Dissertação Thiely Fonseca de Almeida

Livre

0
0
56
1 year ago
Preview
Full text

  ❚❍■❊▲❨ ❋❖◆❙❊❈❆ ❉❊ ❆▲▼❊■❉❆ ❋▲❯❳❖❙ ❉❊ ❆◆❖❙❖❱ ❊ ❖ ●❘❯P❖ ❋❯◆❉❆▼❊◆❚❆▲

  ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐✲ ❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳

  ❱■➬❖❙❆ ▼■◆❆❙ ●❊❘❆■❙ ✲ ❇❘❆❙■▲

  ✷✵✶✼

  ❚❍■❊▲❨ ❋❖◆❙❊❈❆ ❉❊ ❆▲▼❊■❉❆ ❋▲❯❳❖❙ ❉❊ ❆◆❖❙❖❱ ❊ ❖ ●❘❯P❖ ❋❯◆❉❆▼❊◆❚❆▲

  ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐✲ ❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳

  ❆P❘❖❱❆❉❆✿ ✷✶ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✼✳ ❆♥❞rés ▼❛✉r✐❝✐♦ ▲ó♣❡③ ❇❛rr❛❣❛♥

  ❆❧❡①❛♥❞r❡ ▼✐r❛♥❞❛ ❆❧✈❡s ❇✉❧♠❡r ▼❡❥í❛ ●❛r❝í❛

  ✭❈♦♦r✐❡♥t❛❞♦r✮ ❊♥♦❝❤ ❍✉♠❜❡rt♦ ❆♣❛③❛ ❈❛❧❧❛

  ✭❖r✐❡♥t❛❞♦r✮

  ❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛♦s ♠❡✉s ♣❛✐s✱ ❊❧② ❡ ❋át✐♠❛✳

  ✧❆ ❢é ❡♠ ❉❡✉s ♥♦s ❢❛③ ❝r❡r ♥♦ ✐♥❝rí✈❡❧✱ ✈❡r ♦ ✐♥✈✐sí✈❡❧ ❡ r❡❛❧✐③❛r ♦ ✐♠♣♦ssí✈❡❧✧✳

  ❆❣r❛❞❡❝✐♠❡♥t♦s

  ❆❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r t❡r ♠❡ ❞❛❞♦ ❢♦rç❛s ❡ ♥ã♦ t❡r ♠❡ ❞❡✐①❛❞♦ ❞❡s❛♥✐♠❛r ❞✐❛♥t❡ ❞♦s ❞❡s❛✜♦s ❡♠♦❝✐♦♥❛✐s ❡♥❢r❡♥t❛❞♦s ♥❡ss❡s ❞♦✐s ❛♥♦s ❡ ♣❡❧❛s ♠❛r❛✈✐❧❤♦s❛s ♣❡ss♦❛s q✉❡ ❝♦❧♦❝♦✉ ♥♦ ♠❡✉ ❝❛♠✐♥❤♦

  ❙♦✉ ♠✉✐tíss✐♠♦ ❣r❛t❛ ❛♦s ♠❡✉s ♣❛✐s✱ ❊❧② ❡ ❋át✐♠❛✱ ♣❡❧♦ ❡①❡♠♣❧♦✱ ❝❛r✐♥❤♦ ❡ ♠♦t✐✈❛çã♦✱ ✈♦❝ês sã♦ ♦s ♠❛✐♦r❡s ❞❡ t♦❞♦s ♦s ♠❡✉s ♣r♦❢❡ss♦r❡s✱ ❛ ♠✐♥❤❛ ❜❛s❡✱ ♥♦ss❛ ❤✐stór✐❛ ❞❡ ❧✉t❛s ❡ ✈✐tór✐❛s ❝♦♠❡ç❛ ❜❡♠ ❛♥t❡s ❞❛q✉✐✳

  ❆♦ ♠❡✉ q✉❡r✐❞♦ ♦r✐❡♥t❛❞♦r ❉r✳ ❊♥♦❝❤ ❆♣❛③❛✱ ♣♦r t♦❞❛ ♣❛❝✐ê♥❝✐❛✱ t❡♠♣♦ ❡ s❛❜❡❞♦r✐❛ ❞❡❞✐❝❛❞♦s ❛♦ ♠❡✉ tr❛❜❛❧❤♦✳ ❋♦✐ ✉♠ ❣r❛♥❞❡ ♣r❛③❡r s❡r s✉❛ ♦r✐❡♥t❛❞❛✳ ❊ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ♣❛ss❛r❛♠ ♥♦ ♠❡✉ ❝❛♠✐♥❤♦ ❡ ♠❡ ❞❡s♣❡rt❛r❛♠ ♦ ✐♥t❡r❡ss❡ ❡♠ ❝♦♥t✐♥✉❛r ❡st✉❞❛♥❞♦ ♠❛t❡♠át✐❝❛✳ ❆♦ ♠❡✉ ❝♦♦r✐❡♥t❛❞♦r✱ ❇✉❧♠❡r✱ ❛❣r❛❞❡ç♦ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ♣❡❧❛s ❝♦♥tr✐❜✉✐çõ❡s ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✳

  ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ✐r♠ã♦s✱ ❚❤✐❛❣♦ ❡ ❚❤❛②♥❛r❛✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ❢♦rç❛ ❡ ❛❧❡❣r✐❛✳ ❆ ♠✐♥❤❛ s♦❜r✐♥❤❛✱ ▼❛r✐❛ ❱❛❧❡♥t✐♥❛✱ ♠♦t✐✈❛çã♦ ❞✐ár✐❛ ♣❛r❛ s❡r ❡①❡♠♣❧♦✳ ❆♦ ♠❡✉ q✉❡r✐❞♦ ❛✈ô✱ ❏♦ã♦ ❆❧♠❡✐❞❛✱ ❛❣r❛❞❡ç♦ ❝❛r✐♥❤♦s❛♠❡♥t❡ ♣❡❧♦ ❛♣♦✐♦✱ ♣♦r s❡r s❡♠♣r❡ ♣r❡s❡♥t❡ ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❡ ♣♦r s❡r ♦ ♠❡❧❤♦r ❛✈ô✱ ❞♦ ♠✉♥❞♦✳

  ❆♦s ❣r❛♥❞❡s ❡ ✐♥❝rí✈❡✐s ❛♠✐❣♦s q✉❡ ✜③ ❞✉r❛♥t❡ ♦ ♠❡str❛❞♦ q✉❡ ♠❡ ❛❥✉❞❛r❛♠ ♥❛s ❞✐s❝✐♣❧✐♥❛s ❡ ♥♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❞❡ t♦❞♦s ♦s ❞✐❛s✱ ❡♠ ❡s♣❡❝✐❛❧ à ❘♦sâ♥❣❡❧❛✱ q✉❡ ♠❡ ❛❝♦♠♣❛♥❤❛ ❡ ✐♥❝❡♥t✐✈❛ ❡♠ t❡♦r✐❛s s♦❜r❡ ❛ ✈✐❞❛✱ ♦ ✉♥✐✈❡rs♦ ❡ t✉❞♦ ♠❛✐s✳

  ❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉▼❆✲❯❋❱✱ ♣♦r ❝♦❧❛❜♦r❛r❡♠ ❝♦♠ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❡ ♣❡❧♦s ❡✜❝✐❡♥t❡s s❡r✈✐ç♦s ♣r❡st❛❞♦s✳

  ❆❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❋✐♥❛❧♠❡♥t❡✱ ❛❣r❛❞❡ç♦ à ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ✐♥❞✐s♣❡♥sá✈❡❧ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

  ❙✉♠ár✐♦

   ✈✐

   ✈✐✐

   ✶

   ✷

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

   ✸✹

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

  ✹✻

  ❘❡s✉♠♦

  ❆▲▼❊■❉❆✱ ❚❤✐❡❧② ❋♦♥s❡❝❛✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❥✉❧❤♦ ❞❡ ✷✵✶✼✳ ❋❧✉①♦s ❆♥♦s♦✈ ❡ ♦ ●r✉♣♦ ❋✉♥❞❛♠❡♥t❛❧✳ ❖r✐❡♥t❛❞♦r✿ ❊♥♦❝❤ ❍✉♠❜❡rt♦ ❆♣❛③❛ ❈❛❧❧❛✳ ❈♦♦r✐❡♥t❛❞♦r✿ ❇✉❧♠❡r ▼❡❥í❛ ●❛r❝í❛✳ t : M −→ M ◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿ ❙❡ ϕ é ✉♠ 1 (M )

  ✢✉①♦ ❆♥♦s♦✈ ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠ ❡♥tã♦ π t❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧✳ ❊st❡ r❡s✉❧t❛❞♦ ❢♦✐ ♣r♦✈❛❞♦ ♣♦r ▼❛r❣✉❧✐s✱ ●✳ ❆✳✱ ♥♦ ❛♣ê♥❞✐❝❡ ❞❡ ♣❛r❛ 3✲✈❛r✐❡❞❛❞❡s✳ ❆q✉✐ s❡❣✉✐♠♦s ❛ ♣r♦✈❛ ❢❡✐t❛ ♣♦r P❧❛♥t❡ ❡ ❚❤✉rst♦♥ ❡♠ ♣❛r❛ n✲✈❛r✐❡❞❛❞❡s✳

  ❆❜str❛❝t

  ❆▲▼❊■❉❆✱ ❚❤✐❡❧② ❋♦♥s❡❝❛✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❏✉❧②✱ ✷✵✶✼✳ ❆♥♦s♦✈ ✢♦✇s ❛♥❞ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣✳ ❆❞✈✐s❡r✿ ❊♥♦❝❤ ❍✉♠❜❡rt♦ ❆♣❛③❛ ❈❛❧❧❛✳ ❈♦✲❛❞✈✐s❡r✿ ❇✉❧♠❡r ▼❡❥í❛ ●❛r❝í❛✳ t : M −→ M ❲❡ ✇✐❧❧ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✿ ■❢ ϕ ✐s ❛♥ ❆♥♦s♦✈ ✢♦✇ t❤❡♥ π 1 (M )

  ❤❛s ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤✳ ❚❤✐s r❡s✉❧t ✇❛s ✜rst ♣r♦✈❡❞ ❜② ▼❛r❣✉❧✐s✱ ●✳ ❆✳✱ ✐♥ t❤❡ ❛♣♣❡♥❞✐① ♦❢ ❢♦r ✸✲♠❛♥✐❢♦❧❞s✳ ❲❡ ❢♦❧❧♦✇ t❤❡ ♣r♦♦❢ ♦❢ P❧❛♥t❡ ❛♥❞ ❚❤✉rst♦♥ ♠❛❞❡ ✐♥ ❢♦r n✲♠❛♥✐❢♦❧❞s✳

  ■♥tr♦❞✉çã♦

  ✳ ◆❡st❡ tr❛❜❛❧❤♦✱ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛✱ s❡♠ t : M → M

  ❜♦r❞♦✱ ❞❡ ❞✐♠❡♥sã♦ n ≥ 3✳ ❙❡❥❛ ϕ ✉♠ ✢✉①♦ ❆♥♦s♦✈✳ ■st♦ s✐❣♥✐✜❝❛ s u X q✉❡ ♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ T M ❡♠ M s❡ ❞❡❝♦♠♣õ❡ ❡♠ três s✉❜✜❜r❛❞♦s E ✱ E ✱ E ✱ s u ✐♥✈❛r✐❛♥t❡s ♣❡❧❛ ❞❡r✐✈❛❞❛ ❞♦ ✢✉①♦✱ ❝♦♥tr❛✐♥❞♦ ♦ s✉❜✜❜r❛❞♦ E ❡ ❡①♣❛♥❞✐♥❞♦ E ✳ s u ❖s ✜❜r❛❞♦s E ✱ E sã♦ ✉♥✐❝❛♠❡♥t❡ ✐♥t❡❣rá✈❉✐r❡♠♦s q✉❡ ✉♠ ✢✉①♦ s u

  ) = 1 ) = 1 ❆♥♦s♦✈ t❡♠ ❝♦❞✐♠❡♥sã♦ ✉♠ q✉❛♥❞♦ dim(E ♦✉ dim(E ✳

  ❉❛❞♦ ✉♠ ❣r✉♣♦ S ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ❛ ❢✉♥çã♦ Γ(n) é ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ ❞❡ ♣❛❧❛✈r❛s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ n✳ ❆ ❞❡✜♥✐çã♦ ❢♦r♠❛❧ s❡rá ❞❛❞❛ ❡♠ ❉✐③❡♠♦s q✉❡ S t❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ s❡ ❞❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ n ✜♥✐t♦ ❞❡ ❣❡r❛❞♦r❡s t❡♠✲s❡ Γ(n) ≥ Ae ♣❛r❛ ❛❧❣✉♠❛s ❝♦♥st❛♥t❡s A > 0✱ a > 0✳ 1 (M )

  ❊✱ π ❞❡♥♦t❛ ♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ✈❛r✐❡❞❛❞❡ M✳ ❆q✉✐ ♣r♦✈❛♠♦s ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ❞❡✈✐❞♦ ❛ P❧❛♥t❡ t : M −→ M ❚❡♦r❡♠❛ ✿ ❙❡ ϕ é ✉♠ ✢✉①♦ ❆♥♦s♦✈ ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠✱ ❡♥tã♦

  π (M ) 1 t❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦ t❡♦r❡♠❛ ❞❡ P❧❛♥t❡ ♣r♦✈❛ q✉❡ ♥ã♦ ♣♦❞❡ ❡①✐st✐r ✢✉①♦s ❞❡ 3 3

  ❆♥♦s♦✈ ♥❛ ❡s❢❡r❛ S ✱ ♥♦ t♦r♦ T ✳ ▼❛s ♣♦r ♦✉tr♦ ❧❛❞♦✱ ❡①✐st❡♠ ❡①❡♠♣❧♦s ❝❧áss✐❝♦s ❞❡ ✢✉①♦s ❞❡ ❆♥♦s♦✈ ♥✉♠❛ ✸✲✈❛r✐❡❞❛❞❡✳ ❯♠ ❞❡ss❡s é ❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❆♥♦s♦✈✳

  ❊st❛ ❞✐ss❡rt❛çã♦✱ ❡stá ♦r❣❛♥✐③❛❞❛ ❝♦♠♦ s❡❣✉❡✿ ◆♦ ❈❛♣ít✉❧♦ 1 s❡rã♦ ✐♥tr♦❞✉③✐❞❛s ♥♦çõ❡s ❜ás✐❝❛s s♦❜r❡✿ ✈❛r✐❡❞❛❞❡s

  ❞✐❢❡r❡♥❝✐á✈❡✐s ❡ s♦❜r❡ ❡st❛s ❡st✉❞❛♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s✱ ❞❡✜♥✐♠♦s ✢✉①♦s ❞❡ ❆♥♦s♦✈ ❡ ❢❛r❡♠♦s ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠❛ s✉s♣❡♥sã♦ ❞❡ ✢✉①♦ ❆♥♦s♦✈✳ ❆❧é♠ ❞✐ss♦✱ ❡st✉❞❛♠♦s ❣r✉♣♦s ❞❡ ❤♦♠♦t♦♣✐❛✱ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧✱ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❣r✉♣♦s✱ ❝♦♠ ❡①❡♠♣❧♦s s♦❜r❡ ♦s t✐♣♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❡ ♣♦r ✜♠✱ ❢❛❧❛r❡♠♦s ❡♠ ❡s♣❛ç♦s ❞❡ r❡❝♦❜r✐♠❡♥t♦✳ ❉❛r❡♠♦s ✉♠❛ ♥♦çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❢♦❧❤❡❛çõ❡s✱ ❤♦❧♦♥♦♠✐❛ ❡ ❢❛r❡♠♦s ✉♠ ❡s❜♦ç♦ ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❞❡ ❍❛❡✢✐❣❡r✱ ❛s q✉❛✐s s❡rã♦ ✉t✐❧✐③❛❞❛s ❡♠ t♦❞♦ tr❛❜❛❧❤♦✳

  ❋✐♥❛❧♠❡♥t❡✱ ♥♦ ❈❛♣ít✉❧♦ 2 ✉s❛♥❞♦ ❛s ♥♦t❛çõ❡s ❡ ❞❡✜♥✐çõ❡s ❞♦ ❝❛♣ít✉❧♦ 1 ♠♦str❛♠♦s ♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧✳

  ❈❛♣ít✉❧♦ ✶ Pr❡❧✐♠✐♥❛r❡s

  ◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ✐♥tr♦❞✉③✐❞♦s ❝♦♥❝❡✐t♦s s♦❜r❡ ✈❛r✐❡❞❛❞❡s✱ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s✱ ✢✉①♦s ❛ss♦❝✐❛❞♦s à ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ♥✉♠❛ ✈❛r✐❡❞❛❞❡ ❡ ✜♥❛❧♠❡♥t❡ ♦s ✢✉①♦s ❞❡ ❆♥♦s♦✈✳ ❚❡♠✲s❡ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧✱ ❛❥✉❞❛r ♦ ❧❡✐t♦r ❛ s❡ ❢❛♠✐❧✐❛r✐③❛r ❝♦♠ ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s✱ q✉❡ sã♦ ✉t✐❧✐③❛❞♦s ♥♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✳

  ✶✳✶ ◆♦çõ❡s s♦❜r❡ ❱❛r✐❡❞❛❞❡s ❉✐❢❡r❡♥❝✐á✈❡✐s

  ◆❡st❛ s❡çã♦ ✈❛♠♦s ❞❡✜♥✐r ❝♦♥❝❡✐t♦s ❡ ❛♣r❡s❡♥t❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s s♦❜r❡ ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✱ ♦s q✉❛✐s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❡ ❉❡✜♥✐çã♦ ✶✳✶✳ ❯♠ ❛t❧❛s ❞❡ ❞✐♠❡♥sã♦ m s♦❜r❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ M é ✉♠❛ m ❝♦❧❡çã♦ U ❞❡ s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s x : U −→ R ❡♠ M✱ ❝✉❥♦s ❞♦♠í♥✐♦s U

  ❝♦❜r❡♠ M✳ ❖s ❞♦♠í♥✐♦s U ❞♦s s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s x ∈ U sã♦ ❝❤❛♠❛❞♦s ❛s ✈✐③✐♥❤❛♥ç❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ U✳ k

