On the ergodic decomposition for a cocycle

(1)

HAL Id: hal-00456611

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Submitted on 22 Mar 2018

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On the ergodic decomposition for a cocycle

Jean-Pierre Conze, Albert Raugi

To cite this version:

Jean-Pierre Conze, Albert Raugi. On the ergodic decomposition for a cocycle. Colloquium Mathe-

maticum, 2009, 117 (1), pp.121-156. �hal-00456611�

(2)

(X,X, µ, τ) ϕ

X G m_G

τ_ϕ X×G τ_ϕ: (x, g)→(τ x, ϕ(x)g)

(ϕn)n∈Z ϕ

τ_ϕ µ(dx)⊗m_G(dg)

τ_ϕ µ_χ(dx)⊗χ(g)mG(dg) µ_χ(dx) χ◦ϕ

χ G

τ

ϕ

λ

χ

(3)

H

τ x

= ϕ(x) H

x

(ϕ(x))

⁻¹

(X,

^X

, µ, τ ) (X,

^X

) µ

σ

^X

τ X µ

τ

G

B_G

σ

m

G

(dg) dg G e

ϕ X G τ

ϕ

X × G

τ

ϕ

: (x, g) → (τ x, ϕ(x)g).

G (ϕ

n

)

n∈Z

(X, µ, τ ) (ϕ, τ )

ϕ

n

(x) =

 



ϕ(τ

ⁿ⁻¹

x) · · · ϕ(x), n > 0,

e, n = 0,

ϕ(τ

ⁿ

x)

⁻¹

· · · ϕ(τ

⁻¹

x)

⁻¹

, n < 0.

µ τ τ

ϕ

λ

1

:= µ ⊗ m

G

(ϕ

n

) G (X, µ, τ )

(4)

χ G G ]0, + ∞ [

∀ g, g

^#

∈ G, χ(g g

^#

) = χ(g ) χ(g

^#

) µ

χ

χ ◦ ϕ σ X

(τ µ

χ

)(dx) = χ(ϕ(τ

⁻¹

x)) µ

χ

(dx), λ

χ

(dx, dg) := µ

χ

(dx) ⊗ χ(g)m

G

(dg)

σ X × G τ

ϕ

τ

ϕ

λ

χ

X × G

(ϕ

n

)

n∈Z

G

F (G) G

• F (G)

G F (G) G

U ( O , C ) = { S ∈ F (G) : ∀ U ∈ O , S ∩ U ) = ∅ S ∩ C = ∅} ,

O G C G

(F

n

) G

F

(i) ξ : N +→ N (g

n

)

n∈N

g

n

∈ F

ξ(n)

n ≥ 0 (g

n

)

n∈N

g ∈ G g F

(ii) g ∈ F (g

n

)

n∈N

g

n

∈ F

n

n ≥ 0

(5)

{ S ∈ F (G) :

S ⊆ F } F ∈ F (G) G d

G G (g

n

)

n∈N

G { d(g

n

, · ), n ∈ N} F (G)

• H m

_H

(dγ)

dγ H δ

u

u ∈ H e

ρ

1

ρ

2

H ρ

1

∗ ρ

2

ρ

₁

⊗ ρ

₂

(g, g

^#

) ∈ H × H −→

g g

^#

∈ H

ϕ X G τ

ϕ

λ τ

ϕ

X × G

J J_ϕ

σ τ

ϕ X

×

^B^G

X × G τ

ϕ

f τ

ϕ

λ τ

ϕ

g f = g λ

τ

ϕ

G (ϕ, τ ) (ψ, τ) (X, µ, τ )

µ u : X → G

ϕ(x) = u(τ x) ψ(x) (u(x))

⁻¹

for µ − a.e. x.

u ϕ

^(u,µ)

∼ ψ

(ϕ, τ ) µ µ ψ ≡ e

τ

ϕ

λ

χ

λ

χ

= µ

χ

⊗ (χm

G

) χ G µ

χ

σ

χ ◦ ϕ τ X χ ≡ 1 µ

χ

τ

h X × G

$

X×G

h(x, g) µ

_χ

(dx) χ(g) m

_G

(dg) = 1.

h µ

χ

σ X G

P

^h

h λ

_χ

σ

^J

τ

ϕ

P

^h

X × G

(6)

f X × G P

^h

f E

^{h λ}^χ

[f |

^J

]

M

^h

X × G

∀ (x, g) ∈ X × G, M

^h

f(x, g) = P

^h

(f /h)(x, g), f X × G

h h

^#

M

^h^!

((x, g), .) = P

^h

(h/h

^#

)(x, g) M

^h

((x, g), .)

λ

χ

(x, g) ∈ X × G M

^h

((x, g), .) X × G τ

ϕ

• E

^hλχ

[ · ] = E

^hλχ

[ E

^hλχ

[ · |

^J

]]

λ

χ

(dy, dt) =

$

X×G

M

^h

((x, g), (dy, dt))h(x, g) λ

χ

(dx, dg), λ

χ

τ

ϕ

λ

χ

• (µ

x

)

x∈X

σ τ X σ

(X,

^X

) (X,

^X

) x ∈ X µ

x

σ

X

A ∈

^X

x → µ

x

(A) ∈ [0, + ∞ ]

X

• (H

x

)

x∈X

G x → H

x

X F (G)

• η : X × G +→ R

^∗+

x ∈ X χ

x

( · ) := η(x, · ) H

x

• u : X × X +→ G x ∈ X u

x

(.) = u(x, .)

µ

χ

x ∈ X g ∈ G

H

τ x

= ϕ(x)H

x

ϕ(x)

⁻¹

,

ψ(y) := u

_x

(τ y)

⁻¹

ϕ(y) u

_x

(y) ∈ H

_x

, µ

_x

− a.e. y, τ µ

_x

(dy) = χ

_x

(ψ(τ

⁻¹

y)) µ

_x

(dy),

χ

x

(γ) = χ

τ x

(ϕ(x) γ (ϕ(x))

⁻¹

), ∀ γ ∈ H

x

,

ζ

x

(y) := (u

x

(y))

⁻¹

u

τ x

(y) ϕ(x) ∈ H

x

, µ

x

− a.e. y,

µ

τ x

(dy) = c(x) χ

x

(ζ

x

(y)) µ

x

(dy), c(x).

