^α_κτ-a.e. . In partic-ular Wτ= log(pτp⁻¹|Sτ^′u|) has finite expectation with respect to^∗Qb^α_κτ.

Proof. Since Λ^∗_a(Σ) ⊃ M, Lemma 5.11 gives κ(X+) > 0. In order to deal only with positive values of the R^∗-valued Z-cocycle W_n(u, p, bω) = |S^′nu|pnp⁻¹under^∗Qb^α_κ, it is convenient to consider the two sided Markov chain xn_k(k ∈ Z, x_◦∈ X₊) induced on X₊by xn, hence n₁is the first return time of xn to X₊and n1≤ τ since p⁻¹p_τ> 0. We note that the nor-malized restriction κ₊of κ is a stationary ergodic measure for x_nk. Also the relativized Markov kernel^∗Pbα on E = X × R^∗₊induces a fibered measurable Markov kernel^∗Pb_α,+

on X₊× R^∗₊ with projection^∗Qb₊^α on X₊, which satisfies ^∗P_α,+(κ₊⊗ ℓ) = κ+⊗ ℓ. Since p⁻¹p_nk > 0,in(u,p,r) coordinates, the corresponding bilateral Markov chain can be writ-ten as (xn_k,rn_k) with rn_k = r p⁻¹ pn_k|S^′n_ku|. We denote by Ω^#₊the subset of Ω^#defined sequence of return times to Ω^#₀. This allow us to proceed as in ([36] Lemma 2) with the R^∗-valued Z-cocycle |S^′nu| pⁿp⁻¹. Since κ^#is^aθ-invariant and κb ^#(Ω^#₀) > 0 we can apply Kac’s recurrence theorem to Ω^#₀,^aθb^τand Ω^#(see [50]), hence the normalized restriction κ^#₀ of κ^# to Ω^#₀ is^aθb^τ-ergodic and stationary, the return time τ has finite expectation

1

nlog^p_p^τn converges to zero in κ^#₀-measure, hence using the κ^#₀− a.e. convergence of

1

nlog|S^′τ_nu|, we get the κ^#₀-a.e. convergence of _n¹W_τ^a

nto γ^α_τ. In particular E^α₀(W_τ^a) = γ^ατ ∈ ]0,∞[.

In order to relate κ^#₀ and the kernel ^∗Qb^α,τ we consider the Markov kernel adjoint to

∗Qb^α₊(x,·) with respect to κ₊, and we denote by ^∗Qb^α₋⊗ δx the corresponding Markov measure on H^Z−× X₊ with Z₋= −N ∪ {0}. Also we write ^∗Qb^α_x = δx⊗^∗Qb^α₊ where ^∗Qb^α₊ is supported on H^Nand κ^#₊=R_∗Qb^α₋⊗ δx⊗^∗Qb^α₊dκ₊(x), in particular

κ^#(Ω^#0)κ^#₀= Z

(1_Ω^#

0)^∗Qb^α₋⊗ δx⊗^∗Qb^α₊dκ₊(x)

with Ω^#₀= Ω⁻₀× H^Nand Ω⁻₀ ⊂ H^Z−× X+. We denote by κ^τ the projection of κ^#₀ on X₊, hence κ^τhas density u(x) given by κ^#(Ω^#₀)u(x) = (^∗Qb^α₋⊗ δx)(Ω⁻₀) with respect to κ₊. It follows that the projection of κ^#₀on X₊× bΩ can be expressed as

Z

u(x)δ_x⊗^∗Qb^α₊dκ₊(x) = Z

δ_x⊗^∗Qb^α₊dκ^τ(x) =^∗Qb^α_κτ.

Since κ^#₀is invariant and ergodic with respect to the bilateral shift^aθb^τ, the same is valid for^∗Qb^α_κτ with respect to the associated unilateral shift^aθb^τ. Since the kernel x →^∗Qb^α_x commutes with^aθb^τand^∗Qb^α,τ, the^∗Qb^α,τ-invariance and ergodicity of κ^τfollows. Also we have E^α₀(τ) =R

E^α_x(τ)dκ^τ(x) and the above convergences are valid^∗Qb^α_κτ-a.e. .

