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❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s ■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❋✉❞❛♥ ❯♥✐✈❡rs✐t② ❙❤❛♥❣❤❛✐✱ ❈❤✐♥❛ ✐❧✐♠♦♥❝❤❡♥❦♦❅❣♠❛✐❧✳❝♦♠ ❚❤❡ ✹✸r❞ ❙②♠♣♦s✐✉♠ ♦♥ ❚r❛♥s❢♦r♠❛t✐♦♥ ●r♦✉♣s ◆♦✈❡♠❜❡r ✶✽✱ ✷✵✶✻ ❍✐♠❡❥✐✱ ❏❛♣❛♥ ■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s

❆ ♣♦❧②t♦♣❡ ✐♥ ✐s ❝❛❧❧❡❞ ❉❡❧③❛♥t ✐❢ ✐ts ♥♦r♠❛❧ ❢❛♥ ✐s s♠♦♦t❤✳ ❚❤❡♦r❡♠ ✭❚✳❉❡❧③❛♥t✬✽✽✮ ❚❤❡r❡ ✐s ❛ ✶✲✶ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ❝♦♠♣❛❝t s②♠♣❧❡❝t✐❝ t♦r✐❝ ♠❛♥✐❢♦❧❞s ✭✉♣ t♦ ❡q✉✐✈❛r✐❛♥t s②♠♣❧❡❝t♦♠♦r♣❤✐s♠✮ ❛♥❞ ❉❡❧③❛♥t ♣♦❧②t♦♣❡s ✭✉♣ t♦ ❧❛tt✐❝❡ ✐s♦♠♦r♣❤✐s♠✮✳ ❚♦r✐❝ ✈❛r✐❡t✐❡s ❚❤❡♦r❡♠ ✭▼✳❆t✐②❛❤❀ ❱✳●✉✐❧❧❡♠✐♥✱ ❙✳❙t❡r♥❜❡r❣✬✽✷✮ ▲❡t ( M , ω ) ❜❡ ❛ ✷ d ✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❛❝t ❝♦♥♥❡❝t❡❞ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞ ✇✐t❤ ❛ ❤❛♠✐❧t♦♥✐❛♥ ❛❝t✐♦♥ ♦❢ ❛ ❝♦♠♣❛❝t t♦r✉s T n ✳ ❚❤❡♥ t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ♠♦♠❡♥t ♠❛♣ µ : M � R n ✐s ❛ ❝♦♥✈❡① ♣♦❧②t♦♣❡ P ✇❤✐❝❤ ✐s t❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ µ ( M T ) ✳ ■❢ d = n ❛♥❞ t❤❡ t♦r✉s ❛❝t✐♦♥ ✐s ❡✛❡❝t✐✈❡✱ t❤❡♥ ( M , ω ) ✐s ❛ s②♠♣❧❡❝t✐❝ t♦r✐❝ ♠❛♥✐❢♦❧❞✳ ■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s

❚♦r✐❝ ✈❛r✐❡t✐❡s ❚❤❡♦r❡♠ ✭▼✳❆t✐②❛❤❀ ❱✳●✉✐❧❧❡♠✐♥✱ ❙✳❙t❡r♥❜❡r❣✬✽✷✮ ▲❡t ( M , ω ) ❜❡ ❛ ✷ d ✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❛❝t ❝♦♥♥❡❝t❡❞ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞ ✇✐t❤ ❛ ❤❛♠✐❧t♦♥✐❛♥ ❛❝t✐♦♥ ♦❢ ❛ ❝♦♠♣❛❝t t♦r✉s T n ✳ ❚❤❡♥ t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ♠♦♠❡♥t ♠❛♣ µ : M � R n ✐s ❛ ❝♦♥✈❡① ♣♦❧②t♦♣❡ P ✇❤✐❝❤ ✐s t❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ µ ( M T ) ✳ ■❢ d = n ❛♥❞ t❤❡ t♦r✉s ❛❝t✐♦♥ ✐s ❡✛❡❝t✐✈❡✱ t❤❡♥ ( M , ω ) ✐s ❛ s②♠♣❧❡❝t✐❝ t♦r✐❝ ♠❛♥✐❢♦❧❞✳ ❆ ♣♦❧②t♦♣❡ P ✐♥ R n ✐s ❝❛❧❧❡❞ ❉❡❧③❛♥t ✐❢ ✐ts ♥♦r♠❛❧ ❢❛♥ ✐s s♠♦♦t❤✳ ❚❤❡♦r❡♠ ✭❚✳❉❡❧③❛♥t✬✽✽✮ ❚❤❡r❡ ✐s ❛ ✶✲✶ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ❝♦♠♣❛❝t s②♠♣❧❡❝t✐❝ t♦r✐❝ ♠❛♥✐❢♦❧❞s ( M , ω, µ ) ✭✉♣ t♦ ❡q✉✐✈❛r✐❛♥t s②♠♣❧❡❝t♦♠♦r♣❤✐s♠✮ ❛♥❞ ❉❡❧③❛♥t ♣♦❧②t♦♣❡s µ ( M ) ✭✉♣ t♦ ❧❛tt✐❝❡ ✐s♦♠♦r♣❤✐s♠✮✳ ■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s