  ❉❡✜♥✐çã♦ ✶✳✷✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C ✱ é ✉♠ ♣❛r ♦r❞❡♥❛❞♦ (M, U) ♦♥❞❡ M é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❞❡ ❍❛✉s❞♦r✛✱ ❝♦♠ ❜❛s❡ r ❡♥✉♠❡rá✈❡❧ ❡ U ✉♠ ❛t❧❛s ♠á①✐♠♦ ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C s♦❜r❡ M✳ ❉❡✜♥✐çã♦ ✶✳✸✳ ❙❡❥❛ M ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❯♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ♦✉ ❝❛rt❛ ❧♦❝❛❧ ❡♠ M é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ x : U −→ x(U) ❞❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ m ❛❜❡rt♦ U ⊂ M s♦❜r❡ ✉♠ ❛❜❡rt♦ x(U) ⊂ R ✳ ❉✐③❡♠♦s q✉❡✱ m é ❛ ❞✐♠❡♥sã♦ ❞❡ 1 m x : U −→ x(U ) (p), ..., x (p)) i i ✳ P❛r❛ ❝❛❞❛ p ∈ U t❡♠✲ s❡ x(p) = (x ✳ ❖s ♥ú♠❡r♦s x = x (p), i = 1, ..., m sã♦ ❝❤❛♠❛❞❛s ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ p ∈ M ♥♦ s✐st❡♠❛ x✳

  ❊①❡♠♣❧♦ ✶✳✹✳ ❈♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s m m m , U ⊂ R

  ❙❡❥❛♠ M = R ✉♠ ❛❜❡rt♦ ❡ x : U −→ R ❛ ❛♣❧✐❝❛çã♦ ❞❡ ✐♥❝❧✉sã♦✱ x(p) = p ✳ ❆s ❝♦♦r❞❡♥❛❞❛s ✐♥tr♦❞✉③✐❞❛s ❡♠ U ♣❡❧♦ s✐st❡♠❛ x sã♦ ❞❡♥♦♠✐♥❛❞❛s

  ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✳

  ✸ ✶✳✶✳ ◆❖➬Õ❊❙ ❙❖❇❘❊ ❱❆❘■❊❉❆❉❊❙ ❉■❋❊❘❊◆❈■➪❱❊■❙ ❉❡✜♥✐çã♦ ✶✳✺✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ M ♥♦ q✉❛❧ ❡①✐st❡ ✉♠ ❛t❧❛s ❞❡ ❞✐♠❡♥sã♦ m

  ❝❤❛♠❛✲ s❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛ ❞❡ ❞✐♠❡♥sã♦ m s❡✱ s♦♠❡♥t❡ s❡✱ ❝❛❞❛ m ♣♦♥t♦ ❞❡ M t❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠ ❛❜❡rt♦ ❞♦ R m n

  , N ❉❡✜♥✐çã♦ ✶✳✻✳ ✭❆♣❧✐❝❛çõ❡s ❉✐❢❡r❡♥❝✐á✈❡✐s✮ ❙❡❥❛♠ M ✈❛r✐❡❞❛❞❡s ❞❡ r

  (r ≥ 1) ❝❧❛ss❡ C ✳ ❉✐③ ✲s❡ q✉❡ ✉♠❛ ❛♣❧✐❝❛çã♦ f : M −→ N é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ m em M, y : ♣♦♥t♦ p ∈ M s❡ ❡①✐st❡♠ s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s x : U −→ R n −1

  V −→ R em N : x(U ) −→ n ✱ ❝♦♠ p ∈ U ❡ f(U) ⊂ V t❛✐s q✉❡ y ◦ f ◦ x y(V ) ⊂ R

  é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ x(p)✳ M f N x p f(p) y Rm x(p) x(U)

y o f o x

-1 Rn y(f(p)) y(V) ❋✐❣✉r❛ ✶✳✶✿ ❆♣❧✐❝❛çã♦ ❉✐❢❡r❡♥❝✐á✈❡❧ f : M → N ❡♥tr❡ M ❡ N✳ xy = y ◦ f ◦ x −1

  ❆ ❛♣❧✐❝❛çã♦ f é ❞❡♥♦♠✐♥❛❞❛ ❛ ❡①♣r❡ssã♦ ❞❡ f ♥❛s ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s x, y✳ ❖❜s❡r✈❡ q✉❡✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ f : M −→ N é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ p ∈ M✳ ❖❜s❡r✈❛çã♦ ✶✳✼✳ ❈♦♠♦ ❛s ♠✉❞❛♥ç❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡♠ M e N sã♦ r ❞✐❢❡♦♠♦r✜s♠♦s ❞❡ ❝❧❛ss❡ C ✱ ❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ✐♥❞❡♣❡♥❞❡ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s x, y✱ ♣♦✐s ♣❛r❛ q✉❛✐sq✉❡r ♦✉tr♦ ♣❛r ❞❡ s✐st❡♠❛s ❞❡ ′ ′ ′ ′ ′ ′ m n

  −→ R −→ R , y : U : V

  ❝♦♦r❞❡♥❛❞❛s x ✱ t❡♠♦s x ❡♠ M ❡ y ❡♠ N✱ ❝♦♠ ′ ′ ′ ′ ′ −1 ′ ′ p ∈ U ) ⊂ V x y = y ◦ f ◦ (x ) ✱ f(U ✱ ❛ ❛♣❧✐❝❛çã♦ f s❡rá ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ x (p) ✳

  ❉❡✜♥✐çã♦ ✶✳✽✳ ❉✐③❡♠♦s q✉❡ f : M −→ N é ❞✐❢❡r❡♥❝✐á✈❡❧ s❡ f ❢♦r ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ M✳ k

  (k ≤ r) ❉❡✜♥✐çã♦ ✶✳✾✳ ❉✐③❡♠♦s q✉❡ f : M −→ N é ❞❡ ❝❧❛ss❡ C s❡✱ ♣❛r❛ ❝❛❞❛ m n p ∈ M

  ✱ ❡①✐st❡♠ s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s x : U −→ R ❡♠ M✱ y : V −→ R −1 : x(U ) −→ y(V )

  ❡♠ N✱ ❝♦♠ p ∈ U ❡ f(U) ⊂ V t❛✐s q✉❡ y ◦ f ◦ x é ❞❡ ❝❧❛ss❡ k C

  ✳ k ❙❡❣✉❡✲ s❡ ❞❛ ❞❡✜♥✐çã♦ q✉❡✱ ✉♠❛ ❛♣❧✐❝❛çã♦ f : M −→ N é ❞❡ ❝❧❛ss❡ C q✉❛♥❞♦✱ ❡①✐st❡♠ ✉♠ ❛t❧❛s U s♦❜r❡ M ❡ ✉♠ ❛t❧❛s B s♦❜r❡ N✱ t❛✐s q✉❡✱ ♣❛r❛ ❝❛❞❛ k y ∈ B

  ❡①✐st❡ x ∈ U r❡❧❛t✐✈❛♠❡♥t❡ ❛♦s q✉❛✐s ❛ ❡①♣r❡ssã♦ ❞❡ f é ❞❡ ❝❧❛ss❡ C ✳ ■st♦ ′ ′ m : U −→ R

  ✐♠♣❧✐❝❛ q✉❡✱ ♣❛r❛ t♦❞❛ ❝❛rt❛ x ❞♦ ❛t❧❛s ♠á①✐♠♦ ❞❡ M ❡ ♣❛r❛ t♦❞❛ ′ ′ n ′ ′ : V −→ R ) ⊂ V

  ❝❛rt❛ y ❞♦ ❛t❧❛s ♠á①✐♠♦ ❞❡ N✱ ❝♦♠ f(U ✱ ❛ ❡①♣r❡ssã♦ ❧♦❝❛❧

  ✹ ′ ′ ′ ′ k ′ ′ ′ ′ ✶✳✷✳ ❋▲❯❳❖❙ ❆◆❖❙❖❱ f x ,y x ,y : x (U ∩ U ) −→ y (V ∩ V ) s❡rá ❞❡ ❝❧❛ss❡ C ✳ ❊♥tã♦ f ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ ′ ′ ′ ′ −1 ′ −1 −1 ′ −1 f x y = y ◦ f ◦ (x ) = y ◦ y ◦ y ◦ f ◦ x ◦ x ◦ (x ) ′ −1 ′ −1 ′ ′ k

  ◦ y ◦ (x ◦ (x ◦ f ◦ ϕ ∈ C = (y ) ◦ f xy ) ) = ϕ yy xy xx k

  ◗✉❛♥❞♦ ❞✐ss❡r♠♦s q✉❡ f : M −→ N é ❞❡ ❝❧❛ss❡ C ❛❞♠✐t✐r❡♠♦s✱ ❛♦ ♠❡♥♦s r , r ≥ k

  ✐♠♣❧✐❝✐t❛♠❡♥t❡✱ q✉❡ M ❡ N sã♦ ❞❡ ❝❧❛ss❡ C ✳ ❆ ❝♦♠♣♦st❛ ❞❡ ❞✉❛s k ❛♣❧✐❝❛çõ❡s f : M −→ N ❡ g : N −→ P ❞❡ ❝❧❛ss❡ C é t❛♠❜é♠ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ r ❝❧❛ss❡ C ✳ −1

  ❉❡✜♥✐çã♦ ✶✳✶✵✳ ✿❯♠ ❞✐❢❡♦♠♦r✜s♠♦ f : M −→ N é ✉♠❛ ❜✐❥❡çã♦ ❝♦♠ f −1 k ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❙❡ ❛♠❜❛s✱ f e f sã♦ ❞❡ ❝❧❛ss❡ C ✱ ❞✐③❡♠♦s q✉❡ f é ✉♠ k ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❝❧❛ss❡ C ✳ m ❉❡✜♥✐çã♦ ✶✳✶✶✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❉❡✜♥✐♠♦s ♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡ M ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦✿

  T M = {(p, v); p ∈ M, v ∈ T M } pm ❉❡✜♥✐çã♦ ✶✳✶✷✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❯♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ✭♦✉ ❡str✉t✉r❛ ❘✐❡♠❛♥♥✐❛♥❛✮ é ✉♠❛ ❧❡✐ q✉❡ ❢❛③ ❝♦rr❡s♣♦♥❞❡r p ❛ ❝❛❞❛ ♣♦♥t♦ p ∈ M ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♠ T M ✳ p

  ❙❡❥❛ g ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛✱ ❞❡♥♦t❛r❡♠♦s ♣♦r g(p; u, v) ♦✉ hu, vi ♦ p ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞♦s ✈❡t♦r❡s u, v ∈ T M ✳ p ❉❡✜♥✐çã♦ ✶✳✶✸✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ♦✉ ♥♦r♠❛ ❞❡ ✉♠ ✈❡t♦r v ∈ T M ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❝♦♠♦✿ q

  | v | p = hv, vi p . ❊st❛ ♥♦r♠❛ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ♥♦r♠❛ ❘✐❡♠❛♥♥✐❛♥❛✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ♦♥❞❡ ❡stá ❞❡✜♥✐❞❛ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ é

  ❞❡♥♦♠✐♥❛❞❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✳ ❉❡✜♥✐çã♦ ✶✳✶✹✳ ❉✉❛s ✈❛r✐❡❞❛❞❡s V ❡ W sã♦ tr❛♥s✈❡rs❛✐s ❡♠ M s❡ ♣❛r❛ q q q✉❛❧q✉❡r ♣♦♥t♦ q ∈ V ∩ W t❡♠♦s q✉❡ ♦s ❡s♣❛ç♦s t❛♥❣❡♥t❡s ❞❡ T V ❡ T W ❣❡r❛♠ T M q

  ✳

  ✶✳✷ ❋❧✉①♦s ❆♥♦s♦✈

  ◆❡st❛ s❡çã♦ ✈❛♠♦s ❞❡✜♥✐r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ ❛♣r❡s❡♥t❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s s♦❜r❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s✱ ♦s q✉❛✐s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ✱

  ✺ r ✶✳✷✳ ❋▲❯❳❖❙ ❆◆❖❙❖❱ , r > 0

  ❉❡✜♥✐çã♦ ✶✳✶✺✳ ❯♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❞❡ ❝❧❛ss❡ C ✱❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ m M

  é ✉♠❛ ❛♣❧✐❝❛çã♦ X : M → T M✱ t❛❧ q✉❡ ❛ ❝❛❞❛ ♣♦♥t♦ p ∈ M ❛ss♦❝✐❛ ♦ ✈❡t♦r r X(p) ∈ T M p

  ❡ X ∈ C ✳ ❉❡✜♥✐çã♦ ✶✳✶✻✳ ❯♠ ✢✉①♦ é ✉♠❛ ❛♣❧✐❝❛çã♦✱ ϕ : R × M → M✱ t❛❧ q✉❡ ♣❛r❛ t♦❞♦ t, s ∈ R

  ❡ t♦❞♦ p ∈ M ✈❛❧❡✿ ✶✳ ϕ(t + s, p) = ϕ(t, ϕ(s, p)) ✷✳ ϕ(0, p) = p t t (p) = ϕ(t, p)

  ❉❡♥♦t❛♠♦s ♣♦r ϕ ♦ ✢✉①♦ ❛ss♦❝✐❛❞♦ ❛♦ ❝❛♠♣♦ X✱ ♦♥❞❡ ϕ ✳ 2 3 ⊂ R

  ❊①❡♠♣❧♦ ✶✳✶✼✳ ❙❡❥❛ S ❛ ❡s❢❡r❛ ✉♥✐tár✐❛ ❝♦♠ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠✳ ❙❡❥❛♠ 2 P = (0, 0, 1) = (0, 0, −1) N S ❡ P ✱ ♦ ♣ó❧♦ ♥♦rt❡ ❡ ♦ ♣♦❧♦ s✉❧ ❞❛ ❡s❢❡r❛ S ✱ 2 r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡✜♥✐♠♦s ♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ❞❡ ❝❧❛ss❡ C ❡♠ S ♣♦r✱ 2 2 X(x, y, z) = (−xz, −yz, x + y )

  ❊ ♦ ✢✉①♦ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ ❡♠ p N

  

p

S

2

  ❋✐❣✉r❛ ✶✳✷✿ ❋❧✉①♦ ❡♠ S t ❉❡✜♥✐çã♦ ✶✳✶✽✳ ❆ ór❜✐t❛ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M ♣❛r❛ ✉♠ ✢✉①♦ ϕ é ♦ ❝♦♥❥✉♥t♦ O(p) = {ϕ t (p); t ∈ R} ❉❡✜♥✐çã♦ ✶✳✶✾✳ ❆ ór❜✐t❛ ♣♦s✐t✐✈❛ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M ♣❛r❛ ✉♠ ✢✉①♦ ϕ é ♦ +

  (p) = {ϕ(p); t ≥ 0} ❝♦♥❥✉♥t♦ O ϕ ✳ ❊ ❛ ór❜✐t❛ ♥❡❣❛t✐✈❛ ❞❡ p ∈ M ♣❛r❛ ✉♠ ✢✉①♦ ϕ t (p) = {ϕ(p); t ≤ 0} ϕ

  é ♦ ❝♦♥❥✉♥t♦ O ✳ ❉❡✜♥✐çã♦ ✶✳✷✵✳ ❙❡ X(p) = 0 ❡♥tã♦ ❞✐③❡♠♦s q✉❡ p ∈ M é ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ ❞♦ ❝❛♠♣♦ X✳ ❉❡✜♥✐çã♦ ✶✳✷✶✳ ❖ ❝♦♥❥✉♥t♦ ω✲❧✐♠✐t❡ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M é ❞❛❞♦ ♣♦r✿ ω(p) = {x ∈ M ; ∃ t n → ∞ quando n → ∞ tal que lim ϕ t (p) = x}. n →∞ n ❆ss✐♠✱ ω(p) é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❡ ❛❝✉♠✉❧❛çã♦ ❞❛ ór❜✐t❛ ♣♦s✐t✐✈❛ ❞❡ p✳ ❉❡✜♥✐♠♦s t❛♠❜é♠ ♦ ❝♦♥❥✉♥t♦ α✲❧✐♠✐t❡ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M ❞❛❞♦ ♣♦r α(p) = {x ∈ M ; ∃ t n → −∞ quando n → ∞ tal que lim ϕ t (p) = x}. n →∞ n ❊♥tã♦✱ ♦ ❝♦♥❥✉♥t♦ α(p) é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❡ ❛❝✉♠✉❧❛çã♦ ❞❛ ór❜✐t❛ ♥❡❣❛t✐✈❛

  ✻ ✶✳✷✳ ❋▲❯❳❖❙ ❆◆❖❙❖❱

  ❉❡✜♥✐çã♦ ✶✳✷✷✳ ❉✐③❡♠♦s q✉❡ x é r❡❝♦rr❡♥t❡ s❡ x ∈ ω(x)✳ ❉❡✜♥✐çã♦ ✶✳✷✸✳ ❖s ❝♦♥❥✉♥t♦s✿ ss

  W (p) = {q ∈ M ; d(ϕ t (p), ϕ t (q)) → 0, q✉❛♥❞♦ t → ∞} ❡ uu

  W (p) = {q ∈ M ; d(ϕ t (p), ϕ t (q)) → 0, q✉❛♥❞♦ t → −∞} sã♦ ❝❤❛♠❛❞♦s r❡s♣❡❝t✐✈❛♠❡♥t❡ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❡ ✐♥stá✈❡❧ ❢♦rt❡ ❞♦

  ♣♦♥t♦ p ♣❛r❛ ♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X✳ ❖♥❞❡ d ❞❡♥♦t❛ ❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❞❛ ✈❛r✐❡❞❛❞❡ M✳ ❉❡✜♥✐çã♦ ✶✳✷✹ ✭❈♦♥❥✉♥t♦ ❍✐♣❡r❜ó❧✐❝♦✮✳ ❯♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ✐♥✈❛r✐❛♥t❡ H ⊂ M t

  é ❞✐t♦ ❤✐♣❡r❜ó❧✐❝♦ ♣❛r❛ X ♦✉ ♣❛r❛ ϕ s❡✿ u X s p = E ⊕ E ⊕ E ✶✳ T M p p p ✱ ♣❛r❛ t♦❞♦ p ∈ H t ) p E = E t p ∗ ∗ ✷✳ D(ϕ ϕ t (p) ✱ ∗ = s, X, u ✭Dϕ ✲ ✐♥✈❛r✐❛♥t❡✮❀ s u

  , E ✸✳ E p p ✈❛r✐❛♠ ❝♦♥t✐♥✉❛♠❡♥t❡ ❝♦♠ p ❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s λ, C > 0✱ t❛✐s q✉❡✱ ∀t ≥ 0✱ t (p)v| ≤ Ce |v|, ∀v ∈ E e ∀p ∈ H −λt p s