(7)

M

^h

f (x, g) =

%

X

( %

Hx

f(y, u

x

(y) γ g) χ

_x

(γ) m

Hx

(dγ) µ

x

(dy)

%

X

( %

Hx

h(y, u

x

(y) γ g) χ

_x

(γ ) m

Hx

(dγ) µ

x

(dy) .

m

Hx

x ∈ X H

x

$

Hx∩{d(e,·)≤1}

χ

x

(γ) m

Hx

(dγ) = 1,

K(x, dt) := m

Hx

(dt) (X,

^X

) (G,

^BG

) λ

χ

= µ

χ

⊗ (χ m

G

)

λ

χ

(dy, dt) =

$

X×G

M

^h

((x, g), (dy, dt)) h(x, g) λ

χ

(dx, dg).

λ

_χ

τ

_ϕ

f λ

_χ

f = P

^h

f τ

ϕ

H G a : X → G

H

_x

= a(x)H(a(x))

⁻¹

µ

_χ

x ∈ X G

˜

χ

x

(γ ) :=

χ

x

(a

x

γa

⁻¹_x

)

M

^h

f (x, g) =

%

X

( %

H

f(y, u

_x

(y) a(x) γ (a(x))

⁻¹

g) ˜ χ

_x

(γ) dγ) µ

_x

(dy)

%

X

( %

H

h(y, u

x

(y) a(x) γ (a(x))

⁻¹

g) ˜ χ

_x

(γ) dγ) µ

x

(dy) .

G H

x

H G

χ

x

χ

M

^h

f (x, g) =

%

X

( %

H

f(y, u

x

(y) γ g) χ(γ) dγ) µ

x

(dy)

%

X

( %

H

h(y, u

_x

(y) γ g) χ(γ) dγ) µ

_x

(dy) .

• ϕ µ

χ

H G u : X → G

ψ := (u ◦ τ)

⁻¹

ϕ u µ

_χ

H τ

_ψ

: (x, h) → (τ x, ψ(x)h)

µ

χ

⊗ (χ m

H

)

(8)

(χ ◦ u) µ

χ

⊗ χ m

H

τ

ψ

µ

χ

⊗ (χdg) H

x

H H

x

= u(x) H u(x)

⁻¹

x

0

∈ X { x ∈ X : µ

x

∼ µ

x0

} µ

χ

(ϕ, τ ) µ

χ

τ

ϕ

f f (x, g) = F

f

((u(x))

⁻¹

g), µ

χ

⊗ m

G

F

f

H

G λ

χ

M

^h

f (x, g) =

%

X

( %

H

f(y, u(y) γ (u(x))

⁻¹

g) χ(γ) dγ) χ(u(y)) µ

χ

(dy)

%

X

( %

H

h(y, u(y) γ (u(x))

⁻¹

g) χ(γ) dγ) χ(u(y)) µ

_χ

(dy) . H

x

= u(x) H u(x)

⁻¹

χ

x

(γ) = χ(u(x) γ u(x)

⁻¹

) u

x

(y) = u(y) u(x)

⁻¹

µ

x

(dy) = χ(u(y)) µ

χ

(dy)

(ϕ, τ ) µ

χ

µ

χ

x µ

x

µ

χ

⊗ (χ m

G

) µ

χ

G µ

_χ

µ

χ

x ∈ X µ

x

1

_[0,β]

− 1

_[0,β]

(. + r) β r

• u (ϕ

n

)

G G = &

n

U

n

K

n

= U

n

G = &

n∈N

K

n

K G n ∈ N K ⊂ K

n

u X × X G

K G X

X

K

= { x ∈ X : u

x

(y) H

x

⊂ K H

x

, µ

x

y ∈ X } = { x ∈ X : Supp (u

x

(µ

x

)) ⊂ K H

x

} .

X

K

x ∈ X

K

⇒ τ x ∈ X

_K_(ϕ(x))−1

&

n∈N

X

_K_n

τ X µ

_χ

µ

χ

(9)

K G µ

χ

(X

K

) > 0 &

n∈N

X

Kn

µ

χ

u

u n ∈ N

µ

χ

x ∈ X

n

= X

Kn

\ X

K_n−1

, u

x

(y) ∈ K

n

, µ

x

y ∈ X.

{ x : G/H

_x

is compact }

K G F → K.F F (G)

G x → H

x

x → K.H

x

(x, y) ∈ X × X +→ u(x, y) H

x

∈ F (G) g ∈ G F ∈ F (G) +→ d(g, F ) ∈ R

⁺

{ (x, y) ∈ X × X : d(g, KH

x

) ≤ d(g, u(x, y)H

x

) }

(g

n

)

n∈N

G

X

_K

= { x ∈ X : ∀ n ∈ N , ν

_(x,e)

( { y ∈ X : d(g

_n

, K H

_x

) ≤ d(g

_n

, u(x, y) H

_x

) } ) = 1 } . X

K

x ∈ X

K

⇒ τ x ∈ X

K(ϕ(x))⁻¹

K G n ∈ N K ⊂ K

_n

&

n∈N

X

Kn

τ µ

χ

µ

χ

(X

K

) > 0 K G

&

n∈N

X

Kn

µ

χ

u

{ x ∈ X : G/H

x

is compact } = '

n∈N

{ x ∈ X : K

n

.H

x

= G } ; τ

µ

χ

µ

χ

µ

χ

x ∈ X

&

n∈N

K

_n

H

_x

= G &

n∈N

X

_K_n

µ

_χ

µ

χ

K G µ

χ

(X

K

) > 0 u

µ

_x

µ

_χ

x ∈ X

G K G µ

_χ

(X

_K

) > 0

(10)

