❊♥❞♦♠♦r♣❤✐s♠s ❛♥❞ ❚r❛♥s❢❡r ❖♣❡r❛♦rs ❊♥❞♦♠♦r♣❤✐s♠s ✈s✳ ❚r❛♥s❢❡r ❖♣❡r❛t♦rs✿ ❲❤❛t✬s P✳❙✳ ▼✉❤❧② t❤❡ ❞✐✛❡r❡♥❝❡❄ ❚❤❡ ❙❡tt✐♥❣ ❈ ∗ ✲ ❝♦rr❡s♣♦♥❞❡♥❝❡s ❛♥❞ t❤❡✐r ❈✉♥t③✲ P❛✉❧ ❙✳ ▼✉❤❧② ✶ P✐♠s♥❡r ❆❧❣❡❜r❛s ❚❤❡ ❙tr✉❝t✉r❡ ♦❢ ● ❛♥❞ ❈ ∗ ( ● ) ✶ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❚❤❡ ✉♥✐✈❡rs❛❧ ✐s♦♠❡tr② ✐♥ ❯♥✐✈❡rs✐t② ♦❢ ■♦✇❛ ❈ ∗ ( ● ) ▼✐♥✐♠❛❧ ❯♥✐t❛r② ❇❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ▼❛r✐✉s ■♦♥❡s❝✉ ❊①t❡♥s✐♦♥s ♦❢ ❙ ✐♥ ❛ ▼♦r✐t❛ ❊q✉✐✈❛❧❡♥t ❈ ∗ ✲❛❧❣❡❜r❛ ❖♣❡r❛t♦r ❚❤❡♦r② ❛♥❞ ❖♣❡r❛t♦r ❆❧❣❡❜r❛s✱ ❇●❯✱ ▼❛② ✷✸✱ ❆❞❞✐t✐♦♥❛❧ ❘❛t✐♦♥❛❧❡ ✷✵✶✷ ❘❡❝♦❧❧❡❝t✐♦♥s ❢r♦♠ ❑▼❘❲ ❬❑▼❘❲✾✽❪ ❙t❛❜✐❧✐③✐♥❣ ●
❖✉t❧✐♥❡ ❊♥❞♦♠♦r♣❤✐s♠s ❛♥❞ ❚r❛♥s❢❡r ❖♣❡r❛♦rs ❚❤❡ ❙❡tt✐♥❣ ✶ P✳❙✳ ▼✉❤❧② ❊♥❞♦♠♦r♣❤✐s♠s ❛♥❞ tr❛♥s❢❡r ♦♣❡r❛t♦rs ❚❤❡ ❙❡tt✐♥❣ ❈ ∗ ✲❝♦rr❡s♣♦♥❞❡♥❝❡s ❛♥❞ t❤❡✐r ❈✉♥t③✲P✐♠s♥❡r ❆❧❣❡❜r❛s ❈ ∗ ✲ ❚❤❡ ❙tr✉❝t✉r❡ ♦❢ ● ❛♥❞ ❈ ∗ ( ● ) ❝♦rr❡s♣♦♥❞❡♥❝❡s ❛♥❞ t❤❡✐r ❈✉♥t③✲ ❚❤❡ ✉♥✐✈❡rs❛❧ ✐s♦♠❡tr② ✐♥ ❈ ∗ ( ● ) P✐♠s♥❡r ❆❧❣❡❜r❛s ❚❤❡ ❙tr✉❝t✉r❡ ♦❢ ● ❛♥❞ ❈ ∗ ( ● ) ▼✐♥✐♠❛❧ ❯♥✐t❛r② ❊①t❡♥s✐♦♥s ♦❢ ❙ ✐♥ ❛ ▼♦r✐t❛ ❊q✉✐✈❛❧❡♥t ❚❤❡ ✉♥✐✈❡rs❛❧ ✷ ✐s♦♠❡tr② ✐♥ ❈ ∗ ✲❛❧❣❡❜r❛ ❈ ∗ ( ● ) ▼✐♥✐♠❛❧ ❆❞❞✐t✐♦♥❛❧ ❘❛t✐♦♥❛❧❡ ❯♥✐t❛r② ❊①t❡♥s✐♦♥s ♦❢ ❘❡❝♦❧❧❡❝t✐♦♥s ❢r♦♠ ❑▼❘❲ ❬❑▼❘❲✾✽❪ ❙ ✐♥ ❛ ▼♦r✐t❛ ❊q✉✐✈❛❧❡♥t ❙t❛❜✐❧✐③✐♥❣ ● ❈ ∗ ✲❛❧❣❡❜r❛ ❆❞❞✐t✐♦♥❛❧ ❘❛t✐♦♥❛❧❡ ❘❡❝♦❧❧❡❝t✐♦♥s ❢r♦♠ ❑▼❘❲ ❬❑▼❘❲✾✽❪ ❙t❛❜✐❧✐③✐♥❣ ●