  ❛✮ |Dϕ −λt u |w|, ∀w ∈ E ∀p ∈ H

  (p)(w)| ≤ Ce e ❜✮ |Dϕ −t p t

  ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠ s✉❜❝♦♥❥✉♥t♦ ✐♥✈❛r✐❛♥t❡ H ⊂ M ♣❛r❛ ✉♠ ✢✉①♦ ϕ é ❤✐♣❡r❜ó❧✐❝♦ s❡ ❡①✐st❡ ✉♠❛ ❞❡❝♦♠♣♦s✐çã♦ ❝♦♥tí♥✉❛ ❞♦ ❡s♣❛ç♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ u X s s

  T M p = E ⊕ E ⊕ E p p p p u s♦❜r❡ H ❝♦♥s✐st✐♥❞♦ ❞❡ ✉♠ s✉❜✜❜r❛❞♦ ❝♦♥trát✐❧ E ✱ ✉♠ X s✉❜✜❜r❛❞♦ ❡①♣❛♥s♦r E p ❡ ✉♠ s✉❜✜❜r❛❞♦ E p ❣❡r❛❞♦ ♣♦r X✳ ❚❡♦r❡♠❛ ✶✳✷✺✳ ✭❚❡♦r❡♠❛ ❞❛ ❱❛r✐❡❞❛❞❡ ❊stá✈❡❧ ♣❛r❛ ❋❧✉①♦s✮✳ ❙❡❥❛ H ⊂ M t

  ✉♠ ❝♦♥❥✉♥t♦ ❤✐♣❡r❜ó❧✐❝♦ ✐♥✈❛r✐❛♥t❡ ♣❛r❛ ✉♠ ✢✉①♦ ϕ ✳ ❊♥tã♦ ❡①✐st❡ ǫ > 0 ss uu (p) (p) t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ ♣♦♥t♦ p ∈ H ❡①✐st❡♠ ❞♦✐s ❞✐s❝♦s ♠❡r❣✉❧❤❛❞♦s W ǫ ❡ W ǫ p p s u

  ♦s q✉❛✐s sã♦ t❛♥❣❡♥t❡s ❛ E ❡ E ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r t ❉❡✜♥✐çã♦ ✶✳✷✻✳ ❯♠ ✢✉①♦ ϕ ❞❡✜♥✐❞♦ ♥❛ ✈❛r✐❡❞❛❞❡ M é ❆♥♦s♦✈ s❡ M é ✉♠ t ❝♦♥❥✉♥t♦ ❤✐♣❡r❜ó❧✐❝♦ ♣❛r❛ ϕ ✳ ❉❡✜♥✐çã♦ ✶✳✷✼✳ ❯♠ ✢✉①♦ ❆♥♦s♦✈ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ M é ❞✐t♦ ❞❡ ❝♦❞✐♠❡♥sã♦ s u

  ) = 1 ) = 1 ✉♠ s❡ dim(E ♦✉ dim(E ✳ ❉❡✜♥✐çã♦ ✶✳✷✽✳ ❯♠ ❞✐❢❡♦♠♦r✜s♠♦ f : M → M é ❞✐t♦ ❆♥♦s♦✈ s❡ ❡①✐st❡ ✉♠❛ u s

  ⊕ E ❞❡❝♦♠♣♦s✐çã♦ ❝♦♥tí♥✉❛ T M = E t❛❧ q✉❡✿ u s s s

  ) = E ✶✳ ❆♠❜♦s E ❡ E sã♦ ✐♥✈❛r✐❛♥t❡s ♣♦r Df✱ ♦✉ s❡❥❛✱ Df(p)(E x f ❡ u u (p)

  Df (p)(E ) = E p f (p) ✱ ♣❛r❛ t♦❞♦ p ∈ M✳

  ✼ ✶✳✷✳ ❋▲❯❳❖❙ ❆◆❖❙❖❱

  ❋✐❣✉r❛ ✶✳✸✿ ❆s♣❡❝t♦ ❧♦❝❛❧ ❞❡ ✉♠ ✢✉①♦ ❆♥♦s♦✈ ✷✳ ❊①✐st❡♠ ❝♦♥st❛♥t❡s C > 0 ❡ 0 < λ < 1 t❛✐s q✉❡ ♣❛r❛ t♦❞♦ p ∈ M n n s

  |Df (p)(v)| ≤ Cλ |v|, v ∈ E , n > 0 −n n u p |Df (p)(v)| ≤ Cλ |v|, v ∈ E , n > 0 p

  ❆❜❛✐①♦✱ t❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❜❡♠ ❝♦♥❤❡❝✐❞♦ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ ❆♥♦s♦✈✳ ❊①❡♠♣❧♦ ✶✳✷✾✳ ✭❉✐❢❡♦♠♦r✜s♠♦ ❆♥♦s♦✈✮✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ❛♣❧✐❝❛çã♦ 2 2 L A : R → R

  ❞❛❞❛ ♣❡❧❛ ♠❛tr✐③ 2 1 A = 1 1 2 2

  ) = Z ❖ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ss❛ ♠❛tr✐③ é ✉♠✱ A(Z ✳ ❆ ❡①♣r❡ssã♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s é 2 L A (u, v) = (2u + v, u + v) ′ ′ ′ ′ ′ ′ ✳ ❉❡✜♥✐♠♦s ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♠ R ✿ 2

  (u, v) ∼ (u , v ), se (u − u , v − v ) ∈ Z A (u, v) ∼ L A (u , v ) 2 2 2 ❉❡ss❛ ❢♦r♠❛✱ L ✳ ■ss♦ 2 L A : T = R /Z → T A (u, v) = (2u + v, u + v) ❞❡✜♥❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ˆ ✱ ❞❛❞❛ ♣♦r L ✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦✿ R R 2 L A ˆ // 2

  

x x

2 // 2 A

  T T L A ◆❡st❡ ❝❛s♦✱ L é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❆♥♦s♦✈✳ A

  ❉❡♠♦♥str❛çã♦✳ ❇❛st❛ ♠♦str❛r q✉❡ L é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❆♥♦s♦✈✳ ❉❡ ❢❛t♦✱ ♥♦t❡ q✉❡ x é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧✱ ♣♦✐s x é ❛ ❛♣❧✐❝❛çã♦ ♣r♦❥❡çã♦✳ ❈♦♠♦ ❛s ❞❡r✐✈❛❞❛s A ❞❛ ♠❛tr✐③ ❛ss♦❝✐❛❞❛ ❛ L ❞✐❢❡r❡♠ ❛♣❡♥❛s ♣♦r ✐s♦♠♦r✜s♠♦✱ ✐r❡♠♦s ❞❡t❡r♠✐♥❛r s❡✉s ❛✉t♦✈❛❧♦r❡s✳ 3+ √ √ 5 3− 5

  λ u = e λ s = 2 s < 1 < λ u , E 2 s u ❖✉ s❡❥❛✱ 0 < λ ✳ ❙❡❥❛ E ♦s s✉❜✜❜r❛❞♦s ❛ss♦❝✐❛❞♦s ❛♦s s , λ u

  ❛✉t♦✈❛❧♦r❡s λ ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ 2 s s u u w w = w ⊕ E = w ⊕ E P❛r❛ t♦❞♦ w = (u, v) ∈ R ✱ t❡♠♦s✿ E ❡ E ✳

  ✽ 2 s u ✶✳✷✳ ❋▲❯❳❖❙ ❆◆❖❙❖❱

  = E ⊕ E L A ❆✜r♠❛çã♦✿ ❆ ❞❡❝♦♠♣♦s✐çã♦ R é ✐♥✈❛r✐❛♥t❡ ♣❡❧❛ ❞❡r✐✈❛❞❛ ❞❡ ˆ ✳ ❉❡ ❢❛t♦✱ 2 s u

  ⊕ E 2 T R = E w P❛r❛ t♦❞♦ w ∈ R ✳ ❈♦♠♦ é ❧✐♥❡❛r✱ s s s s s

  DL A (w)(E ) = L A (w + E ) = L A (w) + L A (E ) = L A (w) + E = E w L A (w) ❆♥❛❧♦❣❛♠❡♥t❡✱ u u u u u

  DL A (w)(E ) = L A (w + E ) = L A (w) + L A (E ) = L A (w) + E = E w 2 s u L A (w) ⊕ E

  = E P♦rt❛♥t♦✱ T R é ✐♥✈❛r✐❛♥t❡✳ s s s s s s s s w w w w ∈ E = w + v ∈ E k = kv k P♦r ✜♠✱ ✜①❡ v ❡♥tã♦ v ✱ ♣❛r❛ t♦❞♦ v ❡ kv ✱ 2

  ♦♥❞❡ k.k é ❛ ♥♦r♠❛ ❡♠ T R w ✳ ❉❛í✱ s s s s DL A (w)(v ) = L A (w + v ) = L A (z) + L A (v ) = L A (w) + λ s .v w ▲♦❣♦✱ s s s kDL kv k = λ kv k A (w)(v )k L (w) = λ s s w w A w u u u u u u u u w w w w ∈ E = w +v ∈ E k = kv k

  ❆♥❛❧♦❣❛♠❡♥t❡✱ v ❡♥tã♦ v ✱ ♣❛r❛ t♦❞♦ v ❡ kv ✳ ▲♦❣♦✱ u u u kDL A (w)(v )k L = λ u kv k = λ u kv k w w A (w) w s u s < 1 < λ u

  Pr♦✈❛♥❞♦ q✉❡ E ❝♦♥tr❛í ❡ E ❡①♣❛♥❞❡✱ ❞❡s❞❡ q✉❡ 0 < λ ✳ ❋❧✉①♦ ❞❡ P♦✐♥❝❛ré

  ∈ γ ❙❡❥❛ γ ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ ❞❡ ✉♠ ❝❛♠♣♦✳ P♦r ✉♠ ♣♦♥t♦ x ❝♦♥s✐❞❡r❡

  ✉♠❛ tr❛♥s✈❡rs❛❧ ♦✉ s❡çã♦ tr❛♥s✈❡rs❛❧ Σ ❛♦ ❝❛♠♣♦ X✳ ❆ ór❜✐t❛ ♣♦r x ✈♦❧t❛ ✐♥t❡rs❡❝t❛r Σ ♥♦ t❡♠♣♦ τ ✱ ♦♥❞❡ τ é ♦ ♣❡rí♦❞♦ ❞❡ γ✳ P❡❧❛

  ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦ ❝❛♠♣♦✱ ❛ ór❜✐t❛ ♣♦r ✉♠ ♣♦♥t♦ ❡♠ Σ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣ró①✐♠♦ ❛ x t❛♠❜é♠ ✈♦❧t❛ ❛ ✐♥t❡rs❡❝t❛r Σ ❡♠ ✉♠ t❡♠♣♦ ♣ró①✐♠♦ ❛ τ✳

  ❙❡❥❛ V ⊂ Σ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛ ❞❡ x ✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❛♣❧✐❝❛çã♦ P : V → Σ q✉❡ ❝❛❞❛ ♣♦♥t♦ x ∈ V ❛ss♦❝✐❛ P (x)✱ s❡♥❞♦ P (x) ♦ ♣r✐♠❡✐r♦ ♣♦♥t♦ ♦♥❞❡ ❛ ór❜✐t❛ ❞❡ x ✈♦❧t❛ ❛ ✐♥t❡rs❡❝t❛r Σ✳ ❊st❛ ❛♣❧✐❝❛çã♦ é ❞❡♥♦♠✐♥❛❞❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ P♦✐♥❝❛ré ♦✉ ❢✉♥çã♦ ❞❡ Pr✐♠❡✐r♦ ❘❡t♦r♥♦ ❞❡ P♦✐♥❝❛ré ❛ss♦❝✐❛❞❛ ❛ ór❜✐t❛ γ ❡ à tr❛♥s✈❡rs❛❧ Σ✳

  ✾ ✶✳✷✳ ❋▲❯❳❖❙ ❆◆❖❙❖❱

x

P(x) x Σ

  ❋✐❣✉r❛ ✶✳✹✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ P♦✐♥❝❛ré ❙✉s♣❡♥sã♦ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❆♥♦s♦✈

  ❖ ✢✉①♦ s✉s♣❡♥sã♦ é ❞❛❞♦ ♣❡❧♦ ❞✐❢❡♦♠♦r✜s♠♦ f : M → M✳ P❛r❛ ❝♦♥str✉í✲ ❧♦ ✐❞❡♥t✐✜❝❛♠♦s ♦s ♣♦♥t♦s ❞♦ ❜♦r❞♦ M ×R ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ (x, s + 1) ∼ (f (x), s) f = M × R/ ∼

  ♣❛r❛ 0 ≤ s ≤ 1✳ ❙❡❥❛ M ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❡ x : M × R → M f ❛ ♣r♦❥❡çã♦ ♥❛t✉r❛❧✳ r f

  Pr♦♣♦s✐çã♦ ✶✳✸✵✳ ❙❡ f é ❞❡ ❝❧❛ss❡ C ♦ q✉♦❝✐❡♥t❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ q✉❡ t❡♠ r ❡str✉t✉r❛ C ✳ ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❛ ❡q✉❛çã♦ ❡♠ M × R✿ x = 0 t s = 1 t

  ❊ss❛ ❡q✉❛çã♦ ✐♥❞✉③ ✉♠ ✢✉①♦ ϕ ❡♠ M × R q✉❡ ♣♦r s✉❛ ✈❡③ ✐♥❞✉③ ✉♠ ✢✉①♦ ϕ f f f ❡♠ M ✱ ❝♦♠ ❝❛♠♣♦ ❛ss♦❝✐❛❞♦ ❛ X ✳ ❖ ✢✉①♦ s❛t✐s❢❛③✿ 1

  ϕ (x, 0) = (x, 1) ∼ (f (x), 0) f ▲♦❣♦✱ f é ❛ ❛♣❧✐❝❛çã♦ ❞❡ P♦✐♥❝❛ré s♦❜r❡ ❛ tr❛♥s✈❡rs❛❧ ❣❧♦❜❛❧ M × {0} ⊂ M ♣❛r❛ t ♦ ✢✉①♦ ϕ f ✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❞❡t❛❧❤❡s s♦❜r❡ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❞✐❢❡♦♠♦r✐✜s♠♦✱ ❡ ❛ ♣r♦✈❛ f ❞❡ q✉❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧✳ Pr♦♣♦s✐çã♦ ✶✳✸✶✳ ✭❙✉s♣❡♥sã♦ ❆♥♦s♦✈✮✳ ❙❡❥❛ f : M → M ✉♠ f t f ❞✐❢❡♦♠♦r✜s♠♦ ❆♥♦s♦✈✱ ❛ s✉s♣❡♥sã♦ ϕ é ✉♠ ✢✉①♦ ❆♥♦s♦✈ ❡♠ M ✳ s u

  E ⊕ ˆ E ❉❡♠♦♥str❛çã♦✳ ❉❡♥♦t❛♠♦s ♣♦r T M = ˆ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❝♦♥tí♥✉❛ ❛ss♦❝✐❛❞❛ ❛ f✳ Pr❡❝✐s❛♠♦s ❝♦♥str✉✐r ✉♠❛ ❞❡❝♦♠♣♦s✐çã♦✿ ss uu

  T M f = E ⊕ T X f ⊕ E

  ✶✵ ✶✳✷✳ ❋▲❯❳❖❙ ❆◆❖❙❖❱

  ❋✐❣✉r❛ ✶✳✺✿ ❙✉s♣❡♥sã♦ ❞❡ ✉♠ ❉✐❢❡♦♠♦r✜s♠♦ ❆♥♦s♦✈ P❛r❛ t❛❧✱ ❞❡✜♥✐♠♦s ♣❛r❛ (x, 0) ∈ M × 0✿ ss s uu u

  E = ˆ E , E = ˆ E (x,0) (x,0) x x ◗✉❛♥❞♦ (x, t) ∈ M × [0, 1]✱ ❞❡✜♥✐♠♦s✿ ss t s

  E = Dϕ (x, 0)( ˆ E ) uu t u (x,t) f x E = Dϕ (x, 0)( ˆ E ) (x,t) f x f ss uu

  = E ⊕ T X f ⊕ E ❆ss✐♠✱ t❡♠♦s ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡s❡❥❛❞❛ T M ✳ ■r❡♠♦s t

  ♠♦str❛r ❛❣♦r❛ q✉❡ ❡st❛ ❞❡❝♦♠♣♦s✐çã♦ s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s ❆♥♦s♦s ♣❛r❛ ϕ f ✳ ss ss s t , t ∈ R = Dϕ (x, 0)( ˆ E )

  ❱❛♠♦s ❛♥❛❧✐s❛r E ✳ P♦r ❞❡✜♥✐çã♦✱ E f x ✱ ❛ss✐♠ −t ss s (x,t ) (x,t ) Dϕ (x, t )(E = ( ˆ E ) f x ✳ ❋✐①❡ t > 0 t −t −t (x,t ) t +t t +t

  Dϕ (x, t ) = Dϕ (x, t ) = Dϕ (0, x).Dϕ (x, t ) f f f f ✳ ▲♦❣♦✱ t t ss +t s Dϕ (x, t )| E = Dϕ (x, 0)| ˆ f (x,t0) f E x

  ❈♦♠♦ ❛ ú❧t✐♠❛ ❡①♣r❡ssã♦ ✈❛❧❡ ♣❛r❛ q✉❛❧q✉❡r t > 0 ✱ ✈❛♠♦s ❛♥❛❧✐s❛r s❡ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❆♥♦s♦✈ ❡♠ M × 0✱ ♦✉ s❡❥❛✱ ♥✉♠ ♣♦♥t♦ ❞❛ ❢♦r♠❛ (x, 0)✳ ❚♦♠❡♠♦s t t = [t]+r(r ∈ [0, 1]) [t] [t]+r r +[t] [t] [ ❡ [.] s✐❣♥✐✜❝❛ ❛ ♣❛rt❡ ✐♥t❡✐r❛✳ ❊♠ M → 0 ♣♦r ❝♦♥str✉çã♦ ❞❡ ϕ f t r

  = f t] (x, 0) = ϕ (x, 0) = ϕ (x, 0) = ϕ (ϕ (x, 0) t❡♠♦s q✉❡ ϕ f ✳ ❈♦♠♦ ϕ f f f f f ✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ✢✉①♦✱ t❡♠♦s✿ t r [t] [t]