G µ

χ

τ µ

χ

µ

χ

( { x ∈ X : ˜ µ

x

(X) <

+ ∞} ) > 0

˜

µ

x

(dy) := (χ(u

x

(y)))

⁻¹

µ

x

(dy),

τ µ

_χ

(ϕ

_n

) µ

_χ

K G µ

χ

x ∈ X ∀ n ≥ 0 ϕ

n

(x) ∈ K

H

x

G

r G %

G

r(t) χ(t) dt = 1 K G r

_K

(g) := min

_u∈K

r(ug) > 0

f X

M

^h

(f ⊗ r)(x, g) = c(x, g)

$

X

f(y)(

$

Hx

r(u

x

(y) γ g) χ

_x

(γ) m

Hx

(dγ)) µ

x

(dy)

≥ c(x, g)

$

X

f (y)(

$

Hx

1

K

(u

x

(y)) r(u

x

(y) γ g) χ

_x

(γ) m

Hx

(dγ )) µ

x

(dy)

= c(x, g)(

$

Hx

r

K

(γ g) χ

_x

(γ) m

Hx

(dγ ))

$

X

f (y)1

K

(u

x

(y)) µ

x

(dy)

µ

χ

(f ) = λ

χ

(f ⊗ r) ≥

$

X

Ψ

K

(x) ( $

X

f(y)1

K

(u

x

(y)) µ

x

(dy) )

µ

χ

(dx), Ψ

_K

(x) := %

G

c(x, g) ( %

Hx

r

_K

(γ g) χ

_x

(γ) m

_H_x

(dγ)) h(x, g) χ(g) dg > 0 n ∈ N

µ

χ

(f ) ≥

$

Xn

Ψ

Kn

(x) µ

x

(f ) µ

χ

(dx)

f = 1

_X

µ

_x

(X) < + ∞ µ

_χ

x ∈ X

_n

µ

_χ

x ∈ X ∪

ⁿ

X

n

X

G

x µ

x

(τ µ

_x

)(dy) = χ(ψ(τ

⁻¹

y)) µ

_x

(dy) µ ˜

_x

(dy) µ

χ

τ µ ˜

x

(dy) = χ(ϕ(τ

⁻¹

y)) ˜ µ

x

(dy).

n ∈ N µ

χ

(f ) ≥

$

Xn

Φ

Kn

(x) ˜ µ

x

(f ) µ

χ

(dx),

(11)

Φ

Kn

(x) = Ψ

Kn

(x) inf

u∈Kn

χ(u) B ∈

^X

B ⊂ X

n

ξ

_B

X

$

1

B

(x)Φ

Kn

(x) ˜ µ

x

(dy) µ

χ

(dx) = ξ

B

(y) µ

χ

(dy).

ξ

B

◦ τ

⁻¹

= ξ

B

µ

χ

µ

χ

τ

ξ

B

µ

χ

ν(B) B → ν(B) ν

(X

n

, X

n

∩

^X

) µ

χ

ξ X

$

1

B

(x)Φ

Kn

(x) ˜ µ

x

(dy) µ

χ

(dx) = ν(B) µ

χ

(dy) = (

$

1

B

(x)ξ(x) µ

χ

(dx)) µ

χ

(dy) µ

_χ

x ∈ X

_n

ξ(x) µ

χ

(dy) = Φ

Kn

(x) ˜ µ

x

(dy).

&

n∈N

X

n

Φ

Kn

Φ

µ

χ

x ∈ X

ξ(x) µ

_χ

(dy) = Φ(x) ˜ µ

_x

(dy).

X

0

= { x ∈ X : ˜ µ

x

(X) < + ∞} x ∈ X

0

µ ˆ

x

˜

µ

x

/˜ µ

x

(X) K G

µ

χ

(f ) ≥

$

X0

Φ

K

(x) ( $

X

f (y)1

K

(u

x

(y)) ˆ µ

x

(dy) )

µ

χ

(dx);

Φ

_K

(x) := Ψ

_K

(x) inf

_u∈K

χ(u) ˜ µ

_x

(X)

h

1

X τ

µ

χ

( { *

k≥0

h

1

◦ τ

^k

< + ∞} ) = 0 µ

χ

x ∈ X

0

∀ n ∈ N , µ ˆ

_x

( { +

k≥0

h

₁

◦ τ

^k

< + ∞} ∩ { u

_x

∈ K

_n

} ) = 0.

n + ∞

µ

χ

x ∈ X

ˆ µ

x

( { +

k≥0

h

1

◦ τ

^k

< + ∞} ) = 0.

h

₁

µ ˆ

_x

x ∈ X

₀

µ

_χ

x ∈ X

₀

τ µ ˆ

x

X

0

X

0

∩ { x ∈ X : ˆ µ

x

( { *

k≥0

h

1

◦ τ

^k

< + ∞} ) = 0 }

x ∈ X

0

τ µ ˆ

x

(12)

[0, 1] ξ

K

ξ

K

(y) µ

χ

(dy) =

$

X0

Φ

K

(x) 1

K

(u

x

(y)) ˆ µ

x

(dy) µ

χ

(dx) ≤

$

X0

Φ

K

(x) ˆ µ

x

(dy) µ

χ

(dx).

[0, 1] ψ

K

ξ

K

(y) µ

χ

(dy) = ψ

K

(y)

$

X0

Φ

K

(x) ˆ µ

x

(dy) µ

χ

(dx).

µ

_χ

µ ˆ

_x

+

n−1 k=0

T

^k

ξ

_K

(y) µ

_χ

(dy) =

$

X0

Φ

_K

(x) +

n−1 k=0

T

^k

ψ

_K

(y) ˆ µ

_x

(dy) µ

_χ

(dx) T

T f (y) = f ◦ τ

⁻¹

(y) χ(ϕ(τ

⁻¹

y)).