❘❡♠❛r❦s ✭❛✮ ✐s ❤♦♠❡♦♠♦r♣❤✐❝✱ ❛s ❛ ♠❛♥✐❢♦❧❞ ✇✐t❤ ❝♦r♥❡rs✱ t♦ t❤❡ s✐♠♣❧❡ ♣♦❧②t♦♣❡ ❀ ✭❜✮ ❚❤❡ ❛❝t✐♦♥ ✐s ❢r❡❡ ♦✈❡r t❤❡ ✐♥t❡r✐♦r ♦❢ ✱ t❤❡ ✈❡rt✐❝❡s ♦❢ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ✜①❡❞ ♣♦✐♥ts ♦❢ t❤❡ t♦r✉s ❛❝t✐♦♥ ♦♥ ❀ ✭❝✮ ❆ ♣r♦❥❡❝t✐✈❡ t♦r✐❝ ♠❛♥✐❢♦❧❞ ✐s ❛ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞✳ ◗✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞s ❉❡✜♥✐t✐♦♥ ▲❡t P ❜❡ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ s✐♠♣❧❡ ♣♦❧②t♦♣❡ ♦❢ ❞✐♠❡♥s✐♦♥ n ✳ ❆ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞ ♦✈❡r P ✐s ❛ s♠♦♦t❤ ✷ n ✲❞✐♠❡♥s✐♦♥❛❧ ♠❛♥✐❢♦❧❞ M ✇✐t❤ ❛ s♠♦♦t❤ ❛❝t✐♦♥ ♦❢ t❤❡ t♦r✉s T n s❛t✐s❢②✐♥❣ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s✿ ✭✶✮ t❤❡ ❛❝t✐♦♥ ✐s ❧♦❝❛❧❧② st❛♥❞❛r❞❀ ✭✷✮ t❤❡r❡ ✐s ❛ ❝♦♥t✐♥✉♦✉s ♣r♦❥❡❝t✐♦♥ π : M � P ✇❤♦s❡ ✜❜❡rs ❛r❡ T n ✲♦r❜✐ts✳ ■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s

◗✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞s ❉❡✜♥✐t✐♦♥ ▲❡t P ❜❡ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ s✐♠♣❧❡ ♣♦❧②t♦♣❡ ♦❢ ❞✐♠❡♥s✐♦♥ n ✳ ❆ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞ ♦✈❡r P ✐s ❛ s♠♦♦t❤ ✷ n ✲❞✐♠❡♥s✐♦♥❛❧ ♠❛♥✐❢♦❧❞ M ✇✐t❤ ❛ s♠♦♦t❤ ❛❝t✐♦♥ ♦❢ t❤❡ t♦r✉s T n s❛t✐s❢②✐♥❣ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s✿ ✭✶✮ t❤❡ ❛❝t✐♦♥ ✐s ❧♦❝❛❧❧② st❛♥❞❛r❞❀ ✭✷✮ t❤❡r❡ ✐s ❛ ❝♦♥t✐♥✉♦✉s ♣r♦❥❡❝t✐♦♥ π : M � P ✇❤♦s❡ ✜❜❡rs ❛r❡ T n ✲♦r❜✐ts✳ ❘❡♠❛r❦s ✭❛✮ M / T ✐s ❤♦♠❡♦♠♦r♣❤✐❝✱ ❛s ❛ ♠❛♥✐❢♦❧❞ ✇✐t❤ ❝♦r♥❡rs✱ t♦ t❤❡ s✐♠♣❧❡ ♣♦❧②t♦♣❡ P ❀ ✭❜✮ ❚❤❡ ❛❝t✐♦♥ ✐s ❢r❡❡ ♦✈❡r t❤❡ ✐♥t❡r✐♦r ♦❢ P ✱ t❤❡ ✈❡rt✐❝❡s ♦❢ P ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ✜①❡❞ ♣♦✐♥ts ♦❢ t❤❡ t♦r✉s ❛❝t✐♦♥ ♦♥ M ❀ ✭❝✮ ❆ ♣r♦❥❡❝t✐✈❡ t♦r✐❝ ♠❛♥✐❢♦❧❞ ✐s ❛ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞✳ ■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s