  Dϕ (x, 0) = Dϕ (f (x)).Df (x), f f ▲♦❣♦✱ t r [t] [t]

  |Dϕ (x, 0)(v)| ≤ kDϕ (f (x)k.kDf (x)(v)k f f [t] s ≤ K.C.λ |v|, v ∈ ˆ E , ∀ x ∈ M. uu t x ❆♥❛❧♦❣❛♠❡♥t❡✱ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ✈❛❧❡ ♣❛r❛ E ✳ ❙❡❣✉❡ ❡♥tã♦✱ q✉❡ ϕ f é ✉♠ f ✢✉①♦ ❞❡ ❆♥♦s♦✈ ♣❛r❛ M ✱ ❝♦♠♦ q✉❡rí❛♠♦s✳ t (p)

  ✶✶ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ t : R → R t 2 2 ❢✉t✉r❛ ❞❡ ✉♠ ♣♦♥t♦ p ♣❛r❛ ✉♠ ✢✉①♦ ϕ ✳ ❙✉♣♦♥❤❛ q✉❡ ϕ ♣♦ss✉✐ ✉♠ 2

  ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡s✳ ❙❡❥❛♠ p ∈ R ❡ ω(p) ♦ ❝♦♥❥✉♥t♦ ω − limite ❞❡ p ✳ ❊♥tã♦✱ ♦❝♦rr❡ ✉♠❛ ❞❛s ❛❧t❡r♥❛t✐✈❛s✿

  ✶✳ ❙❡ ω(p) ♣♦ss✉✐ s♦♠❡♥t❡ ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s✱ ❡♥tã♦ ω(p) é ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛✳

  ✷✳ ω(p) é ✉♠❛ ór❜✐t❛ ❢❡❝❤❛❞❛✳ ✸✳ ❙❡ ω(p) ♥ã♦ ❝♦♥té♠ ♣♦♥t♦s r❡❣✉❧❛r❡s✱ ❡♥tã♦ ω(p) é ✉♠ ♣♦♥t♦ s✐♥❣✉❧❛r✳

  ❉❡♠♦♥str❛çã♦✳ ❱❡r

  ✶✳✸ ◆♦çõ❡s ❞❡ ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛

  ❖ ♦❜❥❡t✐✈♦ ❞❡st❛ s❡çã♦ é ❞❡✜♥✐r ♦ ♣r✐♠❡✐r♦ ❣r✉♣♦ ❞❡ ❤♦♠♦t♦♣✐❛✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ✐♥❞✐❝❛♠♦s ❛s r❡❢❡rê♥❝✐❛s ❡ ●r✉♣♦s ❞❡ ❍♦♠♦t♦♣✐❛

  ❙❡❥❛♠ X ❡ Y ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✱ f, g : X → Y ❞✉❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s ❡ I = [0, 1]✳ ❉❡✜♥✐çã♦ ✶✳✸✸✳ ❉✐③❡♠♦s q✉❡ f, g : X → Y sã♦ ❤♦♠♦tó♣✐❝❛s s❡ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛

  F : X × I → Y t❛❧ q✉❡ F (x, 0) = f (x) e F (x, 1) = g(x)

  ♣❛r❛ t♦❞♦ x ∈ X✳ ❆ ❛♣❧✐❝❛çã♦ F é ❝❤❛♠❛❞❛ ❞❡ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ❡ g✳ ❊s❝r❡✈❡✲ s❡✱ ♥❡st❡ ❝❛s♦✱ F : f ≃ g✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ f ≃ g✳ ❆ ❤♦♠♦t♦♣✐❛ é ♣❡♥s❛❞❛ ❝♦♠♦ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❞❡❝♦♠♣♦s✐çã♦ ❝♦♥tí♥✉♦ ❞❛

  ❛♣❧✐❝❛çã♦ f ♥❛ ❛♣❧✐❝❛çã♦ g✳ ❖ ♣❛râ♠❡tr♦ t ♣♦❞❡ s❡r ✐♠❛❣✐♥❛❞♦ ❝♦♠♦ s❡♥❞♦ ♦ t❡♠♣♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ ❞❡❢♦r♠❛çã♦✳ Pr♦♣♦s✐çã♦ ✶✳✸✹✳ ❆ r❡❧❛çã♦ ❞❡ ❤♦♠♦t♦♣✐❛✱ f ≃ g✱ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ♥♦ ❝♦♥❥✉♥t♦ ❞❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s ❞❡ X ❡♠ Y ✳ ❉❡♠♦♥str❛çã♦✳ ✶✮ ❘❡✢❡①✐✈❛✳ P❛r❛ t♦❞❛ f : X −→ Y ❝♦♥tí♥✉❛✱ ❛ ❛♣❧✐❝❛çã♦ F : X × I −→ Y

  ✱ ❞❛❞❛ ♣♦r F (x, t) = f(x)✱ é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ❡ f✱ ♣♦rt❛♥t♦✱ ≃ é ✉♠❛ r❡❧❛çã♦ r❡✢❡①✐✈❛✳

  ✷✮ ❙✐♠étr✐❝❛✳ ❉❡✜♥❛♠♦s H : M ×I −→ N ♣♦r H(x, t) = F (x, 1−t)✳ ♦❜t❡♠♦s ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ❡ g✳ ❉❡st❛ ❢♦r♠❛✱ f ≃ g ⇒ g ≃ f✱ ✐st♦ é✱ ❛ r❡❧❛çã♦ ❞❡

  ✶✷ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ ✸✮ ❚r❛♥s✐t✐✈❛✳ ❙❡ F : f ≃ g ❡ H : g ≃ h ❡♥tã♦ ❞❡✜♥❛♠♦s✱ K : X × I −→ Y

  ♣♦♥❞♦ K(x, t) = F (x, 2t) s❡ 0 ≤ t ≤ 1/2 ❡ K(x, t) = H(x, 2t − 1) s❡ 1/2 ≤ t ≤ 1✳ ❆ ❛♣❧✐❝❛çã♦ K é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ❡ h✳ ▲♦❣♦✱ f ≃ g✱ g ≃ h✱ ♦✉ s❡❥❛✱ ❛ r❡❧❛çã♦ ≃ é tr❛♥s✐t✐✈❛✳ ❉❡✜♥✐çã♦ ✶✳✸✺✳ ❆s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ s❡❣✉♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❤♦♠♦t♦♣✐❛ sã♦ ❞❡♥♦♠✐♥❛❞❛s ❝❧❛ss❡s ❞❡ ❤♦♠♦t♦♣✐❛✳ ❯♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ f : X → Y ❝❤❛♠❛✲ s❡ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❤♦♠♦tó♣✐❝❛ q✉❛♥❞♦ ❡①✐st❡ g : Y → X ❝♦♥tí♥✉❛ t❛❧

  = id X = id Y q✉❡ g ◦ f ∼ ❡ f ◦ g ∼ ✳ ❉✐③✲ s❡ ❡♥tã♦✱ q✉❡ g é ♦ ✐♥✈❡rs♦ ❤♦♠♦tó♣✐❝♦ ❞❡ f ❡ q✉❡ ❛s ✈❛r✐❡❞❛❞❡s X ❡ Y t❡♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❤♦♠♦t♦♣✐❛✳ ❊s❝r❡✈❡✲ s❡✱ ♥❡ss❡ ❝❛s♦ f : X ≡ Y ♦✉ X ≡ Y ✳ n ❊①❡♠♣❧♦ ✶✳✸✻✳ ❆ ❡s❢❡r❛ ✉♥✐tár✐❛ S t❡♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❤♦♠♦t♦♣✐❛ q✉❡ R n +1

  − {0} ✳

  ❉❡✜♥✐çã♦ ✶✳✸✼✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ❞✐t♦ ❝♦♥trát✐❧ q✉❛♥❞♦ t❡♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ ✉♠ ♣♦♥t♦✳ Pr♦♣♦s✐çã♦ ✶✳✸✽✳ X é ❝♦♥trát✐❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡ é ❤♦♠♦tó♣✐❝❛ ❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥st❛♥t❡ ❞❡ X✳ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ X é ❝♦♥trát✐❧✳ P♦r ❞❡✜♥✐çã♦✱ X é ❝♦♥trát✐❧ q✉❛♥❞♦

  = {p} ♣♦ss✉í ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ ✉♠ ♣♦♥t♦✱ ✐st♦ é✱ X ∼ ✳ ❊♥tã♦✱ ❡①✐st❡♠ f : X −→ {p} X e g◦f =

  ❡ g : {p} −→ X ✐♥✈❡rs❛s ❤♦♠♦tó♣✐❝❛s✱ ❡♥tã♦ g◦f ≃ id constante ✳ X ≃ constante X

  ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡✱ id ♣♦rt❛♥t♦ id ❡ ❛ ❝♦♥st❛♥t❡ sã♦ = {p}

  ✐♥✈❡rs❛s ❤♦♠♦tó♣✐❝❛s✳ ▲♦❣♦✱ X ∼ ✳ n ≃ {0}

  ❊①❡♠♣❧♦ ✶✳✸✾✳ R ✳ ❈♦r♦❧ár✐♦ ✶✳✹✵✳ ❯♠ ❡s♣❛ç♦ ❝♦♥trát✐❧ X é ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s✳ X

  = c ❉❡♠♦♥str❛çã♦✳ ❋✐①❡♠♦s p ∈ X ❡ ❞❡✜♥❛ ❛ ❤♦♠♦t♦♣✐❛ F : id ✱ ❝♦♠♦ ♥❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✳ ❆ss✐♠✱ ♣❛r❛ ❝❛❞❛ x ∈ X✱ ❞❡✜♥✐♠♦s✿

  α x : I → X t 7−→ F (x, t) ✉♠ ❝❛♠✐♥❤♦ q✉❡ ❧✐❣❛ x ❛ p ❡ ❝♦♥s✐❞❡r❡ X → {p} ❛ ❛♣❧✐❝❛çã♦ ❝♦♥st❛♥t❡✳ ▲♦❣♦✱ X é ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s✳ Pr♦♣♦s✐çã♦ ✶✳✹✶✳ ❙❡ X ♦✉ Y é ❝♦♥trát✐❧ ❡♥tã♦ t♦❞❛ ❛♣❧✐❝❛çã♦ f : X → Y é ❤♦♠♦tó♣✐❝❛ ❛ ✉♠❛ ❝♦♥st❛♥t❡✳ ❈♦r♦❧ár✐♦ ✶✳✹✷✳ ❙❡ X é ❝♦♥trát✐❧ ❡ Y é ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s ❡♥tã♦ ❞✉❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s q✉❛✐sq✉❡r f, g : X → Y sã♦ ❤♦♠♦tó♣✐❝❛s✳ ❙❡ Y é ❝♦♥trát✐❧✱ ❡♥tã♦ q✉❛❧q✉❡r q✉❡ s❡❥❛ X✱ ❞✉❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s f, g : X → Y sã♦ s❡♠♣r❡ ❤♦♠♦tó♣✐❝❛s✳

  ✶✸ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ ❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ♠♦str❛r q✉❡ s❡ X é ❤♦♠♦tó♣✐❝♦ ❡ Y é ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s✱ ❡♥tã♦ q✉❛✐sq✉❡r ❞✉❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s f, g : X → Y sã♦ X = c ∼ ❤♦♠♦tó♣✐❝❛s✳ P♦r ❤✐♣ót❡s❡✱ t❡♠♦s q✉❡ X é ❝♦♥trát✐❧✱ ❛ss✐♠ F : id ✱ ❝♦♠♦ ♥❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ ❞❡✜♥❛ ❛ ❤♦♠♦t♦♣✐❛✿

  H : X × Y → Y H(x, t) = f (F (x, t)) ⇒ H : f ∼

  = c K(x, t) = g(F (x, t)) ⇒ K : g ∼ = c 1

  (x) = g(p) ♦♥❞❡ c(x) = f(p) ❡ c 1 ✱ ♣❛r❛ t♦❞♦ x ∈ X✳ ❈♦♠♦✱ N é ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s✱ ❡①✐st❡ α ❝❛♠✐♥❤♦ ❝♦♥st❛♥t❡ ❧✐❣❛♥❞♦ f(p) a g(p)✱ ❞❡✜♥❛✿

  G : Y × I → Y (y, t) 7−→ α(t)

  = c 1 = g P♦rt❛♥t♦✱ c ∼ ❡ f ∼ ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣❛r❛ ♦ ♦✉tr♦ ❝❛s♦ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳ ❉❡✜♥✐çã♦ ✶✳✹✸✳ ❉❛❞❛s f, g : X −→ Y ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s✱ ❞✐③✲ s❡ q✉❡ f é ❤♦♠♦tó♣✐❝❛ ❛ g r❡❧❛t✐✈❛♠❡♥t❡ ❛ ✉♠ s✉❜❡s♣❛ç♦ A ⊂ X✱ ❡ ❡s❝r❡✈❡✲ s❡ f ≃ g (rel.A) q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❤♦♠♦t♦♣✐❛ F : f ≃ g t❛❧ q✉❡ F (x, t) = f(x) = g(x) ♣❛r❛ t♦❞♦ x ∈ A

  ✳ ❊✈✐❞❡♥t❡♠❡♥t❡✱ ♣❛r❛ q✉❡ s❡ t❡♥❤❛ f ≃ g(rel.A) é ♥❡❝❡ssár✐♦ ❛♥t❡s ❞❡ t✉❞♦✱ q✉❡ f(x) = g(x) ♣❛r❛ t♦❞♦ x ∈ A✳ n − {0}

  ❊①❡♠♣❧♦ ✶✳✹✹✳ ❆ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ R é ❤♦♠♦tó♣✐❝❛ ❛ ❛♣❧✐❝❛çã♦ n n x n −1 r : R − {0} −→ R − {0} ✱ t❛❧ q✉❡ r(x) = ✱ r❡❧❛t✐✈❛♠❡♥t❡ ❛♦ s✉❜❡s♣❛ç♦ S ✳ kxk

  ❖ ●r✉♣♦ ❋✉♥❞❛♠❡♥t❛❧ ❱❛♠♦s✱ ❛ ♣❛rt✐r ❞❡ ❛❣♦r❛✱ ❝♦♥s✐❞❡r❛r ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ❝♦♥❝❡✐t♦ ❣❡r❛❧ ❞❡

  ❤♦♠♦t♦♣✐❛✳ ❊st✉❞❛r❡♠♦s ❤♦♠♦t♦♣✐❛s ❞❡ ❝❛♠✐♥❤♦s✱ ✐st♦ é✱ ❞❡ ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s α : J −→ X

  , s ] 1 ✱ ❞❡✜♥✐❞❛s ♥✉♠ ✐♥t❡r✈❛❧♦ ❝♦♠♣❛❝t♦ J = [s ✳ ❉❡❞✐❝❛r❡♠♦s ❛t❡♥çã♦

  ) = α(s ) ❡s♣❡❝✐❛❧ ❛♦s ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s✿ ❛q✉❡❧❡s ❡♠ q✉❡ α(s 1 ✳ ❙❛❧✈♦ q✉❛♥❞♦ ❞✐t♦ ❡①♣❧✐❝✐t❛♠❡♥t❡ ♦ ❝♦♥trár✐♦✱ s✉♣♦r❡♠♦s ♥♦ss♦s ❝❛♠✐♥❤♦s ❞❡✜♥✐❞♦s ♥♦ ✐♥t❡r✈❛❧♦ ✉♥✐tár✐♦ I = [0, 1]✳

  ❈♦♠♦ ♦ ✐♥t❡r✈❛❧♦ I é ❝♦♥trát✐❧✱ t♦❞♦ ❝❛♠✐♥❤♦ α → X é ❤♦♠♦tó♣✐❝♦ ❛ ✉♠❛ ❝♦♥st❛♥t❡✳ ❊①✐❣✐r❡♠♦s q✉❡✱ ❞✉r❛♥t❡ ❛ ❤♦♠♦t♦♣✐❛✱ ♦s ❡①tr❡♠♦s ❞♦ ❝❛♠✐♥❤♦ s❡❥❛♠ ♠❛♥t✐❞♦s ✜①♦s✳ ❆ss✐♠✱ ❝♦♥s✐❞❡r❛r❡♠♦s ❛ ❢r♦♥t❡✐r❛ ∂I = {0, 1} ❞♦ ✐♥t❡r✈❛❧♦ ❡ ❛s ❤♦♠♦t♦♣✐❛s ❞❡ ❝❛♠✐♥❤♦s s❡rã♦ t♦❞❛s r❡❧❛t✐✈❛s ❛♦ s✉❜❡s♣❛ç♦ ∂I✳ ❉❡✜♥✐çã♦ ✶✳✹✺✳ ❉✐r❡♠♦s q✉❡✱ α, β : I −→ X sã♦ ❝❛♠✐♥❤♦s ❤♦♠♦tó♣✐❝♦s q✉❛♥❞♦ t✐✈❡r♠♦s α ≃ β (rel.∂I)✳ ❆❜r❡✈✐❛r❡♠♦s ❡st❛ ❛✜r♠❛çã♦ ❝♦♠ ❛ ♥♦t❛çã♦ α ∼ = β = β

  ✳ ❆ss✐♠✱ ✉♠❛ ❤♦♠♦t♦♣✐❛ F : α ∼ ❡♥tr❡ ❝❛♠✐♥❤♦s é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ F : I × I −→ X t❛❧ q✉❡✱

  ✶✹ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ F (s, 0) = α(s), F (s, 1) = β(s),

  F (0, t) = α(0) = β(0), F (1, t) = α(1) = β(1), q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ s, t ∈ I✳

  P❛r❛ q✉❡ s❡ t❡♥❤❛ s❡♥t✐❞♦ ❛✜r♠❛r q✉❡ α ≃ β✱ é ♥❡❝❡ssár✐♦✱ ❛♥t❡s ❞❡ ♠❛✐s ♥❛❞❛✱ q✉❡ ♦s ❝❛♠✐♥❤♦s α ❡ β ♣♦ss✉❛♠ ❛s ♠❡s♠❛s ❡①tr❡♠✐❞❛❞❡s✿

  α(0) = α(1) = x e α(1) = β(1) = x 1 ❋✐❣✉r❛ ✶✳✻✿ ❍♦♠♦t♦♣✐❛ ❡♥tr❡ ❝❛♠✐♥❤♦s

  ❖s ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s α, β : I → X sã♦ ❤♦♠♦tó♣✐❝♦s ✭α ≃ β✮ q✉❛♥❞♦ ❡①✐st❡ ∈ M

  ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ F : I × I → X t❛❧ q✉❡✱ ♣♦♥❞♦ α(0) = α(1) = x ✱ t❡♠✲ s❡ F (s, 0) = α(s)