τ µ

χ

µ ˆ

x

x ∈ X

0

f X

( +

ⁿ⁻¹

k=0

T

^k

f /

n−1

+

k=0

T

^k

1 )

n∈N

µ

χ

µ

χ

(f) µ ˆ

x

µ ˆ

x

(f ) x ∈ X

0

L

¹

f

$

X

f(y)

*

_n−1

k=0

T

^k

ξ

K

(y)

*

_n−1

k=0

T

^k

1(y) µ

_χ

(dy)

_n

−→

→+∞

µ

_χ

(ξ

_K

) µ

_χ

(f ) µ

χ

x ∈ X

α

n

(x) =

$

X

f (y)

*

n−1

k=0

T

^k

ψ

K

(y)

*

n−1

k=0

T

^k

1(y) µ ˆ

x

(dy) −→

n→+∞

µ ˆ

x

(ψ

K

) ˆ µ

x

(f ).

Φ

K

µ

χ

(α

_n

)

$

X0

Φ

K

(x) α

n

(x) µ

χ

(dx) −→

n→+∞

$

X0

Φ

K

(x) ˆ µ

x

(ψ

K

) ˆ µ

x

(f) µ

χ

(dx).

µ

χ

(dy) =

$

X0

Φ ,

K

(x) ˆ µ

x

(dy) µ

χ

(dx),

(13)

Φ ,

K

(x) = Φ

K

(x) ˆ µ

x

(ψ

K

)/µ

χ

(ξ

K

) B ∈

^X

B ⊂ X

0

ξ

_B

ξ

B

(y) µ

χ

(dy) =

$

B

Φ ˆ

K

(x) ˆ µ

x

(dy) µ

χ

(dx).

ξ

B

◦ τ

⁻¹

= ξ

B

µ

χ

µ

χ

τ ξ

B

µ

χ

ν(B) ν

(X

n

, X

n

∩

^X

) µ

χ

ξ X

$

1

B

(x) ˆ Φ

K

(x) ˆ µ

x

(dy) µ

χ

(dx) = ν(B) µ

χ

(dy) = (

$

B

ξ(x)µ

χ

(dx)) µ

χ

(dy) µ

χ

x ∈ X

0

ξ(x) µ

χ

(dy) = ˆ Φ

K

(x) ˆ µ

x

(dy).

τ µ

χ

K µ

χ

x ∈ X ϕ

n

(x) ∈ K, n ∈ N

µ

χ

(f) ∈ ]0, + ∞ [ +

n≥0

f(τ

ⁿ

x)1

K

(ϕ

n

(x)) = +

n≥0

f (τ

ⁿ

x) = + ∞ , µ

χ

.

τ

ϕ

λ

χ

x ∈ X

0

X

0

µ

χ

g ∈ G τ

ϕ

M

^h

((x, g), · )

x ∈ X

0

s ∈ Supp (u

x

(µ

x

)) t ∈ H

x

V

W s t µ

_x

y ∈ X *

n≥0

1

_V

(τ

ⁿ

y) 1

_W

(ϕ

_n

(y)) = + ∞

u

x

(τ

ⁿ

y) ψ

n

(y) = ϕ

n

(y) u

x

(y) ⊂ K u

x

(y), µ

x

y ∈ X, µ

χ

x ∈ X µ

x

y ∈ X s t ∈ K u

x

(y)

s (t

_n

) H

_x

µ

_χ

x ∈ X

µ

x

y ∈ X ∀ n ≥ 0, t

n

∈ s

⁻¹

K u

x

(y) H

x

⊂ s

⁻¹

K u

x

(y)

G µ

χ

x ∈ X µ

x

y ∈ X

Supp (u

_x

(µ

_x

)) ⊂ K u

_x

(y) H

_x

µ

_χ

x ∈ X

K

x

G Supp (u

x

(µ

x

)) ⊂ K

x

H

x

K G

K ⊂ K

n

n &

n∈N

X

Kn

µ

χ

u

n ∈ N µ

χ

(f) ≥

$

Xn

Ψ

Kn

(x) µ

x

(f) µ

χ

(dx).

(14)

[0, 1] ξ

$

Xn

Ψ

Kn

(x) µ

x

(dy) µ

χ

(dx) = ξ(y) µ

χ

(dy).

x ∈ X χ

x

µ

x

x ∈ X τ

ξ ◦ τ

⁻¹

dτ µ

χ

/dµ

χ

= ξ µ

χ

ξ µ

_χ

τ { ξ > 0 } µ

_χ

τ

µ

χ

B ∈

^X

B ⊂ X

n

[0, 1] ξ

B

$

B

Ψ

_K_n

(x) µ

_x

(dy) µ

_χ

(dx) = ξ

_B

(y) ξ(y) µ

_χ

(dy).

ξ

B

◦ τ

⁻¹

= ξ

B

µ

χ

ξ

B

µ

χ

ν(B) ν (X

n

, X

n

∩

^X

)

µ

χ

ψ X

$

B

Ψ

Kn

(x) µ

x

(dy) µ

χ

(dx) = ν(B) ξ(y) µ

χ

(dy) = (

$

B

ψ(x) µ

χ

(dx)) ξ(y) µ

χ

(dy) µ

_χ

x ∈ X

_n

ψ (x) ξ(y) µ

_χ

(dy) = Ψ

_K_n

(x) µ

_x

(dy).

G G ϕ

ψ

K G µ ⊗ m

K

τ

ψ

G ϕ

(ϕ

n

)

µ τ

(15)

µ τ X

a ∈ G ∪ {∞} (ϕ, τ ) µ

V a B µ(B) > 0 n ∈ Z

µ(B ∩ τ

⁻ⁿ

B ∩ { x : ϕ

_n

(x) ∈ V } ) > 0.