Remark. If S = X+, P =^∗Pb_α,+, π = κ₊,and in the corresponding measurable setting, the measure κ^τ is closely connected with the measure χ of Theorem4.5. The meas-ure κ^τcan be caracterized as the unique^∗Qb^α,τ-stationary measure which is absolutely continuous with respect to κ₊. However, in contrast to [36], the function log|p| is not known to be κ-integrable, but we know that lim

n→∞|S_−n^′ u|p_−n p⁻¹= 0.

The following weak renewal theorem for a general fibered Markov chain will allow us to control potentials of the measurable fibered Markov kernel^∗Pb^τ_αon X₊× R^∗₊. We recall some notation of section 4 as follows.

Let (S,π) be a complete separable metric space, where π is a probability measure. We consider a general measurable Markov chain on S × R with kernel P, we assume that P commutes with the R-translations and we denote Lebesgue measure on R byℓ . We assume that the measure π ⊗ ℓ is P-invariant. Here, in contrast to section 4, our setting is the measurable one;in particular the symbol sup means essential supremum. We write a path of the corresponding Markov chain as (xn,Vn) where xn ∈ S and Vn ∈ R,

we denote by^aP_x the Markov measure on the paths starting from x ∈ S and we write

aP_π=R_a

P_xdπ(x),^aE_x for the corresponding expectation symbol. In this context the following weak analogue of the renewal Theorem4.5holds.

Proposition 5.14. With the above notation and hypotheses, assume thatψ is a com-pactly supported bounded non negative measurable function on S × R, the potential Uψ =^∞Σ

Furthermore ifψ is a non negative measurable function on S × R and lim

t→−∞Uψ(x, t ) = 0, π-a.e. . then ψ = 0, π ⊗ ℓ-a.e. .

Ifψ is a measurable function on S × R which satisfies:

|ψ|b=

ℓ=∞X

ℓ=−∞

sup{|ψ(x,s)| ; x ∈ S, s ∈ [ℓ,ℓ + 1[} < ∞,

then the above convergence is also valid.

Proof. We observe that the maximum principle implies |U ψ| = sup

x,t |U ψ|(x, t) < ∞, since

We estimate each term I_k(t) separately. We have, since the measure π⊗ℓ is P-invariant, I₁(t) =ⁿ²⁻ⁿ_t¹⁺¹(π ⊗ ℓ)(ψ), hence lim

t→∞I₁(t) = 1

γ(π ⊗ ℓ)(ψ).

Furthermore,

|I4(t)| ≤|ψ|

t (n2− n1+ 1) sup

n1≤n≤n2

Z

S(^aP_x{Vn≤ a} +^aP_x{Vn≥ t − a})dπ(x).

Since n⁻¹V_n converges to γ > 0 in probability, the above integral has limit zero, hence

t→∞limI₄(t) = 0.

We have also |I2(t)| ≤^ε_γπ ⊗ ℓ(ψ).

In order to estimate I₃(t) we denote, for n ∈ N and s > 0, ρ^sn= inf{k ≥ n ; −a ≤ Vn−s ≤ a}, where ψ is supported on [−a,a], and we use the interpretation of Uⁿψ as the expected number of visits to ψ after time n :

Uⁿψ(x, s) ≤ |U ψ|^aP_x{ρ^s_n< ∞}.

Taking n = [^(1+ε)t_γ ] = n2we get

I3(t) ≤ |Uψ|

Z

S

aP_x{Vk− t ≤ a, for some k ≥ [(1 + ε)t

γ ]}dπ(x).

Since^V_nⁿ converges to γ > 0 in probability, we get lim

t→∞I₃(t) = 0.