  ✱ F (s, 1) = β(s)✱ F (0, t) = F (1, t) = x ♣❛r❛ q✉❛✐sq✉❡r s, t ∈ I✳

  ❋✐❣✉r❛ ✶✳✼✿ ❈❛♠✐♥❤♦s ❍♦♠♦tó♣✐❝♦s ❉❡✜♥✐çã♦ ✶✳✹✻✳ ❉♦✐s ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s α, β : I −→ X ❞✐③❡♠✲ s❡ ❧✐✈r❡♠❡♥t❡ ❤♦♠♦tó♣✐❝♦s q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ F : I × I −→ X t❛❧ q✉❡ F (s, 0) = α(s), F (s, 1) = β(s)

  ❡ F (0, t) = F (1, t) ♣❛r❛ q✉❛✐sq✉❡r s, t ∈ I✳ ❆ t : I −→ X, F t (s) = ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s✐❣♥✐✜❝❛ q✉❡✱ ♣❛r❛ t♦❞♦ t ∈ I✱ ♦ ❝❛♠✐♥❤♦ F F (s, t)

  ✶✺ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ ❖❜s❡r✈❛çã♦ ✶✳✹✼✳ ■♥❞✐❝❛r❡♠♦s s❡♠♣r❡✱ ❝♦♠ α = [α] ❛ ❝❧❛ss❡ ❞❡ ❤♦♠♦t♦♣✐❛ ❞♦ ❝❛♠✐♥❤♦ α : I −→ X✱ ✐st♦ é✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s ❡♠ X q✉❡ ♣♦ss✉❡♠ ❛s ♠❡s♠❛s ❡①tr❡♠✐❞❛❞❡s q✉❡ α ❡ q✉❡ sã♦ ❤♦♠♦tó♣✐❝♦s ❛ α ❝♦♠ ❡①tr❡♠♦s ✜①♦s ❞✉r❛♥t❡ ❛ ❤♦♠♦t♦♣✐❛✳ ❉❡✜♥✐çã♦ ✶✳✹✽✳ ❙❡❥❛♠ α, β : I −→ X ❝❛♠✐♥❤♦s t❛✐s q✉❡ ♦ ✜♠ ❞❡ α ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ♦r✐❣❡♠ ❞❡ β✳ ❉❡✜♥✐r❡♠♦s ♦ ♣r♦❞✉t♦ α ∗ β ❝♦♠♦ s❡♥❞♦ ♦ ❝❛♠✐♥❤♦ q✉❡ ❝♦♥s✐st❡ ❡♠ ♣❡r❝♦rr❡r ♣r✐♠❡✐r♦ α ❡ ❞❡♣♦✐s β✳ ❈♦♠♦ ♦ t❡♠♣♦ q✉❡ ❞✐s♣♦♠♦s ♣❛r❛ ♣❡r❝♦rr❡r αβ = 1✱ ✐st♦ ♥♦s ♦❜r✐❣❛ ❛ ❞♦❜r❛r ❛ ✈❡❧♦❝✐❞❛❞❡ ❡♠ α ❡ ❡♠ β✳ ❆ss✐♠✱ ❞❡✜♥✐r❡♠♦s αβ : I −→ X ♣♦♥❞♦✿

  α(2s) s❡ 0 ≤ s ≤ 1/2, α ∗ β(s) =

  β(2s − 1) s❡ 1/2 ≤ s ≤ 1 ❈♦♠♦ α(1) = β(0)✱ ❛s r❡❣r❛s ❛❝✐♠❛ ❜❡♠ ❞❡✜♥❡♠ ✉♠❛ ❛♣❧✐❝❛çã♦ αβ : I −→ X t❛❧ q✉❡ αβ | [0, 1/2] e αβ | [1/2, 1] sã♦ ❝♦♥tí♥✉❛s✳ ❙❡q✉❡✲ s❡ q✉❡ αβ é ❝♦♥tí♥✉❛ ❡ ♣♦rt❛♥t♦ é ✉♠ ❝❛♠✐♥❤♦✱ q✉❡ ❝♦♠❡ç❛ ❡♠ α(0) ❡ t❡r♠✐♥❛ ❡♠ β(1)✳ ❉❡✜♥✐çã♦ ✶✳✹✾✳ ❖ ❝❛♠✐♥❤♦ ✐♥✈❡rs♦ ❞❡ α : I −→ X é✱ ♣♦r ❞❡✜♥✐çã♦✱ ♦ −1 −1 : I −→ X (s) = α(1 − s), 0 ≤ s ≤ 1. ❝❛♠✐♥❤♦ α ✱ ❞❛❞♦ ♣♦r α −1

  (s) ❖❜s❡r✈❛çã♦ ✶✳✺✵✳ ❱❡♠♦s q✉❡ α ❝♦♠❡ç❛ ♦♥❞❡ α t❡r♠✐♥❛✱ ❡ t❡r♠✐♥❛ ♥❛ ♦r✐❣❡♠ ❞❡ α✳ ′ ′ ′ ′ −1 ′ −1

  ∼ ⇒ αβ ∼

  = α , β ∼ = β = α β e α = (α ) Pr♦♣♦s✐çã♦ ✶✳✺✶✳ α ∼ ′ ′

  = α e G : β ∼ = β ❉❡♠♦♥str❛çã♦✳ ❙❡ F : α ∼ sã♦ ❤♦♠♦t♦♣✐❛s✱ ❞❡✜♥✐♠♦s H : I × I −→ X

  ♣♦♥❞♦✿ F (2s, t), s❡ 0 ≤ s ≤ 1/2, t ∈ I

  H(s, t) = G(2s − 1, t), s❡ 1/2 ≤ s ≤ 1t ∈ I

  ❈♦♠♦ F (1, t) = G(0, t) = α(1) = β(0) ♣❛r❛ t♦❞♦ t ∈ I✱ ✈❡♠♦s q✉❡ H é ❜❡♠ ❞❡✜♥✐❞❛✳ ❈♦♠♦ H | ([0, 1/2] × I) e H | ([1/2, 1] × I) sã♦ ❝♦♥tí♥✉❛s✱ s❡q✉❡ q✉❡ H é ❝♦♥tí♥✉❛✳ ❉❛í✿ H(s, 0) = F (2s, 0) = α(s), s ∈ [0, 1/2]✱ H(s, 0) = G(2s − 1, 0) = β(s), s ∈ [0, 1/2] ′ ′ ✳ P♦rt❛♥t♦✱ H(s, 0) = αβ(s)✳ ❙❡❣✉✐♥❞♦

  β (s) ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦✱ ♦❜t❡♠♦s H(s, 1) = α ✳ ❆❧é♠ ❞✐ss♦✱ H(0, t) = F (0, t) = ′ ′ ′ ′ α (0) = α(0) = β(0) = β (0) = αβ(0) = α β (0), ❊✱ ′ ′ ′ ′ H(1, t) = F (1, t) = α (1) = α(1) = β(1) = β (1) = αβ(1) = α β (1).

  ▲♦❣♦✱ ′ ′ H(0, t) = αβ(0) = α β (0) ′ ′ H(1, t) = αβ(1) = α β (1) ′ ′

  β ❉❡st❛ ❢♦r♠❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ H é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ αβ e α ✳ ■st♦ é✱ ′ ′ αβ ∼ β

  = α ✳ ❈♦♥s✐❞❡r❡ L : I × I −→ M✱ ❞❡✜♥✐❞❛ ♣♦r L(s, t) = F (1 − s, t)✳ ❊♥tã♦✱ −1

  L(s, 0) = F (1 − s, 0) = α ′ −1 L(s, 1) = F (1 − s, 1) = (α ) −1 ′ −1

  L(0, t) = F (1, t) = α(1) = α (0) = (α ) (0)

  ✶✻ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ −1 ′ −1 L(1, t) = F (0, t) = α(0) = α (1) = (α ) (1) −1 ′ −1

  ∼ = (α )

  ❙❡♥❞♦ L ❝♦♥tí♥✉❛✱ s❡❣✉❡ ❡♥tã♦ q✉❡ L : α ✱ ❝♦♠♦ q✉❡rí❛♠♦s✳ ❉❡✜♥✐çã♦ ✶✳✺✷✳ ❊♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ X✱ s❡❥❛♠ α ✉♠❛ ❝❧❛ss❡ ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ ❝❛♠✐♥❤♦s q✉❡ t❡♠ ♦r✐❣❡♠ ♥✉♠ ♣♦♥t♦ x ∈ X ❡ t❡r♠✐♥❛♠ ♥✉♠ ♣♦♥t♦ y ∈ X✱ ❡ β ✉♠❛ ❝❧❛ss❡ ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ ❝❛♠✐♥❤♦s q✉❡ ❝♦♠❡ç❛♠ ❡♠ y ∈ X ❡ t❡r♠✐♥❛♠ ❡♠ z ∈ X

  ✳ ❉❡✜♥✐r❡♠♦s ♦ ♣r♦❞✉t♦ αβ t♦♠❛♥❞♦ ❝❛♠✐♥❤♦s a ∈ α, b ∈ β ❡ ♣♦♥❞♦ αβ = [ab]

  ❆ss✐♠✱ ♣♦r ❞❡✜♥✐çã♦ [a][b] = [ab]✳ −1 −1 = [a ]

  ❉❡✜♥✐♠♦s t❛♠❜é♠✱ α ✱ ♦♥❞❡ a ∈ α✳ ❖❜s❡r✈❛çã♦ ✶✳✺✸✳ ❆ ♣r♦♣♦s✐çã♦ ♠♦str❛ q✉❡ ♦ ♣r♦❞✉t♦ αβ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛s ❡s❝♦❧❤❛s ❞♦s ❝❛♠✐♥❤♦s γ ∈ α e ρ ∈ β✱ ♣♦rt❛♥t♦ ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳ ❊✱ ❛ s❡❣✉♥❞❛ −1 −1

  = [γ ] ♣❛rt❡ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ ♠♦str❛ q✉❡ ❛ ❝❧❛ss❡ ❞❡ ❤♦♠♦t♦♣✐❛ α é ❛ −1 ♠❡s♠❛✱ s❡❥❛ q✉❛❧ ❢♦r ♦ ❝❛♠✐♥❤♦ γ q✉❡ ❡s❝♦❧❤❡♠♦s ❡♠ α✳ ❆ ❝❧❛ss❡ α é ❝❤❛♠❛❞❛ ✐♥✈❡rs❛ ❞❡ α✳ ❉❡✜♥✐çã♦ ✶✳✺✹✳ ❙❡❥❛♠ α : I −→ X ✉♠ ❝❛♠✐♥❤♦ ❡ ϕ : I −→ I ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❡ I✱ ✐st♦ é✱ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ t❛❧ q✉❡ ϕ(∂I) ⊂ ∂I✳ ❖ ❝❛♠✐♥❤♦ β = α ◦ ϕ : I −→ X ❝❤❛♠❛✲ s❡ r❡♣❛r❛♠❡tr✐③❛çã♦ ❞♦ ❝❛♠✐♥❤♦ α

  ✳ ❆ ♣❛r❛♠❡tr✐③❛çã♦ ϕ ❞✐③✲ s❡ ♣♦s✐t✐✈❛ q✉❛♥❞♦ ϕ(0) = 0 e ϕ(1) = 1, s❡ ❢♦r ♥❡❣❛t✐✈❛ q✉❛♥❞♦ ϕ(0) = 1 e ϕ(1) = 0✱ ❡ tr✐✈✐❛❧ q✉❛♥❞♦ ϕ(0) = ϕ(1)✳ Pr♦♣♦s✐çã♦ ✶✳✺✺✳ ❙❡❥❛ β = α◦ϕ ✉♠❛ r❡♣❛r❛♠❡tr✐③❛çã♦ ❞♦ ❝❛♠✐♥❤♦ α : I −→ X✳ −1

  = α = α ❙❡ ❛ ♣❛r❛♠❡tr✐③❛çã♦ ϕ ❢♦r ♣♦s✐t✐✈❛✱ ❡♥tã♦ β ∼ ❀ s❡ ❢♦r ♥❡❣❛t✐✈❛✱ t❡♠✲ s❡ β ∼ ❀ s❡ ❢♦r tr✐✈✐❛❧ ❡♥tã♦ β = constante✳ ❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ ❞♦✐s ❝❛♠✐♥❤♦s ❡♠ I sã♦ ❤♦♠♦tó♣✐❝♦s ✭❝♦♠ ❡①tr❡♠✐❞❛❞❡s ✜①❛s✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ tê♠ ❛ ♠❡s♠❛ ♦r✐❣❡♠ ❡ ♦ ♠❡s♠♦ ✜♠✳ ❙❡❥❛♠ i, j : I → I

  = i, ϕ ∼ = j ou ϕ ∼ = ❞❛❞❛s ♣♦r i(s) = s e j(s) = 1 − s✳ ❚❡♠✲ s❡ ♣♦rt❛♥t♦ ϕ ∼ constante

  ✱ ❝♦♥❢♦r♠❡ s❡❥❛ ϕ ✉♠❛ r❡♣❛r❛♠❡tr✐③❛çã♦ ♣♦s✐t✐✈❛✱ ♥❡❣❛t✐✈❛ ♦✉ tr✐✈✐❛❧✳ −1 = α ◦ i = α, α ◦ ϕ ∼ = α ◦ j = α ou α ◦ ϕ ∼ =

  ❙❡❣✉❡✲ s❡ ✐♠❡❞✐❛t❛♠❡♥t❡ q✉❡ α ◦ ϕ ∼ constante ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  ❙✉♣♦♥❤❛♠♦s q✉❡ ❞❛❞♦s ✉♠ ❝❛♠✐♥❤♦ α : I → X ❡ ♣♦♥t♦s ✐♥t❡r♠❡❞✐ár✐♦s 0 = s < s < ... < s k = 1 1 i , s i ] ✱ q✉❡ ❞❡t❡r♠✐♥❛♠ ✉♠❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ I ❡♠ k s✉❜✐♥t❡r✈❛❧♦s ❥✉st❛♣♦st♦s [s −1 ✳ P❛r❛ ❝❛❞❛ i = 1, 2, ..., k✱ ♦❜t❡♠♦s ✉♠ ❝❛♠✐♥❤♦ α i : I → X i = (α | [s i , s i ]) ◦ ϕ i i : I → [s i , s i ]

  ✱ ❞❡✜♥✐❞♦ ♣♦r α ✱ ♦♥❞❡ ϕ é ❛ ú♥✐❝❛ i e ϕ i (1) = s i i −1 −1 ❢✉♥çã♦ ❧✐♥❡❛r t❛❧ q✉❡ ϕ(0) = s −1 ✳ ❖s ❝❛♠✐♥❤♦s α sã♦ ♣♦rt❛♥t♦ i , s i ] r❡♣❛r❛♠❡tr✐③❛çõ❡s ❞❛s r❡str✐çõ❡s α | [s −1 ✱ ❛ ✜♠ ❞❡ ❧❤❡s ❞❛r ♦ ❞♦♠í♥✐♦ ♣❛❞rã♦

  I ✳

  Pr♦♣♦s✐çã♦ ✶✳✺✻✳ ❙❡❥❛♠ α, β, µ : I −→ X ❝❛♠✐♥❤♦s t❛✐s q✉❡ ❝❛❞❛ ✉♠ ❞❡❧❡s t❡r♠✐♥❛ ♦♥❞❡ ♦ s❡❣✉✐♥t❡ ❝♦♠❡ç❛✳ ❙❡❥❛♠ α = [α], β = [β], γ = [µ] s✉❛s x y , e ❝❧❛ss❡s ❞❡ ❤♦♠♦t♦♣✐❛✱ x = α(0) ❛ ♦r✐❣❡♠ ❞❡ α✱ y = α(1) s❡✉ ✜♠✱ e ♦s x = [e x ], ε y = [e y ] ❝❛♠✐♥❤♦s ❝♦♥st❛♥t❡s s♦❜r❡ ❡ss❡s ♣♦♥t♦s ❡ ε ❛s ❝❧❛ss❡s ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ss❛s ❝♦♥st❛♥t❡s✳ ❚❡♠✲ s❡ ❡♥tã♦✿

  ✶✼ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ −1 = ε x

  ✶✳ αα −1 α = ε y

  ✷✳ α x α = α = αε y ✸✳ ε ✹✳ (αβ)γ = α(βγ)

  ❉❡✜♥✐çã♦ ✶✳✺✼✳ ❖ ❝♦♥❥✉♥t♦ ❞❛s ❝❧❛ss❡s ❞❡ ❤♦♠♦t♦♣✐❛ ✭❝♦♠ ❡①tr❡♠♦s ✜①♦s✮ ❞♦s ❝❛♠✐♥❤♦s ♥✉♠❛ ✈❛r✐❡❞❛❞❡ X✱ ♠✉♥✐❞♦ ❞❛ ❧❡✐ ❞❡ ❝♦♠♣♦s✐çã♦ ❛❝✐♠❛ ❞❡✜♥✐❞❛✱ ❝❤❛♠❛✲ s❡ ♦ ❣r✉♣♦✐❞❡ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ X✳

  ) ∈ X ❉❡✜♥✐çã♦ ✶✳✺✽✳ ❈♦♥s✐❞❡r❛r❡♠♦s ♣❛r❡s ❞♦ t✐♣♦ (X, x ✱ ♦♥❞❡ x s❡rá ❝❤❛♠❛❞♦ ♦ ♣♦♥t♦ ❜ás✐❝♦ ❞❛ ✈❛r✐❡❞❛❞❡ X✳ ❖s ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s a : (I, ∂I) −→ (X, x ) s❡rã♦ ❝❤❛♠❛❞♦s ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s ❝♦♠ ❜❛s❡ ♥♦ ♣♦♥t♦ x ✳ ❆s ❤♦♠♦t♦♣✐❛s ✭s❛❧✈♦s q✉❛♥❞♦ ❡①♣❧✐❝✐t❛♠❡♥t❡ ♠❡♥❝✐♦♥❛❞♦ ♦ ❝♦♥trár✐♦✮ s❡rã♦ r❡❧❛t✐✈❛s ❛ ∂I✳ 1 (X, x ) ❖ s✉❜❝♦♥❥✉♥t♦ π ❞♦ ❣r✉♣♦✐❞❡ ❢✉♥❞❛♠❡♥t❛❧✱ ❢♦r♠❛❞♦ ♣❡❧❛s ❝❧❛ss❡s ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s ❝♦♠ ❜❛s❡ ❡♠ x ❝♦♥st✐t✉✐ ✉♠ ❣r✉♣♦✱ ❝❤❛♠❛❞♦ ♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ✈❛r✐❡❞❛❞❡ X ❝♦♠ ❜❛s❡ ♥♦ ♣♦♥t♦ x ✳ ❖ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ x ❞❡ss❡ ❣r✉♣♦ é ❛ ❝❧❛ss❡ ❞❡ ❤♦♠♦t♦♣✐❛ ε = ε ❞♦ ❝❛♠✐♥❤♦ ❝♦♥st❛♥t❡ ♥♦ ♣♦♥t♦ x ✳ 1 (X, x ) = {e}

  ◗✉❛♥❞♦ M é ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s ❡ π ❞✐③❡♠♦s q✉❡ X é s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦✳ ❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ♠♦str❛ q✉❡ ♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ✐♥❞❡♣❡♥❞❡ ❞♦ ♣♦♥t♦ ❜❛s❡ ❡s❝♦❧❤✐❞♦✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ♦ ❣r✉♣♦ 1 (X)

  ❢✉♥❞❛♠❡♥t❛❧ ♣♦r π ✱ s❡♠ ♥❡♥❤✉♠❛ r❡❢❡rê♥❝✐❛ ❡①♣❧í❝✐t❛ ❛♦ ♣♦♥t♦ ❜❛s❡✳ ❊①❡♠♣❧♦ ✶✳✺✾✳ 1. ❙❡ X é ❝♦♥trát✐❧ ❡♥tã♦ X é s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦✳ n 2.

  ❙❡ n > 1✱ ❡♥tã♦ S é s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦✳ ❱❡r = id X

  ❖❜s❡r✈❛çã♦ ✶✳✻✵✳ ❙❡ f : X → Y e g : Y → X s❛t✐s❢❛③❡♠ g ◦ f ∼ ❡♥tã♦ f = id Y é ✐♥❥❡t✐✈❛ ❡ g s♦❜r❡❥❡t✐✈❛✳ ❙❡ f ◦ g ∼ ❡♥tã♦ g é ✐♥❥❡t✐✈❛ ❡ f é s♦❜r❡❥❡t✐✈❛✳

  , x ∈ X 1 Pr♦♣♦s✐çã♦ ✶✳✻✶✳ ❙❡❥❛ X ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s✳ ❉❛❞♦s x ❡ γ : I → X e γ(1) = x t❛❧ q✉❡ γ(0) = x 1 ✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡

  π (X, x ) e π (X, x ) 1 1 1 ✳ e γ(1) = x 1

  ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ γ : I → X ❝❛♠✐♥❤♦ t❛❧ q✉❡ γ(0) = x ✳ −1 ∗ α ∗ γ 1

  ❙❡ α é ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ ❞❡ x ❡♥tã♦ γ é ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ ❞❡ x ✳ 1 (X, x ) ❉❛❞♦ ✉♠ ❡❧❡♠❡♥t♦ [α] ∈ π ✱ ❛♣❧✐❝❛çã♦ ❛❝✐♠❛ ✐♥❞✉③ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ′ −1 [α ] = [γ ∗ α ∗ γ] ∈ π (X, x ) 1 1

  ✳ ❉❡♥♦t❛r❡♠♦s ❡ss❛ ❛♣❧✐❝❛çã♦ ♣♦r✿ η : π (X, x ) → π (X, x ) 1 1 1

  [α ] = η([α]) ❱❛♠♦s ♠♦str❛r q✉❡ η é ✉♠ ✐s♦♠♦r✜s♠♦✳

  ✶✮ η é ✉♠ ❤♦♠♦♠♦r✜s♠♦✳ 1 (X, x ) ❈♦♠ ❡❢❡✐t♦✱ ❞❛❞♦s [α], [β] ∈ π ✱ t❡♠♦s✿ −1

  η([α] ∗ [β]) = [γ ] ∗ [α] ∗ [β] ∗ [γ]

  ✶✽ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ −1 −1 = [γ ] ∗ [α] ∗ [γ] ∗ [γ ] ∗ [β] ∗ [γ]

  = η([α]) ∗ η([β]) ❆ss✐♠✱ η é ✉♠ ❤♦♠♦♠♦r✜s♠♦✳

  ✷✮ η é ❜✐❥❡t✐✈❛✳ P❛r❛ ♠♦str❛r q✉❡ η é ❜✐❥❡t✐✈❛✱ ✈❛♠♦s ✐♥tr♦❞✉③✐r ❛ ✐♥✈❡rs❛ ❞❡ η

  ✳ ❙❡❥❛✿ −1 η : π (X, x ) → π (X, x ) 1 1 1 t❛❧ q✉❡ −1 ′ ′ −1 −1 η ([α ]) = [γ ∗ α ∗ γ ]

  ❆✜r♠❛çã♦✿ η é ❛ ✐♥✈❡rs❛ ❞❡ η✳ ❉❡ ❢❛t♦✱ −1 −1 −1

  η η([α]) = η ([γ ∗ α ∗ γ]) −1 −1 ∗ α ∗ γ ∗ γ

  = [γ ∗ γ ] = [α]. ❆ss✐♠✱ −1

  η η = id π 1 (X,x ) ❡ ♣♦r s✐♠❡tr✐❛ −1

  ηη = id π (X,x ) 1 1 P❡❧❛ ú❧t✐♠❛ ♦❜s❡r✈❛çã♦ η é ✐♥❥❡t✐✈❛ ❡ s♦❜r❡❥❡t✐✈❛✳ ❙❡❣✉❡ ❡♥tã♦ q✉❡✱η é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❜✐❥❡t♦r✱ ♣♦rt❛♥t♦✱ ✉♠ ✐s♦♠♦r✜s♠♦✳ ❈♦♠♦ q✉❡rí❛♠♦s✳ ❊s♣❛ç♦s ❞❡ ❘❡❝♦❜r✐♠❡♥t♦ ❉❡✜♥✐çã♦ ✶✳✻✷✳ ❙❡❥❛♠ X ❡ Y ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❯♠❛ ❛♣❧✐❝❛çã♦ f : X → Y é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❧♦❝❛❧ q✉❛♥❞♦ ♣❛r❛ ❝❛❞❛ ♣♦♥t♦ x ∈ X ❡①✐st❡ ✉♠ ❛❜❡rt♦ U U ❝♦♥t❡♥❞♦ x ❡♠ X t❛❧ q✉❡ V = f(U) é ❛❜❡rt♦ ❡♠ Y ❡ f| é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❞❡ U s♦❜r❡ V ✳ ❊①❡♠♣❧♦ ✶✳✻✸✳ ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❛♣❧✐❝❛çã♦✿ 1

  ξ : R −→ S it t 7−→ exp(t) = e = (cost, sent) ξ

  é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❧♦❝❛❧✳ ❉❡✜♥✐çã♦ ✶✳✻✹✳ ❙❡❥❛♠ f : X → Y ✱ g : Z → Y ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s✳ ❯♠ ❧❡✈❛♥t❛♠❡♥t♦ ❞❡ g ✭r❡❧❛t✐✈❛♠❡♥t❡ ❛ f✮ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ eg : Z → X t❛❧ q✉❡ f ◦ eg = g✳

  X → X

  X ❉❡✜♥✐çã♦ ✶✳✻✺✳ ❙❡❥❛ p : e ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✱ ˜ ❡ X ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❯♠ ❛❜❡rt♦ V ⊂ X é ❞✐t♦ ✈✐③✐♥❤❛♥ç❛ ❞✐st✐♥❣✉✐❞❛ s❡✿ −1 [ p (V ) = U α ,

  ✶✾ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ α |α ∈ Γ}

  X ♦♥❞❡ {U é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❛❜❡rt♦s ❞❡ ˜ ✱ ❞♦✐s ❛ ❞♦✐s ❞✐s❥✉♥t♦s✱ t❛✐s q✉❡✿ p| U α : U α → V

  é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✳ ❉❡✜♥✐çã♦ ✶✳✻✻✳ ❯♠❛ ❛♣❧✐❝❛çã♦ ❞❡ r❡❝♦❜r✐♠❡♥t♦ ✭♦✉ s✐♠♣❧❡s♠❡♥t❡ ✉♠ r❡❝♦❜r✐♠❡♥t♦✮ é ✉♠❛ ❢✉♥çã♦✿ p : e X → X

  ❝♦♥tí♥✉❛✱ s♦❜r❡❥❡t✐✈❛ ❡ t❛❧ q✉❡ t♦❞♦ x ∈ X ♣♦ss✉✐ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞✐st✐♥❣✉✐❞❛✳

  X ❖ ❡s♣❛ç♦ ˜ ❝❤❛♠❛✲s❡ ✉♠ ❡s♣❛ç♦ ❞❡ r❡❝♦❜r✐♠❡♥t♦ ❞❡ X ❡✱ ♣❛r❛ ❝❛❞❛ x ∈ X✱ ♦ −1

  (x) ❝♦♥❥✉♥t♦ p ❝❤❛♠❛✲s❡ ❛ ✜❜r❛ s♦❜r❡ x✳ ➚s ✈❡③❡s✱ X é ❝❤❛♠❛❞♦ ❞❡ ❜❛s❡ ❞♦ r❡❝♦❜r✐♠❡♥t♦✳ ❱❡❥❛

  ❋✐❣✉r❛ ✶✳✽✿ ❆♣❧✐❝❛çã♦ ❘❡❝♦❜r✐♠❡♥t♦ 1 ❊①❡♠♣❧♦ ✶✳✻✼✳ ❆ ❛♣❧✐❝❛çã♦ exp : R → S ❞❛❞❛ ♣♦r✿ 2πit exp(t) = e = (cos(2πt), sen(2πt)) é ✉♠ r❡❝♦❜r✐♠❡♥t♦✳ 1 2 1 1

  × R → T = S × S ❊①❡♠♣❧♦ ✶✳✻✽✳ p : S ❞❛❞❛ ♣♦r✿ 2 p(z, t) = (z, exp(t))

  é ✉♠ r❡❝♦❜r✐♠❡♥t♦ ❞♦ t♦r♦ T ✳ ❊①❡♠♣❧♦ ✶✳✻✾✳ ❙❡❥❛ p : R → [0, +∞) t❛❧ q✉❡ p(x) = |x| ♥ã♦ é ✉♠ r❡❝♦❜r✐♠❡♥t♦✳

  X → X ❉❡✜♥✐çã♦ ✶✳✼✵✳ ❉✐③❡♠♦s q✉❡ ♦ r❡❝♦❜r✐♠❡♥t♦ p : e é ✉♠r❡❝♦❜r✐♠❡♥t♦

  X

  ✷✵ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆

  X → X Pr♦♣♦s✐çã♦ ✶✳✼✶✳ ❙❡❥❛♠ p : e ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ r❡❝♦❜r✐♠❡♥t♦ ❡ Z ✉♠

  X ❡s♣❛ç♦ ❝♦♥❡①♦✳ ❉❛❞♦s ❞♦✐s ❧❡✈❛♥t❛♠❡♥t♦s eg, bg : Z → e ❞❡ g : Z → X✱ ♦✉ s❡❥❛✱ p ◦ eg = p ◦ ˆg = g ❡♥tã♦ ♦✉ eg(z) 6= bg(z)✱ ∀z ∈ Z ♦✉ eg = ˆg✳ e g, g e b // ??

  X p Z g

  X ❉❡♠♦♥str❛çã♦✳ ✶✮ ❙❡ ♥ã♦ ❡①✐st❡ z ∈ Z t❛❧ q✉❡ eg(z) = bg(z)✱ ❡♥tã♦ eg(z) 6= bg(z)✱ ∀z ∈ Z

  ✱ ❛ss✐♠ t❡♠♦s ♦ r❡s✉❧t❛❞♦✳ ∈ Z ) = ˆ g(z )

  ✷✮ ❙❡ ❡①✐st❡ z t❛❧ q✉❡ ˜g(z ✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦✿ A = {z ∈ Z|˜ g(z) = ˆ g(z)}

  ∈ A A 6= ∅

  ✱ ♣♦✐s z ✳ M → M

  ✷✳✶✮ A é ❛❜❡rt♦✳ ❉❡ ❢❛t♦✱ s❡❥❛ z ∈ A ❡♥tã♦ ˜g(z) = ˆg(z)✳ ❙❡♥❞♦ p : ˜ ✉♠

  X r❡❝♦❜r✐♠❡♥t♦✱ ❡♥tã♦ p é ❤♦♠❡♦♠♦r✜s♠♦ ❧♦❝❛❧✱ ❛ss✐♠ ❡①✐st❡ U ⊂ ˜ ❛❜❡rt♦ ❝♦♠ g(z) = ˆ ˜ g(z) ∈ U U t❛❧ q✉❡ p| é ❤♦♠❡♦♠♦r✜s♠♦ s♦❜r❡ s✉❛ ✐♠❛❣❡♠✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ p| U

  é ✐♥❥❡t♦r❛✳ P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ˜g ❡ ˆg ❡①✐st❡ ✉♠ ❛❜❡rt♦ W ⊂ Z✱ z ∈ W t❛❧ q✉❡ ˜g(W ) ⊂ U ′ ′ ′ ′

  ∈ W ) = g(z ) = p ◦ ˆ g(z ) ❡ ˆg(W ) ⊂ U✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ z t❡♠♦s p ◦ ˜g(z ✳ ❈♦♠♦ ′ ′ ′ p| U

  ) = ˆ g(z ) ∈ A é ✐♥❥❡t♦r❛ ❡ ˜g(W ) ∪ ˆg(W ) ⊂ U s❡❣✉❡ q✉❡ ˜g(z ✳ ▲♦❣♦✱ z ❡✱

  ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ z ∈ W ⊂ A✳ P♦rt❛♥t♦✱ A é ❛❜❡rt♦✳ ✷✳✷✮ A é ❢❡❝❤❛❞♦✳ ❱❛♠♦s ♠♦str❛r q✉❡ Z − A é ❛❜❡rt♦✱ ❛ss✐♠ t❡r❡♠♦s q✉❡ A é

  ❢❡❝❤❛❞♦✳ ❙❡❥❛ z ∈ Z − A✱ ❡♥tã♦ ˜g(z) 6= ˆg(z) ❡✱ p ◦ ˜g(z) = g(z) = p ◦ ˆg(z)✱ ❝♦♠♦ p : e X → X é ✉♠ r❡❝♦❜r✐♠❡♥t♦ ❡ g(z) ∈ X✱ s❡❣✉❡ q✉❡ g(z) ♣♦ss✉✐ ✉♠❛ ✈✐③✐♥❤❛♥ç❛

  ❞✐st✐♥❣✉✐❞❛ V ✱ ✐st♦ é✱ −1 [ α |α ∈ Γ} p (V ) = U α α ∈Γ

  X ♦♥❞❡ {U é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❛❜❡rt♦s ❞❡ ˜ ✱ ❞♦✐s ❛ ❞♦✐s ❞✐s❥✉♥t♦s✱ t❛✐s q✉❡

  → V p| U α : U α é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✳ −1 [

  (V ) = U α ❚❡♠♦s q✉❡ ˜g(z), ˆg(z) ∈ p ✱ ❡♥tã♦ s❡ ˜g(z), ˆg(z) ♣❡rt❡♥❝❡ss❡ ❛ α α ∈Γ U α

  ✉♠ ♠❡s♠♦ U ✱ t❡rí❛♠♦s p(˜g(z)) = p(ˆg(z)) ❝♦♠ ˜g(z) 6= ˆg(z) ❛ss✐♠ p| ♥ã♦ s❡r✐❛ U α α ✐♥❥❡t♦r❛✱ ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s p| é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ s♦❜r❡ V ✳ ❆ss✐♠✱ ˜g(z) ∈ U β ❡ ˆg(z) ∈ U ❝♦♠ α 6= β✳

  ❈♦♠♦ ˜g ❡ ˆg sã♦ ❝♦♥tí♥✉❛s s❡❣✉❡ q✉❡ ❡①✐st❡ ✉♠ ❛❜❡rt♦ W ⊂ Z✱ ❝♦♠ z ∈ W t❛❧ α β q✉❡ ˜g(W ) ⊂ U ❡ ˆg(W ) ⊂ U ✳ P❛r❛ ❝❛❞❛ w ∈ W t❡♠✲s❡ q✉❡ ˜g(w) 6= ˆg(w)✱ ♣♦✐s U α ∩ U β = ∅

  ✳ ▲♦❣♦✱ z ∈ W ⊂ Z − A ❡♥tã♦ Z − A é ❛❜❡rt♦ ❡✱ ❛ss✐♠ A é ❢❡❝❤❛❞♦✳

  ✷✶ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ Z

  é ❝♦♥❡①♦ ❡✱ A é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦ ❛❜❡rt♦ ❡ ❢❡❝❤❛❞♦ ❞❡ Z ♣♦rt❛♥t♦ A = Z

  ❡✱ ˜g = ˆg ❝♦♠♦ q✉❡rí❛♠♦s✳ −1 X → X (x)

  Pr♦♣♦s✐çã♦ ✶✳✼✷✳ ❙❡ p : e é ✉♠ r❡❝♦❜r✐♠❡♥t♦ ❡♥tã♦ ❛s ✜❜r❛s p sã♦ ❞✐s❝r❡t❛s✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ p é ✉♠ r❡❝♦❜r✐♠❡♥t♦ ❡♥tã♦ p é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❧♦❝❛❧✱ −1

  (x) ❡♠ ♣❛rt✐❝✉❧❛r✱ p é ❧♦❝❛❧♠❡♥t❡ ✐♥❥❡t♦r❛✱ ❛ss✐♠ ♣❛r❛ ❝❛❞❛ ♣♦♥t♦ a ∈ p ❡①✐st❡

  X U ✉♠ ❛❜❡rt♦ U ⊂ e ✱ ❝♦♠ a ∈ U t❛❧ q✉❡ p| é ❧♦❝❛❧♠❡♥t❡ ✐♥❥❡t♦r❛✱ ❛ss✐♠ a é ♦ ú♥✐❝♦ −1

  (x) ♣♦♥t♦ ❞❡ p q✉❡ ♣❡rt❡♥❝❡ ❛ U✳ ❊♥tã♦✱ −1 −1 −1 U ∩ p (x) = {a}

  (x) (x) P♦rt❛♥t♦✱ a ∈ p é ✐s♦❧❛❞♦ ❡✱ ❛ss✐♠ p é ❞✐s❝r❡t♦✳

  X → X

  X ❈♦r♦❧ár✐♦ ✶✳✼✸✳ ❙❡ p : e é ✉♠ r❡❝♦❜r✐♠❡♥t♦ t❛❧ q✉❡ ˜ é ❝♦♠♣❛❝t♦✱ X é ❝♦♥❡①♦ ❡ ❞❡ ❍❛✉s❞♦r✛✱ ❡♥tã♦ ❛s ✜❜r❛s sã♦ ✜♥✐t❛s✳ −1