E (ϕ) (ϕ, τ ) E (ϕ) = E (ϕ) ∩ G

B µ τ

B

B ϕ

^B

(x) := ϕ

_n(x)

(x) n(x) = n

_B

(x) := inf { j ≥ 1 : τ

^j

x ∈ B } x ∈ B n ≥ 1 ϕ

^B_n

(x) := ϕ

^B

(x) ϕ

^B

(τ

B

x) · · · ϕ

^B

(τ

_Bⁿ⁻¹

x)

a ∈ G ∪ {∞}

(ϕ, τ ) B µ(B) > 0

V a µ( { x : ϕ

^B_n

(x) ∈ V } ) > 0 n ∈ Z

τ µ

χ

∞ )∈ E (ϕ) ϕ

G G

E (ϕ) = { e } ϕ

∞ )∈ E (ϕ) B µ

χ

(B) > 0 (ϕ

^B_n

)

n∈Z

ϕ

^B

τ

_B

G ζ

^B

B G

ψ

^B

B G

ϕ

^B

= ζ

^B

◦ τ

B

ψ

^B

(ζ

^B

)

⁻¹

.

(X, µ

_χ

, τ) µ

_χ

y ∈ X

x ∈ B k 0 ≤ k < n

B

(x) y = τ

^k

x ζ X

y = τ

^k

x 0 ≤ k < n

B

(x)

ζ(y) = ϕ

k

(x) ζ

^B

(x) (ψ(y))

⁻¹

,

ψ(y) = e k < n

B

(x) − 1 ψ(y) = ψ

^B

(x) k = n

B

(x) − 1 0 ≤ k < n

B

(x) − 1

k = n

_B

(x) − 1

E (ϕ) = { e } ϕ

ψ K G

φ ψ τ

_ψ

E (ψ) = { e }

K = { e }

λ

χ

(16)

P (ϕ) G τ

ϕ

γ ∈ G τ

ϕ

f f (x, γ g) = f (x, g), λ

χ

(x, g) ∈ X × G)

P (ϕ, µ

_χ

) P (ϕ) E (ϕ) µ

_χ

P (ϕ) = E (ϕ)

(Y, ρ) (g, y) → g.y

G f X Y G ϕ

f (ϕ, τ ) f (τ x) = ϕ(x).f(x) µ

f (ϕ, τ ) a.f (x) = f (x) µ − a.e. ∀ a ∈ E (ϕ)

(Y, ρ)

X

f

:= { x ∈ X : µ( { x

^#

∈ X : ρ(f(x

^#

), f (x)) < ε } ) > 0, for every ε > 0 }

µ f

⁻¹

(suppf (µ)) x ∈ X

f

a ∈ E (ϕ) ε > 0

E

x

= { x

^#

: ρ(f (x

^#

), f (x)) < ε } µ a ∈ E (ϕ) ε

1

> 0 x

1

∈ E

x

n ∈ Z τ

ⁿ

x

1

∈ E

x

d(a, ϕ

n

(x

1

)) < ε

1

d G f

ρ(a.f(x), f(x)) ≤ ρ(af(x), af (x

₁

)) + ρ(af (x

₁

), ϕ

_n

(x

₁

).f(x

₁

)) + ρ(f (τ

ⁿ

x

₁

), f(x)).

ε ε

1

G ρ(af (x), f(x)) = 0

E (ϕ) P (ϕ)

a )∈ E (ϕ) A µ(A) > 0 V e

A ∩ τ

⁻ⁿ

A ∩ { ϕ

n

∈ aV V

⁻¹

} = ∅ , ∀ n ∈ Z .

a τ

ϕ

B = ∪

ⁿ∈Z

τ

_ϕⁿ

(A × V )

h G %

h(g )m

G

(dg) = 1

G Y G

ρ(f

1

, f

2

) = %

X

inf( | f

1

− f

2

| , 1) h dm

G

X × G

X Y f X × G τ

ϕ

E (ϕ) f

E (ϕ) = G λ

χ

τ

ϕ

(17)

γ G P (ϕ) γ H

x

µ

χ

x ∈ X

P (ϕ) E (ϕ) H

(x, g) ∈ X × G c(x, g) = ( $

X

(

$

Hx

h(y, u

x

(y)γg)χ

x

(γ)dγ)µ

x

(dy) )

−1

.

γ ∈ P (ϕ) ⇔ M

^h

((x, γg), · ) = M

^h

((x, g), · ) λ

_χ

(x, g) ∈ X × G.

λ

χ

(x, g) ∈ X × G

c(x, g) µ

x

(dy) δ

_u_x_(y)

∗ (χ

x

m

Hx

) ∗ δ

g

= c(x, γg) µ

x

(dy) δ

_u_x_(y)

∗ (χ

x

m

Hx

) ∗ δ

γg

, µ

x

y ∈ X

c(x, g) δ

ux(y)

∗ (χ

x

m

Hx

) ∗ δ

γ

= c(x, γg) δ

ux(y)

∗ (χ

x

m

Hx

).

H

_x

γ = H

_x

µ

_χ

x ∈ X

ϕ ψ ϕ

^(u,µ)

∼ ψ f τ

_ϕ

f ˜ τ

ψ

f(x, g) = ˜ f(x, u(x)g )

G P (ϕ) = P (ψ)

G ϕ ˜ := ϕ E (ϕ) E ( ˜ ϕ) = { 0 } E ( ˜ ϕ) = { 0 }

ϕ µ

χ

E (ϕ)

E ( ˜ ϕ) = { 0 }

G/ E (ϕ) E ( ˜ ϕ) = { 0 } ϕ G = R E (ϕ) ) = { 0 }

ϕ ϕ

1

ϕ

2

e G

E (ϕ) = { e }

ϕ Z s )∈ Q

e

^2πisϕ

= ψ/ψ ◦ τ ψ ϕ

ϕ

(18)

τ

ϕ

λ τ

ϕ

λ(dy, dg) =

µ(dy)N (y, dg) µ X N

y ∈ X N (y, dg) G

B G y → N (y, B)

H G u X G

ϕ

u

(y) := (u(τ y))

⁻¹

ϕ(y) u(y) ∈ H µ y ∈ X

˜ λ λ (y, g) → (y, (u(y))

⁻¹

g) τ

_ϕ_u

X × H

λ(dy, dh) = ˜ ˜ µ(dy)χ(h) dh,

χ H µ ˜ σ µ

τ µ(dy) = ˜ χ(ϕ

u

(τ

⁻¹

y)) ˜ µ(dy).