❈♦♥✈❡♥t✐♦♥s ❊♥❞♦♠♦r♣❤✐s♠s ❛♥❞ ❚r❛♥s❢❡r ❆ ✲ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✳ ❖♣❡r❛♦rs P✳❙✳ ▼✉❤❧② α : ❆ → ❆ ✲ ❛ ✉♥✐t❛❧✱ ✐♥❥❡❝t✐✈❡ ❡♥❞♦♠♦r♣❤✐s♠✳ ❚❤❡ ❙❡tt✐♥❣ ❉❡✜♥✐t✐♦♥ ❈ ∗ ✲ ❝♦rr❡s♣♦♥❞❡♥❝❡s ❛♥❞ t❤❡✐r ❆ tr❛♥s❢❡r ♦♣❡r❛t♦r ❢♦r α ✐s ❛ ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ♠❛♣ ❈✉♥t③✲ P✐♠s♥❡r ❆❧❣❡❜r❛s τ : ❆ → ❆ s✉❝❤ t❤❛t ❚❤❡ ❙tr✉❝t✉r❡ ♦❢ ● ❛♥❞ ❈ ∗ ( ● ) ❚❤❡ ✉♥✐✈❡rs❛❧ τ ( α ( ❛ ) ❜ ) = ❛ τ ( ❜ ) ✐s♦♠❡tr② ✐♥ ❈ ∗ ( ● ) ▼✐♥✐♠❛❧ ❯♥✐t❛r② ❢♦r ❛❧❧ ❛ , ❜ ∈ ❆ ✳ ❊①t❡♥s✐♦♥s ♦❢ ❙ ✐♥ ❛ ▼♦r✐t❛ ❊q✉✐✈❛❧❡♥t ❈ ∗ ✲❛❧❣❡❜r❛ ❯♥❞❡r ♦✉r ❝♦♥✈❡♥t✐♦♥s✱ t❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦✿ τ ✐s ❛ ❧❡❢t ❆❞❞✐t✐♦♥❛❧ ❘❛t✐♦♥❛❧❡ ✐♥✈❡rs❡ ♦❢ α ✳ ❆❧s♦ τ ( ❜ α ( ❛ )) = τ ( ❜ ) ❛ ✳ ❘❡❝♦❧❧❡❝t✐♦♥s ❢r♦♠ ❑▼❘❲ ❬❑▼❘❲✾✽❪ ❙t❛❜✐❧✐③✐♥❣ ●
❚✇♦ ❈ ∗ ✲❝♦rr❡s♣♦♥❞❡♥❝❡s ❊♥❞♦♠♦r♣❤✐s♠s α ❆ ✲ ❆ ✇✐t❤ ❜✐♠♦❞✉❧❡ str✉❝t✉r❡✱ ❛♥❞ ❚r❛♥s❢❡r ❖♣❡r❛♦rs ❛ · ξ · ❜ := α ( ❛ ) ξ ❜ P✳❙✳ ▼✉❤❧② ❛♥❞ ✐♥♥❡r ♣r♦❞✉❝t ❚❤❡ ❙❡tt✐♥❣ � ξ , η � := ξ ∗ η . ❈ ∗ ✲ ❝♦rr❡s♣♦♥❞❡♥❝❡s ❛♥❞ t❤❡✐r ❈✉♥t③✲ P✐♠s♥❡r ❆❧❣❡❜r❛s ❚❤❡ ❙tr✉❝t✉r❡ ❆ τ ✲ ❆ ✇✐t❤ ❜✐♠♦❞✉❧❡ str✉❝t✉r❡✱ ♦❢ ● ❛♥❞ ❈ ∗ ( ● ) ❚❤❡ ✉♥✐✈❡rs❛❧ ✐s♦♠❡tr② ✐♥ ❛ · ξ · ❜ := ❛ ξα ( ❜ ) ❈ ∗ ( ● ) ▼✐♥✐♠❛❧ ❛♥❞ ✐♥♥❡r ♣r♦❞✉❝t ❯♥✐t❛r② ❊①t❡♥s✐♦♥s ♦❢ ❙ ✐♥ ❛ ▼♦r✐t❛ � ξ , η � := τ ( ξ ∗ η ) . ❊q✉✐✈❛❧❡♥t ❈ ∗ ✲❛❧❣❡❜r❛ ❆❞❞✐t✐♦♥❛❧ ❘❛t✐♦♥❛❧❡ ❘❡❝♦❧❧❡❝t✐♦♥s ❍♦✇ ❛r❡ t❤❡ ❈✉♥t③✲P✐♠s♥❡r ❆❧❣❡❜r❛s O ( α ❆ ) ❛♥❞ O ( ❆ τ ) ❢r♦♠ ❑▼❘❲ ❬❑▼❘❲✾✽❪ ❙t❛❜✐❧✐③✐♥❣ ● r❡❧❛t❡❞❄
❚❤❡ ❈✉♥t③✲P✐♠s♥❡r ❆❧❣❡❜r❛ O ( α ❆ ) ❊♥❞♦♠♦r♣❤✐s♠s ❛♥❞ ❚r❛♥s❢❡r ❖♣❡r❛♦rs ❚❤❡♦r❡♠ P✳❙✳ ▼✉❤❧② ✭❑❤♦s❤❦❛♠ ❛♥❞ ❙❦❛♥❞❛❧✐s ❬❑❙✾✼❪✮ O ( α ❆ ) ≃ ❆ ∞ ⋊ α ∞ Z ✳ ❍❡r❡ ❚❤❡ ❙❡tt✐♥❣ ❆ ∞ ✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡ s②st❡♠ ❈ ∗ ✲ ❝♦rr❡s♣♦♥❞❡♥❝❡s ❛♥❞ t❤❡✐r ❈✉♥t③✲ ··· → ❆ → ❆ → ❆ → ··· P✐♠s♥❡r ❆❧❣❡❜r❛s ❚❤❡ ❙tr✉❝t✉r❡ ♦❢ ● ❛♥❞ ❈ ∗ ( ● ) ✇❤❡r❡ t❤❡ ❝♦♥♥❡❝t✐♥❣ ♠❛♣s ❛r❡ ❛❧❧ α ✱ ❛♥❞ α ∞ ✐s t❤❡ ✐♥❞✉❝❡❞ ❚❤❡ ✉♥✐✈❡rs❛❧ ✐s♦♠❡tr② ✐♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❆ ∞ ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ❆ = ❈ ( ❳ ) ❛♥❞ ❈ ∗ ( ● ) ▼✐♥✐♠❛❧ α ( ❢ ) = ❢ ◦ σ ✱ σ ✲ ❛ ❧♦❝❛❧ ❤♦♠❡♦♠♦r♣❤✐s♠✱ t❤❡♥ ❆ ∞ = ❈ ( ❳ ∞ ) ✱ ❯♥✐t❛r② ❊①t❡♥s✐♦♥s ♦❢ ❳ ∞ = { ( ① ✵ , ① ✶ , ··· ) | ① ♥ − ✶ = σ ( ① ♥ ) } ❛♥❞ α ∞ ( ❢ ) = ❢ ◦ σ ∞ ✱ ❙ ✐♥ ❛ ▼♦r✐t❛ ❊q✉✐✈❛❧❡♥t σ ∞ ( ① ✵ , ① ✶ , ··· ) = ( σ ( ① ✵ ) , σ ( ① ✶ ) , ··· ) ✳ ❈ ∗ ✲❛❧❣❡❜r❛ ❆❞❞✐t✐♦♥❛❧ ❘❛t✐♦♥❛❧❡ ❘❡❝♦❧❧❡❝t✐♦♥s ❢r♦♠ ❑▼❘❲ ❬❑▼❘❲✾✽❪ ❙t❛❜✐❧✐③✐♥❣ ●
❯♥✐✈❡rs❛❧ Pr♦♣❡rt② ❊♥❞♦♠♦r♣❤✐s♠s ❛♥❞ ❚r❛♥s❢❡r ❖♣❡r❛♦rs P✳❙✳ ▼✉❤❧② O ( α ❆ ) ✐s ✉♥✐✈❡rs❛❧ ❢♦r ♣❛✐rs ( π , ❯ ) ✱ ✇❤❡r❡ π ✐s ❛ ❈ ∗ ✲r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❆ ♦♥ ❛ ❍✐❧❜❡rt s♣❛❝❡ ❍ ❛♥❞ ❯ ✐s ❛ ❚❤❡ ❙❡tt✐♥❣ ✉♥✐t❛r② ♦♥ ❍ s✉❝❤ t❤❛t ❈ ∗ ✲ ❝♦rr❡s♣♦♥❞❡♥❝❡s ❛♥❞ t❤❡✐r ❈✉♥t③✲ ❯ π ( ❛ ) ❯ − ✶ = π ( α ( ❛ )) , P✐♠s♥❡r ❛ ∈ ❆ , ❆❧❣❡❜r❛s ❚❤❡ ❙tr✉❝t✉r❡ ♦❢ ● ❛♥❞ ❈ ∗ ( ● ) ❚❤❡ ✉♥✐✈❡rs❛❧ ✐✳❡✳ O ( α ❆ ) ✐s ✉♥✐✈❡rs❛❧ ❢♦r ✉♥✐t❛r② ❝♦✈❛r✐❛♥t ✐s♦♠❡tr② ✐♥ ❈ ∗ ( ● ) r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ( ❆ , α ) ✳ ▼✐♥✐♠❛❧ ❯♥✐t❛r② ❲❡ ❝❛❧❧ s✉❝❤ ❛ ♣❛✐r ( π , ❯ ) ❛ ✉♥✐t❛r② α ✲❝♦✈❛r✐❛♥t ❊①t❡♥s✐♦♥s ♦❢ ❙ ✐♥ ❛ ▼♦r✐t❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ( ❆ , α ) ✳ ❊q✉✐✈❛❧❡♥t ❈ ∗ ✲❛❧❣❡❜r❛ ❆❞❞✐t✐♦♥❛❧ ❘❛t✐♦♥❛❧❡ ❘❡❝♦❧❧❡❝t✐♦♥s ❢r♦♠ ❑▼❘❲ ❬❑▼❘❲✾✽❪ ❙t❛❜✐❧✐③✐♥❣ ●
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