Recurrence, ergodicity and invariant measures for cocycles over a rotation

HAL Id: hal-00456631

https://hal.archives-ouvertes.fr/hal-00456631

Submitted on 22 Mar 2018

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Recurrence, ergodicity and invariant measures for cocycles over a rotation

Jean-Pierre Conze

To cite this version:

Jean-Pierre Conze. Recurrence, ergodicity and invariant measures for cocycles over a rotation. Idris Assani. Ergodic theory: Papers from the Probability and Ergodic Theory Workshops held at the University of North Carolina, American Mathematical Society, pp.45-70, 2009, Contemporary Math- ematics n° 485. �hal-00456631�

(X,A, µ, τ) ϕ

X G τ_ϕ

X×G τ_ϕ : (x, g)→(τ x, ϕ(x)g)

G=R^d ϕ

µ×dg τ_ϕ τ

µ×dg τ_ϕ

N₃

(X,A) τ :X →X

σ µ

τ ϕ X G

ϕ (ϕn)_n∈Z (ϕ, τ) n≥1

ϕn(x) =ϕ(τⁿ⁻¹x)... ϕ(x),

τϕ X×G

τ_ϕ : (x, y)→(τ x, ϕ(x)y).

m_G(dg) dg G τ_ϕ

µ×mG λ0 λχ

µχ

G ϕn ϕn = !_n−1

k=0ϕ◦τ^k n >0

(ϕn) G

(X, µ, τ) (ϕ_n(x)) x

G τϕ

µ×mG

(ϕ, τ) G λ0 λχ

τϕ τϕ

ϕ R (ϕn)n∈Z

µ(ϕ) = 0

G

(X, µ, τ) (ϕ, τ) G

G R^d ϕ

λ0 τϕ

Z^d

λ0

τ_ϕ σ

λ0, λχ τϕ

X ×G

σ X

τ µ

σ µ

τϕ X×G

τϕ

λ0 λχ

R ϕ

τϕ

N3

R^d

ϕ ϕ=!

ici1Ii−β R^d d >1

α (α, β)

(X,A)

τ :X →X σ µ

τ f X τ f f◦τ

G e G

(ϕ, τ) (ψ, τ) (X, µ, τ) µ

u u:X →G

ϕ(x) =u(τ x)ψ(x) (u(x))⁻¹ for µ−a.e. x.

(ϕ, τ) µ u ϕ(x) =u(τ x) (u(x))⁻¹

µ x

σ

τϕ µ

X τ ϕ

"

ϕ dµ = 0

(νι, ι ∈ J) τ

ϕ ϕ=τ uιu⁻¹_ι (X, νι, τ) B

ν_ι(B) = 0 ∀ι µ(B) = 0

τ_ϕ ν_ι ×δ_y0 ι ∈J, y₀ ∈ G

(x, y)→(x, uι(x)y)

τϕ

λ X×G λ(dx, dg) =

µ(dx)N(x, dg) µ X N

x ∈ X N(x, dg)

G x→N(x, B) B G

λ τϕ

ϕ µ ψ H G

ϕ = τ u ψ u⁻¹ λ˜ λ (x, g) → (x,(u(x))⁻¹g) τψ

X×H

λ(dx, dh) = ˜˜ µ(dx)χ(h)dh,

dh H χ H µ˜ σ

µ

τµ(dx) =˜ χ(ψ(τ⁻¹x)) ˜µ(dx).

H = G u(y) ≡ e λ(dy, dg) = ˜µ(dy) χ(g)dg µ˜

X χ G

τµ(dy) =˜ χ(ϕ(τ⁻¹y)) ˜µ(dy).

γ ∈G Rγ X×G (x, g)→Rγ(x, g) =

(x, gγ) λ τ_ϕ R_γ

λ λ

H(λ) := {γ :Rγλ ∼λ} H G

H =G λ

τϕ X

ϕ X R^d λ X×G

λ(X×K)<∞ K G

ζ X

ν τ ν =ζν inf_µ∈E(τ)|"

logζ dµ|= 0 E(τ) " τ

X logζ dx= 0

x ∈ X

!

k∈Zζk(x) < ∞ ζk ζ

ζ x !

k∈Zζk(x) < +∞

x ν({Tⁿx}) = ζn(x) ∀n ∈ Z τ ν =ζν

τ logζ !

k∈Zζk(x) = ∞,∀x ∈ X

ν ζ

ϕ R^d

τ χ R^d

µχ χ◦ϕ

τ µ_χ =χ◦ϕ◦τ⁻¹ µ_χ.