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

⁻¹

y)) ˜ µ(dy)

λ

χ

• h X × G λ

χ

(h) = 1

(X × G,

X

×

^B^G

) h λ

χ

P

^h

h λ

χ

σ

τ

_ϕ J

M

^h

X × G

f X × G

∀ (x, g) ∈ X × G, M

^h

f (x, g) = P

^h

(f /h)(x, g).

λ

_χ

(dy, dt) =

$

X×G

M

^h

((x, g), (dy, dt)) h(x, g) λ

_χ

(dx, dg).

(19)

λ

χ

(x, g) ∈ X × G P

^h

((x, g), · ) τ

ϕ

∀ A ∈

^J

, P

^h

((x, g), A) = 0 1

τ

ϕ

P

^h

((x, g), (dy, dt)) = h ◦ τ

_ϕ⁻¹

(y, t)

h(y, t) P

^h

((x, g), (dy, dt)),

τ

ϕ

M

^h

((x, g), (dy, dt)) = M

^h

((x, g), (dy, dt)).

P

^h

((x, g), (dy, dt)) = ρ((x, g), dy) Q((x, g, y), dt),

ρ (X × G,

^X

⊗

^BG

) (X,

^X

) Q

(X × G × X,

X

⊗

^B^G

⊗

^X

) (G,

B

)

ν

(x,g)

(dy) := ρ((x, g), dy) N

(x,g)

(y, dt) := Q((x, g, y), dt).

(x, g) ∈ X × G ν

(x,g)

A ∈

X

, ν

_(x,g)

(A) = P

^h

((x, g), A × G) { N

_(x,g)

(y, · ) : y ∈ X }

ν

(x,g)

(X × G,

^X

×

^BG

, P

^h

((x, g), · )) U V X G ν

(x,g)

U

N

(x,g)

V U

M

^h

M

^h

((x, g), (dy, dt)) = ρ((x, g), dy) ˜ Q((x, g, y), dt) = ν

_(x,g)

(dy) ˜ N

_(x,g)

(y, dt),

Q((x, g, y ˜ ), dt) = ˜ N

(x,g)

(y, dt) = h(y, t)

⁻¹

N

(x,g)

(y, dt) (X × G × X,

^X

×

^BG

×

^X

) (G,

^BG

)

f µ

_χ

X K

G

$

X×G

-$

X

f(y) ˜ N

(x,g)

(y, K) ν

(x,g)

(dy) .

h(x, g)λ

χ

(dx, dg)

=

$

X×G

f (x)1

K

(g)λ

χ

(dx, dg) < + ∞ .

λ

χ

(x, g) ν

_(x,g)

y N ˜

_(x,g)

(y, K) < + ∞

(K

n

)

n≥0

G &

n∈N

K

n

= G λ

χ

(x, g) ν

(x,g)

y ∀ n ≥ 0, N ˜

(x,g)

(y, K

n

) < + ∞ N ˜

(x,g)

(y, · )

G

(20)

P

^h

λ

χ

(x, g) ∈ X × G N ˜

(x,g)

(y, · ) : y ∈ X }

ν

_(x,g)

(x, g) ∈ X × G M

^h

((x, g), · ) τ

_ϕ

y ∈ X N ˜

(x,g)

(y, · ) G

• τ

ϕ

M

^h

((x, g), · )

M

^h

((x, g), (dy, dγ)) = ˜ µ

_(x,g)

(dy) × [δ

_v_(x,g)_(y)

∗ (

χ

_(x,g)

(γ) m

_H_(x,g)

(dγ) ) ],

H

(x,g)

G χ

(x,g)

H

(x,g)

v

(x,g)

X G µ ˜

(x,g)

σ X

ν

_(x,g)

τ

ϕ

(˜ µ

(x,g)

)(dy) = χ(ϕ

v_(x,g)

(τ

⁻¹

y)) ˜ µ

(x,g)

(dy),

ϕ

v_(x,g)

(y) := (v

(x,g)

(τ y))

⁻¹

ϕ(y) v

(x,g)

(y) ∈ H

(x,g)

, f or µ ˜

(x,g)

− a.e. y ∈ X.

t ∈ G f X × G R

t

(f )(x, g) := f (x, g t)

t ∈ G f X × G

λ

χ

(x, g) ∈ X × G

M

^h

(R

t

(f ))(x, g) = P

^h

(R

t

h/h)(x, g) M

^h

(f )(x, g t).

c

(x,g),t

c

(x,g),t

= P

^h

(R

t

h/h)(x, g).

˜

µ

(x,g)

(dy) × [δ

v_(x,g)

(y) ∗ (

χ

(x,g)

(γ) m

H_(x,g)

(dγ) )

∗ δ

t

]

= c

(x,g),t

µ ˜

(x,gt)

(dy) × [δ

v_(x,gt)

(y) ∗ (

χ

(x,gt)

(γ) m

H_(x,gt)

(dγ) ) ].

σ

X

×

^B^G

λ

χ

(x, g) ∈ X × G m

G

t ∈ G

R

t

( M

^h

((x, g), · ) )

= P

^h

(R

t

h/h)(x, g) M

^h

((x, g t), · )

R

_g−1

( M

^h

((x, g), · ) )

= P

^h

(R

t

h/h)(x, g) R

_{(g t)}−1

( M

^h

((x, g t), · ) )

.