❆ ♠♦♠❡♥t✲❛♥❣❧❡ ♠❛♥✐❢♦❧❞ Z P ■♥ t❤❡ ✇♦r❦ ♦❢ ▼✳❉❛✈✐s ❛♥❞ ❚✳❏❛♥✉s③❦✐❡✇✐❝③✬✾✶ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥str✉❝t✐♦♥ ❛♣♣❡❛r❡❞✳ ❉❡✜♥✐t✐♦♥ ❙✉♣♣♦s❡ P n ✐s ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ s✐♠♣❧❡ ♣♦❧②t♦♣❡ ✇✐t❤ ❢❛❝❡ts F ✶ , . . . , F m ✳ ❉❡♥♦t❡ ❜② T F i ❛ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♦r❞✐♥❛t❡ s✉❜❣r♦✉♣ ✐♥ = T m ❢♦r ❡❛❝❤ ✶ ≤ i ≤ m ❛♥❞ T G = � T F i ⊂ T F ❢♦r ❛ ❢❛❝❡ T F ∼ G = ∩ F i ♦❢ ❛ ♣♦❧②t♦♣❡ P n ✳ ❚❤❡♥ t❤❡ ♠♦♠❡♥t✲❛♥❣❧❡ ♠❛♥✐❢♦❧❞ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ P ✐s ❛ q✉♦t✐❡♥t s♣❛❝❡ Z P = T F × P n / ∼ , ✇❤❡r❡ ( t ✶ , p ) ∼ ( t ✷ , q ) ✐✛ p = q ∈ P ❛♥❞ t ✶ t − ✶ ∈ T G ( p ) ✱ G ( p ) ✐s ❛ ✷ ♠✐♥✐♠❛❧ ❢❛❝❡ ♦❢ P ✇❤✐❝❤ ❝♦♥t❛✐♥s p = q ✳ ■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s

❙✉❝❤ ❛ ♣♦❧②t♦♣❡ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s ❛ ❜♦✉♥❞❡❞ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❤❛❧❢s♣❛❝❡s✿ ① ❛ ① ✵ ❢♦r ✶ ✇❤❡r❡ ❛ ✱ ✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❤②♣❡r♣❧❛♥❡s ❞❡✜♥❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s ❛ ① ✵ ❛r❡ ✐♥ ❣❡♥❡r❛❧ ♣♦s✐t✐♦♥✱ t❤❛t ✐s✱ ❛t ♠♦st ♦❢ t❤❡♠ ♠❡❡t ❛t ❛ s✐♥❣❧❡ ♣♦✐♥t✳ ❲❡ ❛❧s♦ ❛ss✉♠❡ t❤❛t t❤❡r❡ ❛r❡ ♥♦ r❡❞✉♥❞❛♥t ✐♥❡q✉❛❧✐t✐❡s ✐♥ ✱ t❤❛t ✐s✱ ♥♦ ✐♥❡q✉❛❧✐t② ❝❛♥ ❜❡ r❡♠♦✈❡❞ ❢r♦♠ ✇✐t❤♦✉t ❝❤❛♥❣✐♥❣ ✳ ❆ ♠♦♠❡♥t✲❛♥❣❧❡ ♠❛♥✐❢♦❧❞ Z P ❙✐♠♣❧❡ ♣♦❧②t♦♣❡s ◆♦✇ ❝♦♥s✐❞❡r s✐♠♣❧❡ ❝♦♥✈❡① n ✲❞✐♠❡♥s✐♦♥❛❧ ♣♦❧②t♦♣❡s P ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R n ✇✐t❤ s❝❛❧❛r ♣r♦❞✉❝t � , � ✳ ■✈❛♥ ▲✐♠♦♥❝❤❡♥❦♦ ❖♥ t♦♣♦❧♦❣② ♦❢ t♦r✐❝ s♣❛❝❡s ❛r✐s✐♥❣ ❢r♦♠ ✷✲tr✉♥❝❛t❡❞ ❝✉❜❡s