λχ(dx, dy) :=µχ(dx)×χ(y)dy.

τϕ

τϕ

ϕ = 1[0,β]− β τ

α (α, β) τϕ

X×R

µ

(ϕ, τ) µ H G

u : X → G ψ := (u◦τ)⁻¹ϕ u µ H

τψ : (x, h)→(τ x, ψ(x)h) µ⊗mH X×H

τϕ F(u(x)⁻¹g)

F H

µ×dg

µx µ⊗mG

µ G

a∈G=G∪ {∞} (ϕ, τ)

V a B µ(B) > 0

n∈Z

µ(B∩τ⁻ⁿB∩ {x:ϕn(x)∈V})>0.

E(ϕ) (ϕ, τ) E(ϕ) =E(ϕ)∩G

P(ϕ) τ_ϕ

f X×G

E(ϕ) P(ϕ) G

E(ϕ) ={0} ϕ

ϕ ϕ₁ ϕ₂

{e} E(ϕ) = {e}

ϕ Z s.∈ Q

e^2πisϕ =ψ⁻¹ ψ◦τ ψ E(ϕ) ={0} ϕ

E(ϕ) ={0,∞} ϕ

α τα τ

x→ x+α 1 X = R/Z µ0 f

X τ f f◦τ fk f +...+τ^k−1f, k≥1

•

α∈]0,1[ α = [0;a1, ..., an, ...]

(p_n/q_n)_n≥0 q

(qn) α p₋1 = 1

p0 = 0 q₋1 = 0 q0 = 1 n≥1

p_n = a_np_n−1+p_n−2, qn = anq_n−1+q_n−2, (−1)ⁿ = pn−1qn−pnqn−1.

u {u} ((u)) := inf({u},{1−u}) = infn∈Z|u−n|, n≥0

((qnα)) = (−1)ⁿ(qnα−pn),

1 = q_n((q_n+1α)) +q_n+1((q_nα)).

n≥0

((qnα)) + ((qn+1α)) ≤ inf(1 q_n, 2

q_n+1), 1

qn+1 +qn ≤((qnα)) ≤ 1 qn+1

.

α = [0;a1, ..., an, ...]

(an)

α η >0

infk [k^η−ε((kα))] = 0, inf

k [k^η+ε((kα))]>0, ∀ε >0.

(10) ≥ 1

α

α η #

k≥1

1 k^η+δ

1

((kα)) <∞ δ >0

i≥ 1 Si (ki,j)j≥1 ((ki,jα))∈

[2⁻⁽ⁱ⁺¹⁾,2⁻ⁱ[ Ki = minj{ki,1, ki,j+1−ki,j}

0< ε < δ C(ε)

C(ε)

k_i,1^η+ε ≤ ((ki,1α))<2⁻ⁱ, C(ε)

(ki,j+1−ki,j)^η+ε ≤ (((ki,j+1−ki,j)α))<2⁻⁽ⁱ⁻¹⁾ K_i≥ (C(ε)2ⁱ⁻¹)^η+ε1 .

k_i,j ≥jK_i,∀j ≥1 D(ε)

#

k∈Si

1 k^η+δ

1

((kα)) = #

j≥1

1 k^η+δ_i,j

1

((ki,jα)) ≤#

j≥1

2ⁱ⁺¹ (jKi)^η+δ

= 2ⁱ⁺¹ K_i^η+δ

#

j≥1

1

j^η+δ ≤D(ε)2^{i(1−η+δη+ε)};

!

k≥1 1 k^η+δ

1 ((kα))

f X =R/Z V(f)

(fn) p q

|qα−p|<1/q

|

q−1

#

0

f(x+0α)−q

$

X

f dµ0| ≤V(f),∀x∈X.

(qn) α

|

q#n−1 0

f(x+0α)−qn

$

X

f dµ0| ≤V(f),∀x∈X.

f ck(f) = O(¹_k)

L² L²

f O(¹_k)

C α q

α 0!_q−1

k=0f(.+kα)−qµ0(f)02 ≤C

•

f τα

(qn) (qn)

α µ0(f) = 0

n ≥1 n = !-n−1

1 bjqj 0 ≤bj ≤ aj+1 j = 1, ..., 0n−1 b_-n−1 ≥1 0_n

q-n−1 ≤n < q-n.