Since ε is arbitrary we get finally, lim

t→∞I (t ) = 1

γ(π ⊗ ℓ)(ψ).

The second conclusion follows by restriction and truncation of ψ on S × [−a,a].

For the proof of the last assertion we observe that for any ℓ∈ Z, ∆ = |U1_S×[0,1]| = |U 1_{S×[ℓ,ℓ+1[}| <

∞. Writing ψ = ^ℓ=∞Σ

ℓ=−∞

ψ1_{S×[ℓ,ℓ+1[}we get |Uψ| ≤ ∆|ψ|b.

The quantity ψ → |ψ|bis a norm on the space H of measurable functions on S ×R such that |ψ|b < ∞. Since the set of essentially bounded functions supported on S × [−c,c]

for some c > 0 is dense in H and ψ → (π ⊗ ℓ)(ψ) is a continuous functional on H , the above relation extends by density to any ψ ∈ H .

Proof of Proposition 5.9. Assume that for some u ∈ M, C (u) = lim

t→∞t^αPb{〈R,u〉 > t} = 0. For p > 0, with the notations of Lemma 5.12, this means lim

t→∞ψ^α(v, p) = 0. Using Proposition 5.3 we know that this implies lim

t→∞ψ^α(u, p) = 0 for any u = tv ∈ M (t > 0).

Also, using Lemma 5.13, we have, since Λ^∗a(Σ) ⊃ M,

n→∞lim 1

nlog(|Sτ^′_nu| pτ_n

p ) = γ^ατ> 0,^∗Qb^α_κτ− a.e.

Since the canonical Markov measure associated with κ^τand^∗Qb^α,τ is a push-forward of ^∗Qb^α_κτ, this convergence is also valid with respect to this canonical Markov meas-ure. Then, using Lemma 5.12 and ψ^α_τ ≥ 0, we can apply Proposition 5.14 with Vn = log(p⁻¹p_τn)|S^′τ_nu|, γ = γ^ατ > 0, S = X+, to the measurable Markov kernel P =^∗Pb_α^τ on X₊×R^∗₊, to the potential^∞Σ

0(^∗Pb_α^τ)^kψ^α_τof the non negative function ψ^α_τ≤ (^∗e^α⊗h^α)⁻¹and to the^∗Qb^α,τ-stationary measure π = κ^τ; we get ψ^α_τ= 0, κ^τ⊗ ℓ-a.e., hence

b

P{t < p⁻¹〈R,u〉 ≤ t + p⁻¹〈Rτ,u〉,τ < ∞} = 0.

Since p⁻¹〈Rτ,u〉 > 0, this gives p⁻¹〈R,u〉 ≤ 0 κ^τ⊗ bP − a.e. on {τ < ∞}, in particular for some (u, p) ∈ X+, we have p⁻¹〈R,u〉 ≤ 0 i.e. 〈R,u〉 ≤ 0, bP-a.e. on {τ < ∞}. But, since Λ^∗(Σ) ⊃ M, for any u ∈ M the set {〈R,u〉 > 0 ; τ < ∞} = {〈R,u〉 > 0} is not bP-negligible, hence the required contradiction. Since, using Proposition 5.3, we have C (u) = C(σ^α⊗

ℓ^α)(H_u⁺) it follows C > 0. ä

Remark. The proof of Proposition 5.9 given above uses the R^∗-valued multiplicative cocycle Wn. The interpretation of the ladder index τ as a first return time to a subset of Ω^#depends on the reduction of Wnto a positively valued cocycle; hence the use of the seemingly artificial inducing procedure on X₊× R^∗, as was done above.

Remark. The proof of Proposition 5.9 given above uses the R^∗-valued multiplicative cocycle Wn. The interpretation of the ladder index τ as a first return time to a subset of Ω^#depends on the reduction of Wnto a positively valued cocycle; hence the use of the seemingly artificial inducing procedure on X₊× R^∗, as was done above.

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