  (x) ❉❡♠♦♥str❛çã♦✳ P❡❧❛ ♣r♦♣♦s✐çã♦ t❡♠♦s q✉❡ p é ❞✐s❝r❡t♦✱ ♣❛r❛ ❝❛❞❛ −1 x ∈ X (x)

  ✳ ❙✉♣♦♥❤❛ q✉❡ p é ✐♥✜♥✐t♦✳ ❈♦♠♦ X é ❞❡ ❍❛✉s❞♦r✛ t❡♠♦s q✉❡ −1 (x)

  ♦s ♣♦♥t♦s sã♦ ❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s ❡✱ ❝♦♠♦ p é ❝♦♥tí♥✉❛ s❡❣✉❡ q✉❡ p é ❢❡❝❤❛❞♦ −1

  X X (x) ❡♠ e ✳ e é ❝♦♠♣❛❝t♦ ❡♥tã♦ p é ❝♦♠♣❛❝t♦✳ −1

  (x) P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s q✉❡ p é ✉♠ ❝♦♥❥✉♥t♦ ❞✐s❝r❡t♦✱ ❛ss✐♠ ♣❛r❛ ❝❛❞❛ −1 −1 a ∈ p (x) a a a ∩ p (x) = {a}

  ❡①✐st❡ ✉♠ ❛❜❡rt♦ U ✭a ∈ U ✮ t❛❧ q✉❡ U ✳ [ −1

  U a (x) ▲♦❣♦✱ é ✉♠❛ ❝♦❜❡rt✉r❛ ✐♥✜♥✐t❛ ❞❡ p q✉❡ ♥ã♦ ❛❞♠✐t❡ ✉♠❛ a (x) ∈p −1 −1

  (x) s✉❜❝♦❜❡rt✉r❛ ✜♥✐t❛✱ ✉♠ ❛❜s✉r❞♦ ♣♦✐s p é ❝♦♠♣❛❝t♦✳ −1 (x)

  P♦rt❛♥t♦✱ p é ✜♥✐t♦✳ −1 (x)

  ❆ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡ p é ❝❤❛♠❛❞❛ ♦ ♥ú♠❡r♦ ❞❡ ❢♦❧❤❛s ❞♦ r❡❝♦❜r✐♠❡♥t♦✳ ▲♦❣♦✱ ✉♠ r❡❝♦❜r✐♠❡♥t♦ ❝♦♠ ❛s ❤✐♣ót❡s❡s ❞♦ ❝♦r♦❧ár✐♦ ♣♦ss✉✐ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❢♦❧❤❛s✳ 1

  ❊①❡♠♣❧♦ ✶✳✼✹✳ ❆ ❛♣❧✐❝❛çã♦ exp : R → S ❞❛❞❛ ♣♦r✿ 2πit exp(t) = e = (cos(2πt), sen(2πt)) 1 é ✉♠ r❡❝♦❜r✐♠❡♥t♦ ❞❡ ✐♥✜♥✐t❛s ❢♦❧❤❛s✱ ♣♦✐s ♣❛r❛ ❝❛❞❛ ♣♦♥t♦ x ∈ S ✱ ♣♦❞❡ s❡r

  ) ∈ [0, 1] ❡s❝r✐t♦ ❝♦♠♦ x = exp(t ♣❛r❛ ❛❧❣✉♠ t t❡♠♦s q✉❡✿ −1 exp (x) = {t + n ∈ R|n ∈ Z} ✳

  ✷✷ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ ❉❡✜♥✐çã♦ ✶✳✼✺✳ ❙❡❥❛♠ X, Y ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❯♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ ❡ s♦❜r❡❥❡t♦r❛ f : X → Y s❛t✐s❢❛③ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❧❡✈❛♥t❛♠❡♥t♦ ❞❡ ❝❛♠✐♥❤♦s✱ q✉❡ ❛❜r❡✈✐❛r❡♠♦s ♣♦r ✭P▲❈✮✱ q✉❛♥❞♦ ❞❛❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡ ✉♠ ❝❛♠✐♥❤♦ α : J → Y, J = [s , s ] 1

  ) ✱ ❡ ✉♠ ♣♦♥t♦ x ∈ X t❛❧ q✉❡ f(x) = α(s ✱ ❡①✐st✐r ✉♠ ❝❛♠✐♥❤♦

  α : J → X ˜ ) = x t❛❧ q✉❡ ˜α(s ❡ f ◦ ˜α = α. ˜ α // ??

  X f J Y α

  ❊①❡♠♣❧♦ ✶✳✼✻✳ ◆❡♠ t♦❞♦ ❤♦♠♦♠♦r✜s♠♦ ❧♦❝❛❧ s❛t✐s❢❛③ ❛ P▲❈✿ 1 ❈♦♥s✐❞❡r❡ ♦ ❤♦♠❡♦♠♦r✜s♠♦ ❧♦❝❛❧ f : (0, 3π) → S ❞❡✜♥✐❞♦ ♣♦r f(t) = 1

  (cost, sent) ❡ t♦♠❡ ♦ ❝❛♠✐♥❤♦ α : [0, 1] → S ❞❡✜♥✐❞♦ ♣♦r α(t) = (cos4πt, sen4πt).

  ) = α(0) = (1, 0) = f (2π) t❡♠♦s α(s ✳ (0, 3π) // 1 f

  [0, 1] S a P❛r❛ q✉❡ ❢ s❛t✐s❢❛ç❛ ❛ P▲❈✱ é ♥❡❝❡ssár✐♦ q✉❡ ❡①✐st❛ ✉♠ ❝❛♠✐♥❤♦ ˜α : [0, 1] → (0, 3π) q✉❡ ❢❛ç❛ ♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛r ❡ q✉❡ ˜α(0) = 2π✳ P❛r❛ q✉❡ ♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❡ ❞❡✈❡♠♦s t❡r

  α : [0, 1] → (0, 3π) ˜ x 7−→ 4πx, ♣♦✐s ❞❛í f(˜α(x)) = f(4πx) = (cos4πx, sen4πx) = α(x)✳ ▼❛s ♥❡ss❡ ❝❛s♦ t❡♠♦s α(0) = 0 6= 2π ˜

  ✳ P♦rt❛♥t♦ ❢ ♥ã♦ s❛t✐s❢❛③ ❛ P▲❈✳ ❉❡✜♥✐çã♦ ✶✳✼✼✳ ◗✉❛♥❞♦ f : X → Y ❢♦r s♦❜r❡❥❡t✐✈❛ ❡✱ ❞❛❞♦s ❛r❜✐tr❛r✐❛♠❡♥t❡

  ) ✉♠ ❝❛♠✐♥❤♦ α : J → Y ❡ x ∈ X ❝♦♠ f(x) = α(s ✱ ❡①✐st✐r ✉♠ ú♥✐❝♦ ❝❛♠✐♥❤♦ α : J → X ˜ ) = x t❛❧ q✉❡ f ◦ ˜α = α ❡ ˜α(s ✱ ❞✐r❡♠♦s q✉❡ f ❣♦③❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❧❡✈❛♥t❛♠❡♥t♦ ú♥✐❝♦ ❞❡ ❝❛♠✐♥❤♦s ✭P▲❯❈✮ Pr♦♣♦s✐çã♦ ✶✳✼✽✳ ❚♦❞♦ ❡s♣❛ç♦ ❝♦♥❡①♦✱ ❧♦❝❛❧♠❡♥t❡ ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s ❡ ❧♦❝❛❧♠❡♥t❡ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦ ♣♦ss✉✐ ✉♠ r❡❝♦❜r✐♠❡♥t♦ ✉♥✐✈❡rs❛❧✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ♣á❣✐♥❛ ✷✵✶✳

  ❙❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥✐❛♥❛ ❡ M ❛ ❜❛s❡ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ M → M r❡❝♦❜r✐♠❡♥t♦ p : f ✳ ❊♥tã♦ é ❛ ú♥✐❝❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ s♦❜r❡ M ❡ p é

  ✉♠❛ ✐s♦♠❡tr✐❛ ❧♦❝❛❧✳ ◗✉❛♥❞♦ M ♣♦ss✉✐ ❡ss❛ ♠étr✐❝❛✱ ❛ ❝♦❜❡rt✉r❛ s❡rá ❝❤❛♠❛❞❛ ❞❡ r❡❝♦❜r✐♠❡♥t♦ ❘✐❡♠❛♥♥✐❛♥♦✳ ❖s s❡❣✉✐♥t❡s ❢❛t♦s sã♦ ✐❞❡♥t✐✜❝❛❞♦s ❛ ♣❛rt✐r ❞❛ ❞❡✜♥✐çã♦✳ ❱❡❥❛

  ✶✳ p ❧❡✈❛ ❣❡♦❞és✐❝❛ ❡♠ ❣❡♦❞és✐❝❛✳ M

  ✷✸ ✶✳✸✳ ◆❖➬Õ❊❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ ✸✳ ❆ tr❛♥s❢♦r♠❛çã♦ ❞❛s ❝♦❜❡rt✉r❛s sã♦ ✐s♦♠❡tr✐❛s ❡♠ M✳

  M → M ❉❡✜♥✐çã♦ ✶✳✼✾✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛✱ s❡♠ ❜♦r❞♦✱ p : f ❛ ❛♣❧✐❝❛çã♦ r❡❝♦❜r✐♠❡♥t♦ ✉♥✐✈❡rs❛❧✳ ❯♠ ❞♦♠í♥✐♦ ❝♦♠♣❛❝t♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ (p, f M , M ) M

  é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ K ⊂ f t❛❧ q✉❡✿ 1 (M ) M ✶✳ ❆ ✉♥✐ã♦ ❞❡ γK ♣❛r❛ t♦❞♦ γ ∈ π ❝♦❜r❡♠ f ✳ ✷✳ ❆ ❝♦❧❡çã♦ γK sã♦ ❞♦✐s ❛ ❞♦✐s ❞✐s❥✉♥t♦s✳ K : K → M ✸✳ π(K) = M ❡ ❛ r❡str✐çã♦ p| é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ s♦❜r❡ s✉❛

  ✐♠❛❣❡♠✳ ❱❡r ❘❡❝♦❜r✐♠❡♥t♦ ❞✉♣❧♦ ♦r✐❡♥t❛❞♦

  → Y ❉❡✜♥✐çã♦ ✶✳✽✵✳ ❯♠❛ ❛♣❧✐❝❛çã♦ p : X ❝♦♥tí♥✉❛ ✐♥❞✉③ ✉♠

  = p(x ) ❤♦♠♦♠♦r✜s♠♦✭y ✮✿ p : π (X, x ) → π (Y, y ) 1 1

  ❞❡✜♥✐❞♦ ♣♦r✿ p ([α]) = [f ◦ α] p ∗ é ❝❤❛♠❛❞♦ ❤♦♠♦♠♦r✜s♠♦ ✐♥❞✉③✐❞♦ ♣❡❧❛ ❛♣❧✐❝❛çã♦ p✳ ❱❡❥❛ M →

  ❉❡✜♥✐çã♦ ✶✳✽✶✳ ❯♠ r❡❝♦❜r✐♠❡♥t♦ ❞✉♣❧♦ ♦r✐❡♥t❛❞♦ é ✉♠❛ ❛♣❧✐❝❛çã♦ p : f k M

  ✱ ❞❡ ❝❧❛ss❡ C ✱ ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ M

  ✶✳ ▼ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♥❡①❛✱ f é ✉♠❛ ✈❛r✐❡❞❛❞❡ ♦r✐❡♥t❛❞❛ ❡ p é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧❀ −1

  (y) ✷✳ P❛r❛ ❝❛❞❛ y ∈ M✱ ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ p ❝♦♥té♠ ❡①❛t❛♠❡♥t❡ ❞♦✐s ♣♦♥t♦s❀ ′ −1 ′ 1 ) = p(x 2 ) 1 6= x 2 (x 2 ) : p (x 1 ) :

  ✸✳ ❙❡ p(x ❝♦♠ x ❡♥tã♦ ♦ ✐s♦♠♦r✜s♠♦ ❧✐♥❡❛r p f f T x M → T x M 1 2 é ♥❡❣❛t✐✈♦✳ ❱❡❥❛ m m

  → P ❊①❡♠♣❧♦ ✶✳✽✷✳ ◗✉❛♥❞♦ m é ♣❛r✱ ❛ ❛♣❧✐❝❛çã♦ q✉♦❝✐❡♥t❡ π : S é ✉♠ m r❡❝♦❜r✐♠❡♥t♦ ❞✉♣❧♦ ♦r✐❡♥t❛❞♦ ❞♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ P ✳ ◆♦ ❝❛s♦ ❞❡ m í♠♣❛r✱ π ♥ã♦ ❝✉♠♣r❡ ❛ t❡r❝❡✐r❛ ❝♦♥❞✐çã♦ ❞❛ ❞❡✜♥✐çã♦

  : π ( f x ) → π (M, x ) 1 1 Pr♦♣♦s✐çã♦ ✶✳✽✸✳ ❆ ❛♣❧✐❝❛çã♦ p ∗ M , e ✐♥❞✉③✐❞❛ ♣❡❧❛ 1 ( f x ) → π (M, x ) 1 ❛♣❧✐❝❛çã♦ r❡❝♦❜r✐♠❡♥t♦ p : π M , e é ✐♥❥❡t✐✈❛✳ ❆ ✐♠❛❣❡♠

  (π ( f x ) (M, x ) 1 1 ❞♦ s✉❜❣r✉♣♦ p ∗ M , e ❡♠ π ❝♦♥s✐st❡ ♥❛s ❝❧❛ss❡s ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ ❝❛♠✐♥❤♦s ❞✐st✐♥t♦s✱ ❢❡❝❤❛❞♦s ❡♠ M ❝♦♠ ♣♦♥t♦ ❜❛s❡ ❡♠ x ❉❡♠♦♥str❛çã♦✳ ❱❡r ♣á❣✐♥❛ 61✳

  ✷✹ ✶✳✹✳ ❈❘❊❙❈■▼❊◆❚❖ ❊❳P❖◆❊◆❈■❆▲

  ✶✳✹ ❈r❡s❝✐♠❡♥t♦ ❊①♣♦♥❡♥❝✐❛❧

  ◆❡st❛ s❡çã♦✱ ❞❡✜♥✐r❡♠♦s ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ♣❛r❛ ❣r✉♣♦s ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦s ❡ ✈❡r❡♠♦s ❝♦♠♦ ♦ ❝r❡s❝✐♠❡♥t♦ ❞♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ♣♦❞❡ t❡r s❡♥t✐❞♦ ❣❡♦♠étr✐❝♦ ❞❡♥tr♦ ❞❛ ✈❛r✐❡❞❛❞❡✳ ❊✱ ♣♦r ✜♠✱ ✈❡r❡♠♦s ❡①❡♠♣❧♦s ❞❡ ❣r✉♣♦s ❝♦♠ ❞✐st✐♥t♦s t✐♣♦s ❞❡ ❝r❡s❝✐♠❡♥t♦✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s✱ 1 , ..., g m }

  ❉❡✜♥✐çã♦ ✶✳✽✹✳ ❙❡❥❛ S ✉♠ ❣r✉♣♦✱ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ♣♦r G = {g ✳ ❙❡ ε ε i ε i 1 j = ±1 i ∈ G j g = g g ...g i i i 1 2 n ♦♥❞❡ ε ❡ g ✳ ❉❡✜♥✐♠♦s ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ g em G✳ ❈♦♠♦✱

  | + |ε | + ... + |ε | l 1 (g) = |ε 1 2 n P❛r❛ ❝❛❞❛ n ∈ N ✱ ❞❡✜♥✐♠♦s ❛ ❢✉♥çã♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ S✱ ♣♦r✿

  Γ(n) = #{g ∈ S; l (g) ≤ n} 1 ❱❡❥❛

  ❉❡✜♥✐çã♦ ✶✳✽✺✳ ❉✐③❡♠♦s q✉❡ S t❡♠ ❝r❡s❝✐♠❡♥t♦ ♣♦❧✐♥♦♠✐❛❧ s❡ Γ(n) ≥ p,

  ♦♥❞❡ p é ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ n✳ ❡ q✉❡ S t❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ s❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s A, a > 0 t❛✐s q✉❡ an

  Γ(n) ≥ Ae , ❝♦♠ n ≥ 1✳ ❖❜s❡r✈❛çã♦ ✶✳✽✻✳ ❖ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❛ ❢✉♥çã♦ ✐♥❞❡♣❡♥❞❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s✳

  ❱❡❥❛ 1 (M ) (M ) 1 ❉❡✜♥✐çã♦ ✶✳✽✼✳ ◗✉❛♥❞♦ S = π ✳ ❉❡✜♥✐♠♦s ♣❛r❛ α ∈ π ✱ l 2 (α) = inf{|α|; α ∈ α} +

  ♦♥❞❡ |α| é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ❞❛ ❝✉r✈❛ α✳ P❛r❛ ❝❛❞❛ r ∈ R ✱ ❞❡✜♥✐♠♦s ❛ ❢✉♥çã♦✱

  P (r) = #{α ∈ π (M ); l (α) ≤ r} 1 2 Pr♦✈❛r❡♠♦s ♥♦ ▲❡♠❛ q✉❡ ♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ t❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ P t❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧✱ ✐st♦ é✱ ❡①✐st❡♠ br ❝♦♥st❛♥t❡s B > 0✱ b > 0 t❛❧ q✉❡ P (r) ≥ B❡ ✳ 1 2

  × D ❊①❡♠♣❧♦ ✶✳✽✽✳ ❈♦♥s✐❞❡r❛♥❞♦ M = S ♦ ❚♦r♦ ❙ó❧✐❞♦✱ ♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ 1 (M ) = Z (M ) 1

  ✷✺ ✶✳✹✳ ❈❘❊❙❈■▼❊◆❚❖ ❊❳P❖◆❊◆❈■❆▲ ❉❡♠♦♥str❛çã♦✳ P♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡✱ Γ(1) = 3✱ Γ(2) = 5 ✱Γ(3) = 7✱ . . . ✱ Γ(n) = 2n + 1