(qn) 0n=O(logn

θ(n) := !-n−1

1 aj+1 f O(¹_k)

0fn02 ≤C θ(n), ∀n≥1.

f

0fn0∞≤V(f)θ(n),∀n ≥1.

θ(n) α θ(n) =O(logn)

α ε > 0 C(ε) > 0

an≤C(ε)q^ε_n−1 ∀n ≥1 n < (a-n+ 1)q-n−1 <(C(ε)q_-^ε_n−1+ 1)q-n−1

C1(ε)n^1−ε≤C1(ε)n^1+ε1 ≤q-n−1.

α ε >0 θ(n) =O(n^ε)

θ(n)≤0nsup

k≤-n

ak≤C(ε)q^ε_-n−1log n≤C(ε)n^ε log n.

(E,A, µ, τ) µ

ϕ G

mG (Dn) G

(kn)

limn µ(x:ϕ_kn(x).∈D_n) = 0 and lim

n

m_G(D_n.K)

n = 0

K ⊂G (ϕ, τ)

λ µ×mG B

τϕ (τ_ϕ^-B, 0≥0) λ(B) = 0

λ(B)>0 B E×K K G

E_n = {x : ϕ_kn(x) ∈ D_n} B_n = (E_n × K)%

B n₀

λ(Bj)≥ ¹₂λ(B) j ≥n0 λ(∪ⁿj=n0τϕ^kjBj)≥ ¹₂(n−n0)λ(B) E×(D_n.K)

λ(∪ⁿj=n0τ_ϕ^kjB_j)≤m_G(D_n.K).

lim inf

n

mG(Dn.K)

n ≥ 1

2λ(B)>0

N3

R^d

G=N3

g =



1 a c 0 1 b 0 0 1



, a, b, c∈R³.

G R³

(a, b, c).(a^&, b^&, c^&) = (a+a^&, b+b^&, c+c^&+ab^&).

N₃ N3

Φ = (f, g, h) X R³ f, g, h

µ₀(f) = µ₀(g) = µ₀(h) = 0 (f, g)

τΦ X×N3

τΦ(x, g) = (x+α,Φ(x).g) = (x+α, a+f(x), b+g(x), c+h(x) +bf(x)).

k ≥2

τ_Φ^k(x, a, b, c) = (x+kα, a+fk(x), b+gk(x), c+hk(x) +bfk(x) +

k−1

#

j=1

f(τ^jx)gj(x)).

k = q_n α f_k, g_k, h_k V(f)

Φ N3

!n

j=1τ^jf gj

nγ(f, g) α γ(f, g) f =!

p'=0cpep g =!

p'=0dpep f g ep =e^2πip., p∈Z

#n j=1

τ^jf gj = #

p,q'=0

cpdqep+q

*_n−1

#

k=1

e^2πipkα(

k−1

#

j=0

e^2πiqjα) +

=n#

p'=0

cpd₋p (e^−2πipα−1)⁻¹−#

p'=0

cpd₋p (e^−2πipα−1)⁻¹(e^2πipnα−1 e^2πipα−1)

+ #

p,q'=0,p+q'=0

cpdqep+q (e^2πiqα−1)⁻¹(e^2πi(p+q)nα−1

e^2πi(p+q)α−1 − e^2πipnα −1 e^2πipα−1).

γ(f, g) n

γ(f, g) =#

p'=0

cpd_−p

1−e^−2πipα =#

p≥1

[2(cpdp) +3(cpdp)cosπpα sinπpα].

!

p≥12(cpdp) ¹₂µ0(f g)

!

p3(cpdp)^cos_sin^πpα_πpα α

α η < 2 f =g

γ(f, g) !

p≥12(|cp|²) = ¹₂µ0(f²)

γ(f, g) α

x → x+α Φ T¹ N3

(f, g, h) γ(f, g) = 0

g X q≥1 g = 1

q

#q k=1

gk−τ(1 q

#q k=1

gk) + 1 qτ gq.

g =ψ^(q)−τ ψ^(q)+ζ^(q),

ψ^(q) = 1 q

#q k=1

gk, ζ^(q) = 1 qτ gq.

"

ψ^(q)f dµ0

$

ψ^(q)f dµ0=−#

p'=0

cpd₋p

1−e^−2πipα(1 +εp,q), εp,q =−¹_q^e_e⁻−^2πiqpα2πipα−⁻1¹e^−2πipα

|εp,q|= 1

q|e^2πiqpα−1

e^2πipα−1| ≤inf(1,1 q

2 ((pα))).