(21)

λ

χ

(x, g) ∈ X × G M

^h

((x, g), (dy, dt)) c(x, g)

˜

µ

x

(dy) [δ

vx(y)

∗ (χ

x

m ˜

Hx

) ∗ δ

g

](dt),

˜

m

Hx

H

x

m ˜

Hx

m

Hx

λ

χ

(x, g) ∈ X × G P

^h

(1)(x, g) = M

^h

(h)(x, g) = 1 (c(x, g))

⁻¹

=

$

X

( $

Hx

h(y, v

x

(y) γ g) χ

x

(γ ) ˜ m

Hx

(dγ) )

˜ µ

x

(dy)

λ

χ

(x, g) ∈ X × G f X × G

M

^h

(f )(x, g) =

%

X

( %

Hx

f (y, v

x

(y) γ g) χ

x

(γ) ˜ m

Hx

(dγ) )

˜ µ

x

(dy)

%

X

( %

Hx

h(y, v

x

(y) γ g) χ

x

(γ) ˜ m

Hx

(dγ ) )

˜ µ

x

(dy)

.

• M

^h

ν

_(x,g)

(dy) = P

^h

((x, g), dy × G) = c(x, g) ( $

Hx

h(y, v

x

(y)γg) χ

x

(γ) ˜ m

Hx

(dγ) )

˜ µ

x

(dy),

N ˜

(x,g)

(y, dt) = ( $

Hx

h(y, v

x

(y)γg) χ

x

(γ) ˜ m

Hx

(dγ) )

−1

(δ

vx(y)

∗ (χ

x

m ˜

Hx

) ∗ δ

g

)(dt).

v

x

(y)H

x

S(x, y) Q((x, e, y), .) = N

(x,e)

(y, · )

G H

_x

Q((x, e, y), .) , ∗ Q((x, e, y), .)

Q((x, e, y), .) , Q((x, e, y), .)

t +→ t

⁻¹

G x ∈ X +→ H

x

∈ F (G) (x, y) ∈ X × X +→

v

x

(y)H

x

∈ F (G)

F G

{ (x, y) ∈ X × X : v

x

(y)H

x

⊂ F } = { (x, e, y) : Q((x, e, y ), F

^c

) = 0 } . u : X × X +→ G

(x, y ) ∈ X × X u(x, y ) ∈ S(x, y ) v

x

(y)H

x

= u(x, y)H

x

f X × G

$

Hx

f(y, v

x

(y) γ g) χ

x

(γ) ˜ m

Hx

(dγ ) = χ

⁻_x¹

((u(x, y ))

⁻¹

v

x

(y))

$

Hx

f (y, u(x, y) γ g) χ

x

(γ) ˜ m

Hx

(dγ)

(22)

δ

(u(x,y))⁻¹vx(y)

∗ (χ

x

m ˜

Hx

) = χ

⁻¹_x

((u(x, y))

⁻¹

v

x

(y)) (χ

x

m ˜

Hx

),

R((x, g, y), dt) = δ

(u(x,y))⁻¹

∗ N /

x,g

(y, dt) ∗ δ

g⁻¹

X × X G ( $

Hx

h(y, u(x, y) γ g) χ

_x

(γ) ˜ m

_H_x

(dγ) )

₋1

χ

_x

(t) ˜ m

_H_x

(dt).

U G e

( $

Hx

h(y, u(x, y) γ g) χ

_x

(γ) ˜ m

_H_x

(dγ ) )

₋1

$

Hx∩U

χ

_x

(t) ˜ m

_H_x

(dt) = R((x, g, y), U) > 0 γ ∈ H

x

χ

x

(γ) = R((x, e, y), γ U ) R((x, e, y), U) ,

η : X × G +→ R

^∗+

µ

χ

x ∈ X ∀ γ ∈ H

x

, χ

x

(γ) = η(x, γ)

˜ m

Hx

(dt)

%

Hx∩U

χ

x

(γ) ˜ m

Hx

(dγ) = R((x, e, y), dt) R((x, e, y), t U )

X G

m

Hx

H

x

$

Hx∩U

χ

x

(γ ) m

Hx

(dγ) = 1.

M

^h

((x, g), dy, dt) = R((x, g, y ), U ) ν

_(x,g)

(dy) (

δ

_u(x,y)

∗ (χ

x

m

Hx

) ∗ δ

g

) (dt)

R((x, g, y), U) ν

(x,g)

(dy) = c(x, g) χ

x

((u(x, y))

⁻¹

v

x

(y)) (

$

Hx∩U

χ

x

(t) ˜ m

Hx

(dt)) ˜ µ

x

(dy).

χ

x

((u(x, y))

⁻¹

v

x

(y)) ˜ µ

x

(dy) = d(x) µ

x

(dy) (d(x))

⁻¹

= c(x, e) (

$

Hx∩U

χ

x

(t) ˜ m

Hx

(dt))

µ

x

(dy) = R((x, e, y), U ) ν

(x,e)

(dy).

(µ

x

(dy))

x∈X

(X,

^X

)

(23)

M

^h

(f )(x, g) =

%

X

( %

Hx

f (y, u(x, y) γ g) χ

_x

(γ) m

_H_x

(dγ ) )

µ

_x

(dy)

%

X

( %

Hx

h(y, u(x, y) γ g) χ

x

(γ) m

Hx

(dγ) )

µ

x

(dy) .

(x, g) ∈ X × G M

^h

((x, g), · )

• τ

ϕ

(M

^h

((x, g), (dy, dt)) = M

^h

((x, g), (dy, dt)) (τ µ

x

)(dy) (

δ

ϕ(τ⁻¹y)

∗ δ

ux(τ⁻¹y)

∗ (χ

x

m

Hx

) ∗ δ

g

)(dt) = µ

x

(dy) (

δ

ux(y)

∗ (χ

x

m

Hx

) ∗ δ

g

) (dt),

ϕ(τ

⁻¹

y) u

x

(τ

⁻¹

y) H

x

= u

x

(y)H

x

, µ

x

x ∈ X (τ µ

x

)(dy) = χ

x

((u

x

(y))

⁻¹

ϕ(τ

⁻¹

y) u

x

(τ

⁻¹

y)) µ

x

(dy);

M

^h

((x, g), · ) = M

^h

(τ

ϕ

(x, g), · ) ν

(x,g)

= ν

τϕ(x,g)

ν

(x,g)

y ∈ X N ˜

(x,g)

