❯❋❋❊ ❍❆❆●❊❘❯P❙ ❚❍■❘❉ ❚❆▲❑ ❆❚ ▼❆❙❚❊❘❈▲❆❙❙ ❖◆ ❱❖◆ ◆❊❯▼❆◆◆ ❆▲●❊❇❘❆❙ ❆◆❉ ●❘❖❯P ❆❈❚■❖◆❙ ✶✳ ❈r♦ss❡❞ ♣r♦❞✉❝ts ■❢ A ⊆ B ( H ) ✐s ❛ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛ ❛♥❞ G ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ❛♥❞ α : G → ❆✉t ( A ) ✐s ❛ ❛❝t✐♦♥ ♦❢ G ♦♥ A ✱ ❧❡t ( π ( a ) ξ )( g ) = α − 1 ∀ ξ ∈ l 2 ( G, H ) , a ∈ A ✭✶✳✶✮ g ( a ) ξ ( g ) , ❛♥❞ ( λ ( g ) ξ )( h ) = ξ ( h − 1 g ) , ∀ ξ ∈ l 2 ( G, H ) , g ∈ G ✭✶✳✷✮ ❉❡✜♥❡ M := A ⋊ G := ( π ( A ) ∪ λ ( G )) ′′ ✭✶✳✸✮ t❤❡♥ λ ( g ) π ( a ) λ ( g ) ∗ = π ( α g ( a )) ✭✶✳✹✮ ❛♥❞ ✐❢ ✇❡ ✐❞❡♥t✐❢② a ✇✐t❤ π ( a ) ∈ M ✱ ✇❡ ❣❡t M := ( A ∪ λ ( G )) ′′ ✭✶✳✺✮ ❛♥❞ ✭✶✳✻✮ λ ( g ) aλ ( g ) ∗ = α g ( a ) . ✷✳ ●r♦✉♣ ♠❡❛s✉r❡ s♣❛❝❡ ❝♦♥str✉❝t✐♦♥ ❆s ❛ s♣❡❝✐❛❧ ❝❛s❡ t❛❦❡ A = L ∞ ( X, µ ) ✇❤❡r❡ X ✐s ❛ st❛♥❞❛r❞ ❇♦r❡❧ s♣❛❝❡ ❛♥❞ µ ✐s ❛ σ ✲✜♥✐t❡ ♠❡❛s✉r❡✱ ❛♥❞ ❧❡t α g ( f ) = f ( σ − 1 g x ) ❢♦r ❛♥ ❛❝t✐♦♥ σ : g → ❆✉t ( X, [ µ ]) ✱ t❤❡ ❇♦r❡❧ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ X ♣r❡s❡r✈✐♥❣ t❤❡ ♠❡❛s✉r❡ ❝❧❛ss✳ ❉❡✜♥✐t✐♦♥ ✶✳ σ ✐s ❛♥ ❡r❣♦❞✐❝ ❛❝t✐♦♥ ✐✛ ❢♦r ❡✈❡r② G ✲✐♥✈❛r✐❛♥t ❇♦r❡❧s❡t B ⊆ X ❡✐t❤❡r µ ( B ) = 0 ♦r µ ( X \ B ) = 0 ✳ ❉❡✜♥✐t✐♦♥ ✷✳ σ ✐s ❢r❡❡ ✐❢ ❢♦r µ ✲❛❧♠♦st ❛❧❧ x ∈ X g → gx ✐s ❛ ✶✲t♦✲✶✲♠❛♣ ❢r♦♠ G t♦ X ✳ ❚❤❡♦r❡♠ ✸ ✭▼✉rr❛② ✰ ✈♦♥ ◆❡✉♠❛♥♥✱ ≈ ✶✾✹✵✮ ✳ ■❢ σ ✐s ❢r❡❡ ❛♥❞ ❡r❣♦❞✐❝ t❤❡♥ M = L ∞ ( X, µ ) ⋊ α G ✐s ❛ ❢❛❝t♦r ❛♥❞ A = L ∞ ( X, µ ) ✐s ❛ ▼❆❙❆ ✭♠❛①✐♠❛❧ ❛❜❡❧✐❛♥ s❡❧❢❛❞❥♦✐♥t s✉❜❛❧❣❡❜r❛✮ ✐♥ M ✳ ❚❤❡♦r❡♠ ✹ ✭▼✉rr❛② ✰ ✈♦♥ ◆❡✉♠❛♥♥✱ ≈ ✶✾✹✵✮ ✳ ❆ss✉♠❡ ❛ ❢r❡❡ ❛♥❞ ❡r❣♦❞✐❝ ❛❝t✐♦♥ • M ✐s ❛ I ∞ ✲❢❛❝t♦r ✐✛ Ω ✐s ✐♥✜♥✐t❡ ❜✉t ❝♦✉♥t❛❜❧❡ • M ✐s ❛ II 1 ✲❢❛❝t♦r ✐✛ Ω ✐s ✉♥❝♦✉♥t❛❜❧❡ ❛♥❞ t❤❡r❡ ❡①✐st ❛ G ✲✐♥✈❛r✐❛♥t ✜♥✐t❡ ♠❡❛s✉r❡ ν ∈ [ µ ] • M ✐s ❛ II ∞ ✲❢❛❝t♦r ✐✛ Ω ✐s ✉♥❝♦✉♥t❛❜❧❡ ❛♥❞ t❤❡r❡ ❡①✐sts ❛ G ✲✐♥✈❛r✐❛♥t σ ✲ ✜♥✐t❡✱ ❜✉t ♥♦t ✜♥✐t❡ ♠❡❛s✉r❡ ν ∈ [ µ ] ❉❛t❡ ✿ ✷✼✴✵✶✴✷✵✶✵✳ ✶
✷ ❯❋❋❊ ❍❆❆●❊❘❯P❙ ❚❍■❘❉ ❚❆▲❑ ❆❚ ▼❆❙❚❊❘❈▲❆❙❙ ❖◆ ❱❖◆ ◆❊❯▼❆◆◆ ❆▲●❊❇❘❆❙ ❆◆❉ ●❘❖❯P ❆❈❚■❖◆❙ • M ✐s ❛ III ✲❢❛❝t♦r ✐✛ Ω ✐s ✉♥❝♦✉♥t❛❜❧❡ ❛♥❞ t❤❡r❡ ❞♦❡s ♥♦t ❡①✐st ❛ G ✲✐♥✈❛r✐❛♥t σ ✲✜♥✐t❡ ♠❡❛s✉r❡ ν ∈ [ µ ] ❋r♦♠ ♥♦✇ ♦♥ ✇❡ ❧♦♦❦ ❛t t❤❡ II 1 ✲❢❛❝t♦r ❝❛s❡ ♦♥❧②✳ ▲❡t ( X, µ ) ❜❡ ❛ ✉♥❝♦✉♥t❛❜❧❡ st❛♥❞❛r❞ ❇♦r❡❧ s♣❛❝❡ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡✱ ❛♥❞ ❧❡t σ : G → ❆✉t ( X, µ ) ❜❡ ❛ ❇♦r❡❧ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ X ✇❤✐❝❤ ❧❡❛✈❡s µ ✐♥✈❛r✐❛♥t✳ ◆♦✇ ❞❡✜♥❡ M = L ∞ ( X, µ ) ⋊ σ G ✳ ❚❤❡♥ A = L ∞ ( X, µ ) ✐s ❛ ❈❛rt❛♥ ▼❆❙❆ ✐♥ M ✱ ✇❤❡r❡ ❈❛rt❛♥ ♠❡❛♥s t❤❛t ❢♦r N ( A ) = { u ∈ U ( M ) | u A a ∗ = A} ❤♦❧❞s N ( A ) ′′ = M ✳ ❚❤❡♦r❡♠ ✺ ✭❱♦✐❝✉❧❡s❝✉ ≈ ✶✾✾✺✮ ✳ ❋♦r 2 ≤ n < ∞ ✱ L ( F n ) ❤❛s ♥♦ ❈❛rt❛♥ ▼❆❙❆✳ ❍❡♥❝❡ L ( F n ) ❝❛♥♥♦t ❜❡ ♦❜t❛✐♥❡❞ ❜② t❤❡ ❣r♦✉♣ ♠❡❛s✉r❡ s♣❛❝❡ ❝♦♥str✉❝t✐♦♥✳ ✸✳ ❊q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ❆ss✉♠❡ ( A 1 ⊆ M 1 ) ❛♥❞ ( A 2 ⊆ M 2 ) ❜♦t❤ ♦❜t❛✐♥❡❞ ❜❡ ❣r♦✉♣ ♠❡❛✉s✉r❡ s♣❛❝❡ ❝♦♥str✉❝t✐♦♥ ❢r♦♠ ( X i , µ i , Γ i , σ i ) ✇❤❡r❡ t❤❡ ❛❝t✐♦♥s ❛r❡ ❢r❡❡ ❛♥❞ ❡r❣♦❞✐❝✳ ❲❡ ✇r✐t❡ ( A 1 ⊆ M 1 ) ∼ = ( A 2 ⊆ M 2 ) ✐✛ t❤❡r❡ ❡①✐sts ❛ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛ ✐s♦♠♦r♣❤✐s♠ θ : M 1 → M 2 ✇✐t❤ θ A 1 = A 2 ✳ ❉❡✜♥✐t✐♦♥ ✻✳ (Γ 1 , σ 1 ) ✐s ♦r❜✐t ❡q✉✐✈❛❧❡♥t ✭❖❊✮ t♦ (Γ 2 , σ 2 ) ✐❢ t❤❡r❡ ❡①✐st ♥✉❧❧s❡ts N 1 ⊆ X i ❛♥❞ ❛ ❇♦r❡❧ ✐s♦♠♦r♣❤✐s♠ χ : X 1 \ N 1 → A 2 \ N 2 ♠❛♣♣✐♥❣ σ 1 ✲♦r❜✐ts ♦♥t♦ σ 2 ✲♦r❜✐ts✳ ❚❤❡♦r❡♠ ✼ ✭❙✐♥❣❡r ✶✾✺✺✮ ✳ ( A 1 ⊆ M 1 ) ∼ = ( A 2 ⊆ M 2 ) ✐✛ (Γ 1 , σ 1 ) ✐s ❖❊ t♦ (Γ 2 , σ 2 ) ✳ ❉❡✜♥✐t✐♦♥ ✽✳ ▲❡t α : Γ → ❆✉t ( X, µ ) ❛♥❞ x, y ∈ X t❤❡♥ x ∼ α y ✐✛ x ❛♥❞ y ❛r❡ ✐♥ t❤❡ s❛♠❡ Γ ✲♦r❜✐t✱ t❤❛t ✐s y = α γ ( x ) ❢♦r s♦♠❡ γ ∈ Γ ✳ ❚❤❡♦r❡♠ ✾ ✭❉②❡✱ ✶✾✺✾✮ ✳ ❆♥② t✇♦ ❡r❣♦❞✐❝ ❛❝t✐♦♥s ♦❢ Z ♦♥ ( X, µ ) ❛r❡ ❖❊✳ ❚❤❡♦r❡♠ ✶✵ ✭❖r♥st❡✐♥ ✰ ❲❡✐ss✮ ✳ ■❢ Γ 1 , Γ 2 ❛r❡ ❛♠❡♥❞❛❜❧❡ ❣r♦✉♣s ❛❝t✐♥❣ ❡r❣♦❞✐❝❧② ♦♥ ( X i , µ i ) t❤❡♥ t❤❡ ❛❝t✐♦♥s ❛r❡ ❖❊✳ ❚❤❡♦r❡♠ ✶✶ ✭❈♦♥♥❡s ✰ ❲❡✐ss✮ ✳ ■❢ Γ ❤❛s ♣r♦♣❡rt② ❚ t❤❡♥ ✐t ❤❛s ❛t ❧❡❛st t✇♦ ♥♦♥✲❖❊ ❡r❣♦❞✐❝ ❛❝t✐♦♥s ♦♥ ( X i , µ i ) ✳ ❚❤❡♦r❡♠ ✶✷ ✭❋✉r♠❛♥♥✱ ✶✾✽✽✮ ✳ ■❢ Γ ❤❛s ❛♥ ❛❝t✐♦♥ ♦♥ ( X, µ ) ✇❤✐❝❤ ✐s ❖❊ t♦ t❤❡ st❛♥❞❛r❞ ❛❝t✐♦♥ ♦❢ SL (2 , Z ) ♦♥ T n = R n / Z n ❢♦r n ≥ 3 t❤❡♥ Γ ∼ = SL ( n, Z ) ✳ ❉❡✜♥✐t✐♦♥ ✶✸✳ ▲❡t Y ⊆ X ❜❡ ❇♦r❡❧ s♣❛❝❡s ❛♥❞ α ❛ ❢r❡❡ ❛♥❞ ❡r❣♦❞✐❝ ❛❝t✐♦♥ ♦❢ G ♦♥ t❤❡s❡✳ ❲❡ ❞❡✜♥❡ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ R α ❜② x ∼ α y ✐❢ x, y ❛r❡ ✐♥ t❤❡ s❛♠❡ G ✲♦r❜✐t✱ ❛♥❞ R α ⊆ X × X ❜② R α = { ( x, y ) | x ∼ a lphay } ❛♥❞ R α | Y = R α ∩ Y × Y ✳ ❖❜s❡r✈❛t✐♦♥ ✶✹✳ ■❢ G ✐s ❢r❡❡ ❛♥❞ ❡r❣♦❞✐❝ ❛❝t✐♥❣ ♦♥ ( X, µ ) ✱ ❛♥❞ Y, Z ⊆ X ❛r❡ ❇♦r❡❧✱ ✇✐t❤ µ ( Y ) = µ ( Z ) > 0 t❤❡♥ R α | Y ∼ OE R α | Z ✳ ❈♦r♦❧❧❛r② ✶✺✳ ▲❡t A ⊆ M ❝♦♠❡ ❢r♦♠ ( X, µ, G, α ) ✇✐t❤ α ❢r❡❡ ❛♥❞ ❡r❣♦❞✐❝✳ ▲❡t p, q ∈ P ( A ) s✉❝❤ t❤❛t τ ( p ) = τ ( q ) t❤❡♥ ( p A ⊆ p M p ) ∼ = ( q A ⊆ q M q ) ✳ ✹✳ ❋✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣ ♦♣ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ✶✻✳ ▲❡t A ⊆ M ❛♥❞ ❧❡t t ∈ ]0 , 1] ✱ ❝❤♦♦s❡ pt ∈ P ( A ) s✉❝❤ t❤❛t τ ( p ) = t ✳ ❉❡✜♥❡ ( A t ⊆ M t ) ❜② t❤❡ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss ♦❢ ( p t A ⊆ p t M p t ) ✳ ❋♦r t > 0 ❛r❜✐tr❛r② ❞❡✜♥❡ ( A t ⊆ M t ) ❛s t❤❡ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss ♦❢ ( A ⊗ D n ( C )) t/n ⊆ ( M ⊗ M n ( C )) t/n ✇❤❡r❡ D n ⊆ M n ❛r❡ t❤❡ ❞✐❛❣♦♥❛❧ ♠❛tr✐❝❡s ❛♥❞ n ≥ t ✳ ■t ❤❛s t♦ ❜❡ ♣r♦✈❡♥ t❤❛t t❤❡ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ n ✳
Recommend
More recommend