!

p'=0|_1−e^cp−^d2πipα^−p |<∞ limq

$

ψ^(q)f dx=−#

p

cpd₋p

1−e^−2πipα =−γ(f, g).

α Sq

Sq :=#

p≥1

1 p²

1

((pα))inf(1,((qpα)) q((pα))).

Sq ≤ 1 q²

q−1

#

p=1

1 p

1

((pα))² + 1 q¹²

#

p≥q

1 p³²

1

((pα)) = (1) + (2).

(2) !

p≥1 1 p²³

1

((pα)) <∞ (1) _q¹2(!_q−1

p=1 1 p²

1

((pα)))¹²(!_q−1

p=1 1

((pα))³)¹² !_q−1

p=1 1 p²

1 ((pα))

rα 1

1 ≤ r < q [^k_q,^k+1_q [ k = 1, ..., q−1 !q−1 p=1 1

((pα))³)¹² (!_q−1

p=1 q³

p³)¹² ≤Cq³²

α Sq = 0(q⁻¹²)

|

$

f ψ^(q) dµ0−γ(f, g)|= 0(q⁻²¹).

q α ψ^(q) ζ^(q)

0 0

0ψ^(q)0 ≤ θ(q)V(g), V(ψ^(q)) ≤ 1

2qV(g), 0ζ^(q)0 ≤ V(g)

q .

#n k=1

τ^kf gk =

#n k=1

(ψ^(q)τ^kf−τ^k(ψ^(q)f)) +

#n k=1

ζ_k^(q)τ^kf

=

#n k=1

(ψ^(q)τ^kf−τ^k(ψ^(q)f)) +

#n k=1

τ^k−1ζ^(q) τ^kfn−k+1,

0

#n k=1

τ^kf gk−nγ(f, g)0

≤ 0ψ^(q)0 0fn0+0(ψ^(q)f −

$

ψ^(q)f)n0+0ζ^(q)0

#n k=1

0τ^kfn−k+10+n|

$

ψ^(q)f dµ0+γ(f, g)|

≤ 0ψ^(q)0V(f)θ(n) +V(ψ^(q)f)θ(n) +0ζ^(q)0

#n k=1

0fn−k+10+ Cn

√q.

0

#n k=1

τ^kf gk−nγ(f, g)0

≤ (2θ(q)V(g)V(f) + 1

2qV(g)0f0)θ(n) + V(g)

q V(f)nθ(n) + Cn

√q

≤ C(q+n

q)θ(n) + Cn

√q. q=qrn−1 qrn−1 ≤√

n < qrn qrn−1 ≥C(√ n)^1−ε n

q_rn−1 ≤C n (√

n)^1−ε =Cn^12+ε2 n

√q_rn−1 ≤Cn^{1−14(1−ε)}.

C1 ε 0

#n−1 k=0

τ^kf gk−nγ(f, g)0 ≤C1

√n(1 +n^ε2)θ(n) +C n

√qrn−1 ≤2C1n^12+3ε2 +Cn^34+ε4. γ(f, g) = 0 !_n−1

k=0τ^kf gk=O(n^δ) δ > ³₄

O(n^ε) ε > 0

kn=n Dn ε >0 C

Dn:={(x, y, z),0≤ |x| ≤Cn^ε,0≤ |y| ≤Cn^ε,0≤ |z| ≤Cn^(34+ε)}. (Dn)

N₃ γ(f, g) = 0

α γ(f, g).= 0 (Φ_n)

nγ(f, g)

γ(f, g)

ckd₋k= 0,∀k∈Z.

f(x+ ¹₂) ≡ f(x) g(x+ ¹₂) ≡ −g(x)

#n k=1

τ^kf

* _k

#

j=1

τ^jg(

#j -=1

τ^-h) +

.