(y, · ) = ˜ N

τϕ(x,g)

(y, · ) N ˜

_(x,g)

(y, · ) = ˜ N

_τ_ϕ_(x,g)

(y, · )

u

x

(y)H

x

= u

τ x

(y) H

τ x

ϕ(x)

ζ

x

(y) = (u

x

(y))

⁻¹

u

τ x

(y) ϕ(x) ∈ H

x

, H

τ x

= ϕ(x) H

x

(ϕ(x))

⁻¹

χ

τ x

(ϕ(x) ζ

x

(y) (ϕ(x))

⁻¹

) δ

uτ x

(y) ∗ (χ

τ x

m

Hτ x

) ∗ δ

ϕ(x)

= δ

ux(y)

∗ (χ

τ x

(ϕ(x) · (ϕ(x))

⁻¹

) m

Hx

,

m

Hx

= δ

ϕ(x)

∗ m

Hτ x

∗ δ

(

^ϕ(x)

)

⁻¹

H

x

,

m

Hx

= d(x) m

Hx

d(x) x

γ ∈ H

x

χ

x

(γ) = χ

τ x

(ϕ(x) γ (ϕ(x))

⁻¹

)

χ

x

(ζ

x

(y))

$

Hτ x

h(y, u

τ x

γϕ(x)g) χ

τ x

(γ) dγ = d(x)

$

Hx

h(y, u

x

(y)γg) χ

x

(γ) dγ.

(24)

ν

(x,g)

= ν

τϕ(x,g)

˜

µ

τ x

(dy) = c(x) χ

x

(ζ

x

(y)) ˜ µ

x

(dy)

c(x) x

ϕ

ux

H

x

σ µ

x

H

x

• H

x

H

a : X → G H

x

= a(x)H(a(x))

⁻¹

x ∈ X a(x) H

ψ(x) := a(τ x)

⁻¹

ϕ(x) a(x) H

(a(x))

⁻¹

(u

x

(y))

⁻¹

u

τ x

(y) ϕ(x) a(x) ∈ H.

f M

^h

f (x, g) =

%

X

( %

H

f (y, u

x

(y) a(x) γ a(x)

⁻¹

g) χ

x

(a(x) γa(x)

⁻¹

) dγ) µ

x

(dy)

%

X

( %

H

h(y, u

x

(y) a(x) γ a(x)

⁻¹

g) χ

x

(a(x)γa(x)

⁻¹

) dγ) µ

x

(dy) . χ

τ x

(a(τ x) γ a(τ x)

⁻¹

) = χ

x

(a(x)(ψ(x))

⁻¹

γψ(x)a(x)

⁻¹

)

˜

χ

x

(γ) := χ

x

(a(x) γ (a(x))

⁻¹

) χ ˜

τ x

(γ) = ˜ χ

x

((ψ(x))

⁻¹

γψ(x))

G H

τ(x)

= H

x

µ

χ

x ∈ X x ∈ X → H

x

∈

F (G)

H G H

x

= H µ

χ

x ∈ X

γ ∈ H λ

χ

(R

γ

(f)) = χ

⁻¹

(γ) λ

χ

(f ) λ

χ

(x, g) ∈ X × G M

^h

R

γ

(f)(x, g) = χ

⁻¹_x

(γ) M

^h

f(x, g) f = h

∀ γ ∈ H, χ(γ) =

$

X×G

χ

x

(γ) h(x, g) λ

χ

(dx, dg) χ

x

= χ µ

χ

x ∈ X

λ

χ

f

M

^h

f (x, g) =

%

X

( %

H

f(y, u

x

(y) γ g) χ(γ ) dγ) µ

x

(dy)

%

X

( %

H

h(y, u

x

(y) γ g) χ(γ) dγ) µ

x

(dy) .

(25)

ϕ H

1

G µ

1

⊗ m

H1

τ

ϕ

µ

1

⊗ m

H1

τ

ϕ

ϕ µ

1

ψ

H

₂

G u

g

0

∈ G µ

1

x ∈ X

u(x) H

₂

= g

₀

H

₂

H

₁

⊂ u(x) H

₂

(u(x))

⁻¹

= g

₀

H

₂

g

₀⁻¹

.

τ

ψ

µ

2

⊗ m

H2

µ

2

∼ µ

1

τ

ϕ

g

0

∈ G

H

₁

u(x) = H

₁

g

₀

, u(x) H

₂

= g

₀

H

₂

g

⁻₀¹

H

₁

g

₀

= H

₂

. µ

1

µ

2

χ

1

◦ τ

ϕ

χ

2

◦ τ

ψ

χ

1

H

1

χ

2

H

2

µ

1

x ∈ X γ ∈ H

1

χ

₁

(γ) = χ

₂

(g

₀⁻¹

γ g

0

) µ

₁

x ∈ X χ

₁

(u(x) g

₀⁻¹

) = χ

₂

(g

⁻¹₀

u(x))

µ

2

(dx) = χ

₁

(u(x) g

₀⁻¹

) µ

1

(dx)

τ

_ϕ

µ

₂

⊗ (δ

_u(x)

∗ (χ

₂

m

_H₂

)) µ

₁

⊗ ((χ

₁

m

_H₁

) ∗ δ

_g₀

)

H

2

F G g ∈ G

f

^g

(x, t) = F ((u(x))

⁻¹

t g) τ

ϕ

µ

1

⊗ m

H1

F µ

1

x ∈ X g ∈ G t ∈ H

1

−→ F ((u(x))

⁻¹

t u(x) g )

F (g) t = e (u(x))

⁻¹

H

1

u(x) ⊂ H

2

ϕ ψ H

1

H

2

µ

1

x ∈ X (u(τ x))

⁻¹

u(x) ∈ H

2

u(τ x) H

2

= u(x) H

2

(µ

1

, τ) g

0

∈ G µ

1

x ∈ X u(x) H

2

= g

0

H

2

ψ µ

₂

ϕ x ∈ X → (u(x))

⁻¹

∈ G