N₃

p ≥2 (f, g)

p−1

#

l=0

f(y+ l

p)g(x+ l

p) = 0,∀x, y,

ck(f) ck^"(g) = 0 ∀(k, k^&) p k+k^&

f x → x+ ¹_p g

!_p−1

k=0g(x+ ^k_p) = 0

p= 2 f(x)≡f(x+¹₂) g(x)≡ −g(x+¹₂) f(x)≡ −f(x+¹₂) g(x)≡g(x+ ¹₂)

R

ϕ

G

x→x+α mod 1

µ0 µχ µχ

χ R

(X,A, µ, τ) µ ϕ

X G c∈G

(ϕ, τ) δ > 0 (0n)_n≥1 (εn>0)_n≥1 limnεn= 0

limn τ^-nx=x, for µ−a.e. x∈X µ(An)≥δ,∀n≥1, An ={x:d(ϕ-n(x), c)< εn}

(ϕ, τ) τϕ

λ0

f(x, t) τϕ

X ×G f(τ x, ϕ(x) +t) = f(x, t) f τϕ

t f (x, t)→ "

Gf(x, s)h(t−s) ds h

G f t

c {x : f(x, c +t) = f(x, t), t} τ

τ f(x, c+t) =f(x, t)

µ x .= 0

limnτ^-nx=x, ∀x∈X ζ X

limn

$

X |ζ(τ^-nx)−ζ(x)| dµ(x) = 0.

X

µ

u >0 R

limn

$

R

$

X

|f(τ^-nx, t)−f(x, t)| u(t)dµ(x)dt= 0.

$

R

$

X

1An(x)|f(x, t)−f(x, t−c)| u(t) dµ(x)dt

≤

$

R(

$

X|f(x, t)−f(τ^-nx, t)| dµ(x))u(t)dt +

$

R

$

X

1An(x)|f(x, t−ϕ-n(x))−f(x, t−c)| u(t)dµ(x)dt.

limn

$

X

1An(x) (

$

R|f(x, t)−f(x, t−c)| u(t)dt) dµ(x) = 0.

µ(A_n)≥δ >0 x→

$

R|f(x, t)−f(x, t−c)| u(t)dt

≥δ

f τϕ

c θ_c f(x, c+t) =θ(c)f(x, t)

τ x → x+α mod 1 ϕ

(0n) (qn) α

µ χ◦ϕ µχ

λχ

C >0

C⁻¹ ≤ d(τ^qnµ_χ) dµχ ≤C.

ϕ R

µ=µ0 µ=µχ λ

X×R ϕ (ϕqn)

µ τϕ λ

f(x, y) f(τ x, y+ϕ(x)) = f(x, y),forµ−a.e. x

f y u

R

$ $

|f(x, y)−f(τ^qnx, y)| u(y)dy dµ(x)→0

f(τ^qnx, y) =f(x, y−ϕqn(x))

$ $

|f(x, y)−f(x, y−ϕqn(x))| u(y)dy dµ(x)→0.

0 01 (qnk)

µ x

$

|f(x, y)−f(x, y−ϕq_nk(x))|u(y)dy→0.

(ϕq_nk)

(tj(x))_j≥1 N

+∞ (qn_k)

limk ϕtj(x)(x) =c(x), c

$

|f(x, y)−f(x, y−c(x))| u(y)dy= 0.

x >0 y→f(x, y)

τϕ τ

x x

τϕ (qn_k)

(qn) (0nqn)

ϕ = !

ici1Ii −β

G = R ϕ

ϕ(x) = #

i

ci1Ii(x mod 1)−β,

(Ii) [0,1] ci β

µ(ϕ) = 0

V ({qnβ})n≥1 F F ={l∈Z:|0| ≤V(ϕ) + 1}.

(ϕn) ϕn(x) = un(x)− {nβ}, n ≥ 1 un

|ϕqn(x)| ≤ V(ϕ) uqn(x) F

γ ∈ V τ_ϕ λ

u∈ F u−γ Ru−γλ∼λ

γ ({qnβ}ⁿ≥1) (tk)

γk:={qtkβ} →γ

F X×R

λ k ≥0

$ $

F(x, y)dλ(x, y) =

$ $

F(x+qt_kα, y+ϕq_tk(x))dλ

= #

u∈F

$ $

u_qt

k=u

F(x+qt_kα, y+u−γk) dλ

≤ #

u∈F

$ $

F(x+qtkα, y+u−γk) dλ ;

$ $

F(x, y) dλ ≤ #

u∈F

$ $

F(x, y+u−γ) dλ,

u∈ F R_u−γλ

λ

γ ∈ V u ∈ F u−γ

µ=µ₀ µ=µ_χ

u ∈ F (tk) µ{x:uq_tk(x) =u} ≥Card(F)⁻¹

q_tkα 1→0 u−γ

V .={0,1} u−γ u F γ ∈ V

.

= 0