  ✳ ➱ ❢á❝✐❧ ✈❡r q✉❡✱ ❢♦r♠❛♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s í♠♣❛r❡s✳ P♦rt❛♥t♦✱ ♦ ❚♦r♦ ❙ó❧✐❞♦ t❡♠ ❝r❡s❝✐♠❡♥t♦ ♣♦❧✐♥♦♠✐❛❧✳ 1 1

  × S ❊①❡♠♣❧♦ ✶✳✽✾✳ ❈♦♥s✐❞❡r❛♥❞♦ M = S ♦ ❚♦r♦✱ ♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ é π (M ) = Z × Z (M ) 1 1

  ✳ ❖ ❝r❡s❝✐♠❡♥t♦ ❞❡ π é ♣♦❧✐♥♦♠✐❛❧✳ ❉❡ ❢❛t♦✱ ♦❜s❡r✈❛♠♦s q✉❡ Γ(1) = 5✱ Γ(2) = 13✱ Γ(3) = 25 ✱ Γ(4) = 41✱ 2

  Γ(n) = 2n + 2n + 1 ✳

  ▲♦❣♦✱ ❛ ❢✉♥çã♦ ❝r❡s❝✐♠❡♥t♦ ❞♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ ❚♦r♦ é ♣♦❧✐♥♦♠✐❛❧✳ ❉❡✜♥✐çã♦ ✶✳✾✵✳ ❙❡❥❛ S ✉♠ ❣r✉♣♦ ❡ {α ∈ Γ} ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s✉❜❣r✉♣♦s q✉❡ ❣❡r❛♠ S α ∩ S β = {e} α

  ✳ S s❡ α 6= β✳ ❉✐③❡♠♦s q✉❡✱ S é ♦ ♣r♦❞✉t♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣❡❧♦s S ✱ α s❡ ♣❛r❛ ❝❛❞❛ s ∈ S ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ♣❛❧❛✈r❛ r❡❞✉③✐❞❛ ❡♠ S q✉❡ r❡♣r❡s❡♥t❛ ❛ s✳ ❊♠ t❛❧ ❝❛s♦ ❡s❝r❡✈❡♠♦s✿

  Y S = S α α

  ◆♦ ❝❛s♦ ✜♥✐t♦✱ ❡s❝r❡✈❡♠♦s✿ S = S ∗ S ∗ ... ∗ S j 1 2

  ♦♥❞❡✱ j = 1, ..., n✳ S

  é ♦ ♣r♦❞✉t♦ ❧✐✈r❡ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❛❧❛✈r❛s r❡❞✉③✐❞❛s✳ ❊♠ r❡s✉♠♦✱ ♦ ❣r✉♣♦ S é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❡①♣r❡ssõ❡s ❢♦r♠❛✐s✿ ε ε j 1 2 ε ε j s = s s ...s i 1 2 j

  ❖♥❞❡ ❝❛❞❛ s j ♣❡rt❡♥❝❡ ❛ ❛❧❣✉♠ S ❞✐❢❡r❡♥t❡ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❡ t❛❧ q✉❡ ♦s ❡❧❡♠❡♥t♦s ε j i α ❛❞❥❛❝❡♥t❡s ❛ s ♣❡rt❡♥❝❡♠ ❛ ❞✐❢❡r❡♥t❡s S ✳ ❊①❡♠♣❧♦ ✶✳✾✶✳ ❙❡❥❛ M ♦ ❜✐t♦r♦ só❧✐❞♦ s❛❜❡♠♦s q✉❡ ♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ é ❞❛❞♦ 1 (M ) = Z ∗ Z 1 (M ) ♣♦r π ✳ ❖ ❝r❡s❝✐♠❡♥t♦ ❞❡ π é ❡①♣♦♥❡♥❝✐❛❧✳

  ❉❡ ❢❛t♦✱ ❋✐①❡♠♦s γ(n) ❝♦♠♦ s❡♥❞♦ ❛s ♣❛❧❛✈r❛s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✐❣✉❛✐s ❛ n ❡ Γ(n)

  ❛s ♣❛❧❛✈r❛s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♠❡♥♦r❡s ♦✉ ✐❣✉❛✐s à n✳ ❈♦♥s✐❞❡r❡ ♦ ❣rá✜❝♦ ❞❡ ❈❛❧❡②

  P❡❧♦ ❣rá✜❝♦ t❡♠♦s q✉❡ ❛s ♣❛❧❛✈r❛s ❞✐st✐♥t❛s ❞❡ ❝♦♠♣r✐♠❡♥t♦ 1✱ sã♦✿ −1 −1 0, a, b, a , b

  ❆ss✐♠✱ γ(1) = 5 ❡ γ(2) = 12✳ ▲♦❣♦✱ 1 Γ(2) = γ(1) + γ(2) = 4.3 + 4.3 + 1 = 17.

  ❆ss✐♠✱ γ(3) = 36✳ ▲♦❣♦✱ 2 1 Γ(3) = γ(1) + γ(2) + γ(3) = 4.3 + 4.3 + 4.3 + 1 = 53. n n n −1 −2 −3 2 1

  • 4.3 + 4.3 + ... + 4.3 + 4.3 + 4.3 + 1 P♦rt❛♥t♦✱ Γ(n)✱ é✿ Γ(n) = 4.3 n n n n n n −1 −1 −1

  Γ(n) ≥ 4.3 = (3 + 1)3 = 3 + 3 ≥ 3

  ✷✻ ✶✳✺✳ ❚❊❖❘❊▼❆ ❉❊ ❍❆❊❋▲■●❊❘

  ❋✐❣✉r❛ ✶✳✾✿ ●rá✜❝♦ ❞❡ ❈❛❧❡② ❖❜s❡r✈❛çã♦ ✶✳✾✷✳ ❆ ❢♦❧❤❡❛çã♦ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ♦r✐❡♥tá✈❡❧✱ ❝❛s♦ ❝♦♥trár✐♦✱ ❜❛st❛ ♦❜s❡r✈❛r ♦ r❡❝♦❜r✐♠❡♥t♦ ❞✉♣❧♦ ❡ ♠♦str❛r q✉❡ ♦ ♠❡s♠♦ t❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧✳ ❊♥tã♦✱ ❛ ♣ró♣r✐❛ ✈❛r✐❡❞❛❞❡ t❛♠❜é♠ t❡♠ ❡ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ♦r✐❡♥tá✈❡❧✳

  ✶✳✺ ❚❡♦r❡♠❛ ❞❡ ❍❛❡✢✐❣❡r

  ◆❡st❛ s❡çã♦ ❛❜♦r❞❛r❡♠♦s ❝♦♥❝❡✐t♦s ✐♥tr♦❞✉tór✐♦s s♦❜r❡ ❢♦❧❤❡❛çõ❡s ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ✱ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s s♦❜r❡ ❤♦❧♦♥♦♠✐❛ ❡ ♣♦st❡r✐♦r♠❡♥t❡✱ ❢❛r❡♠♦s ♦ ❡s❜♦ç♦ ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❞❡ ❍❛❡✢✐❣❡r✳ ❋♦❧❤❡❛çõ❡s

  ❆ ✐❞❡✐❛ ✐♥t✉✐t✐✈❛ ❞❡ ❢♦❧❤❡❛çã♦ ❝♦rr❡s♣♦♥❞❡ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ♥✉♠❛ ✉♥✐ã♦ ❞❡ s✉❜✈❛r✐❡❞❛❞❡s ❝♦♥❡①❛s✱ ❞✐s❥✉♥t❛s ❡ ❞❡ ♠❡s♠❛ ❞✐♠❡♥sã♦✱ ❞❛s q✉❛✐s s❡ ❛❝✉♠✉❧❛♠ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♦ ❢♦❧❤❛s ❞❡ ✉♠ ❧✐✈r♦✳ ♦s q✉❛✐s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❉❡✜♥✐çã♦ ✶✳✾✸✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦ n ❡ ❝❧❛ss❡ r C

  ✳ ❯♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❝❧❛ss❡ C ❡ ❞✐♠❡♥sã♦ m ❞❡ M✱ é ✉♠ ❛t❧❛s ♠á①✐♠♦ F r ❞❡ ❝❧❛ss❡ C ❡♠ M✱ s❛t✐s❢❛③❡♥❞♦✿

  ✶✳ ❙❡ (U, ϕ) ∈ F ❡♥tã♦ n m −n 1 e U 2 ϕ(U ) = U × U ⊂ R × R 1 2 n m −n ♦♥❞❡ U sã♦ ❞✐s❝♦s ❛❜❡rt♦s ❞❡ R ❡ ❞❡ R r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  ✷✳ ❙❡ (U, ϕ) e (V, Ψ) ∈ F sã♦ t❛✐s q✉❡ U ∩ V 6= ∅

  ✷✼ ✶✳✺✳ ❚❊❖❘❊▼❆ ❉❊ ❍❆❊❋▲■●❊❘

  ❡♥tã♦ ❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s −1 Ψ ◦ ϕ : ϕ(U ∩ V ) → Ψ(U ∩ V )

  é ❞❛❞❛ ♣♦r −1 Ψ ◦ ϕ (x, y) = (h (x, y), h (y)) 1 2

  ❋✐❣✉r❛ ✶✳✶✵✿ ❋♦❧❤❡❛çõ❡s ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ M é ❢♦❧❤❡❛❞❛ ♣♦r F✱ ♦✉ ❛✐♥❞❛ q✉❡ F é ✉♠❛ ❡str✉t✉r❛ r

  ❢♦❧❤❡❛❞❛ ❞❡ ❞✐♠❡♥sã♦ n ❡ ❝❧❛ss❡ C s♦❜r❡ M✳ r ❉❡✜♥✐çã♦ ✶✳✾✹✳ ❙❡❥❛♠ F ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❝❧❛ss❡ C ❞❡ ❞✐♠❡♥sã♦ n✱ 0 < n < m✱ m ❞❡ M ❡ (U, ϕ) ✉♠❛ ❝❛rt❛ ❧♦❝❛❧ ❞❡ F t❛❧ q✉❡ n m −n ϕ(U ) = U × U ⊂ R × R . −1 1 2

  (U ×{c}), c ∈ U 1 2 ❖s ❝♦♥❥✉♥t♦s ❞❛ ❢♦r♠❛ ϕ sã♦ ❝❤❛♠❛❞♦s ♣❧❛❝❛s ❞❡ U✱ ♦✉ ♣❧❛❝❛s ❞❡ F✳ −1 2 (U | : U × {c} → U 1

  ❋✐①❛♥❞♦ c ∈ U ✱ ❛ ❛♣❧✐❝❛çã♦ f = ϕ 1 ×{c}) é ✉♠❛ r s✉❜✈❛r✐❡❞❛❞❡ ❝♦♥❡①❛ ❞❡ ❞✐♠❡♥sã♦ n ❡ ❞❡ ❝❧❛ss❡ C ❞❡ M✳ P♦rt❛♥t♦✱ s❡ α e β sã♦ ♣❧❛❝❛s ❞❡ U ❡♥tã♦ α ∩ β = ∅ ou α = β✳ 1 , ..., α k ❉❡✜♥✐çã♦ ✶✳✾✺✳ ❯♠ ❝❛♠✐♥❤♦ ❞❡ ♣❧❛❝❛s ❞❡ F é ✉♠❛ s❡q✉ê♥❝✐❛ α ❞❡ j ∩ α j 6= ∅ +1 ♣❧❛❝❛s t❛❧ q✉❡ α ✱ ♣❛r❛ t♦❞♦ j ∈ {1, ..., k − 1}✳

  ❉❛❞♦ q✉❡ ❛ ✈❛r✐❡❞❛❞❡ M é ❝♦❜❡rt❛ ♣♦r ♣❧❛❝❛s ❞❡ F✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❡♠ M , ..., α k

  ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✿ ✧pRq s❡ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ ❞❡ ♣❧❛❝❛s α 1 ❝♦♠ p ∈ α e q ∈ α k 1 ✳ ✧❊st❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❛ r❡❧❛çã♦ R sã♦ ❝❤❛♠❛❞❛s

  ❢♦❧❤❛s ❞❡ F✳ P♦r ❞❡✜♥✐çã♦✱ s❡q✉❡✲ s❡ q✉❡ ❛s ❢♦❧❤❛s ❞❡ F sã♦ s✉❜❝♦♥❥✉♥t♦s ❞❡ M

  ✷✽ ✶✳✺✳ ❚❊❖❘❊▼❆ ❉❊ ❍❆❊❋▲■●❊❘

  ❉❡✜♥✐çã♦ ✶✳✾✻✳ ❙❡ F é ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✱ ❡♥tã♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ S ⊂ M é ❞✐t♦ s❡r F✲ s❛t✉r❛❞♦ ♦✉ s✐♠♣❧❡s♠❡♥t❡ s❛t✉r❛❞♦ s❡ é ✉♠❛ ✉♥✐ã♦ ❞❡ ❢♦❧❤❛s ❞❡ F✳ u x s

  ⊕ E ⊕ E ❖❜s❡r✈❛çã♦ ✶✳✾✼✳ ❙❡❥❛ E ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡✳ ❙❛❜❡✲ u x s x u s

  ⊕ E ⊕ E s❡ q✉❡ ♦s s✉❜✜❜r❛❞♦s E ✱ E ✱ E ✱ E sã♦ ✉♥✐❝❛♠❡♥t❡ ✐♥t❡❣rá✈❡✐s ❡ q✉❡ r ❛s ✈❛r✐❡❞❛❞❡s ✐♥t❡❣rá✈❡✐s sã♦ ❞❡ ❝❧❛ss❡ C ✳ ❆s ✈❛r✐❡❞❛❞❡s ✐♥t❡❣rá✈❡✐s ❞❡t❡r♠✐♥❛♠ u s uu ss ❢♦❧❤❡❛çõ❡s ❞❡ M ❛s q✉❛✐s sã♦ ❞❡♥♦t❛❞❛s ♣♦r F ✱ F ✱ F ✱ F ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ u

  (x) ❙❡ x ∈ M ❡♥tã♦ ❛s r❡s♣❡❝t✐✈❛s ❢♦❧❤❛s ❞❡ss❛ ❢♦❧❤❡❛çã♦ ❝♦♥t❡♥❞♦ x sã♦ W ✱ s uu ss W (x) (x) (x)

  ✱ W ✱ W ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡✜♥✐çã♦ ✶✳✾✽✳ ❙❡❥❛ ϕ : G × M −→ M ✉♠❛ ❛çã♦ ❞❡ ❣r✉♣♦s✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ M é ✐♥✈❛r✐❛♥t❡ ✭♣♦r ϕ✮ s❡ ❡❧❡ é ✉♥✐ã♦ ❞❡ ór❜✐t❛s ❞❡ ϕ✳ t ❉❡✜♥✐çã♦ ✶✳✾✾✳ ❯♠ ❝♦♥❥✉♥t♦ ♠✐♥✐♠❛❧ ❞❡ ϕ é ✉♠ s✉❜❝♦♥❥✉♥t♦ N ⊂ M q✉❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡✿

  ✶✳ ◆ã♦ ✈❛③✐♦ ✷✳ ❢❡❝❤❛❞♦ t (N ) = N ✸✳ ■♥✈❛r✐❛♥t❡ ✱ ✐st♦ é✱ ϕ ✳

  ❊ t❛❧ q✉❡ ♥❡♥❤✉♠ s✉❜❝♦♥❥✉♥t♦ ♣r♦♣r✐❛♠❡♥t❡ ❝♦♥t✐❞♦ ❡♠ N ♣♦ss✉✐ ❡ss❛s três ♣r♦♣r✐❡❞❛❞❡s✳ r

  , r ≥ 1 ❉❡✜♥✐çã♦ ✶✳✶✵✵✳ ❯♠❛ ❢♦❧❤❡❛çã♦ F ❞❡ ❝♦❞✐♠❡♥sã♦ s ❡ ❝❧❛ss❡ C ❞❡ M i , f i ), i ∈ I i ❡stá ❞❡✜♥✐❞❛ ♣♦r ✉♠❛ ❝♦❧❡çã♦ ♠á①✐♠❛ ❞❡ ♣❛r❡s (U ♦♥❞❡ ♦s U sã♦ i : U i → R s ❛❜❡rt♦s ❡♠ M ❡ ❛s f sã♦ s✉❜♠❡rsõ❡s s❛t✐s❢❛③❡♥❞♦✿

  S U i = M

  ✶✳ i ∈I r s i j ij ∩ U 6= ∅ ✷✳ ❙❡ U ✱ ❡①✐st❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧ g ❞❡ ❝❧❛ss❡ C ❞❡ R t❛❧ q✉❡ f i = g ij ◦ f j i ∩ U j i ❡♠ U ✳

  ❆s f sã♦ ❝❤❛♠❛❞❛s ❛♣❧✐❝❛çõ❡s ❞✐st✐♥❣✉✐❞❛s ❞❡ F✳ i ◆❡st❛ ❞❡✜♥✐çã♦ ❛s ♣❧❛❝❛s ❞❡ F ❡♠ U sã♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ❞♦s −1 s

  , c ∈ R ❝♦♥❥✉♥t♦s f i ✳ ▲❡♠❛ ✶✳✶✵✶✳ ✭❆♣❧✐❝❛çõ❡s ❉✐st✐♥❣✉✐❞❛s✮✳ ❙❡❥❛ F ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ✉♠❛ i /i ∈ I} ✈❛r✐❡❞❛❞❡ M✳ ❊①✐st❡ ✉♠❛ ❝♦❜❡rt✉r❛ C = {U ❞❡ M ♣♦r ❞♦♠í♥✐♦s ❞❡ i ∩ U j 6= ∅ ❝❛rt❛s ❧♦❝❛✐s ❞❡ F t❛❧ q✉❡ s❡ U ❡stá ❝♦♥t✐❞♦ ♥♦ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❝❛rt❛ ❧♦❝❛❧ ❞❡ F✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❝❛♣✐t✉❧♦ ■■✱ ♣á❣✐♥❛ ✸✶✳ ❉❡✜♥✐çã♦ ✶✳✶✵✷✳ ❙❡❥❛♠ N, S ✈❛r✐❡❞❛❞❡s✳ ❉✐③❡♠♦s q✉❡ ❡st❛s ✈❛r✐❡❞❛❞❡s sã♦ p N + T p S = T p M tr❛♥s✈❡rs❛✐s ♥✉♠ ♣♦♥t♦ p s❡✱ s♦♠❡♥t❡ s❡✱ S ∩ N 6= ∅ ♦✉ T ✳ r

  (r ≥ 1) ❉❡✜♥✐çã♦ ✶✳✶✵✸✳ ❯♠❛ ❢♦❧❤❡❛çã♦ F ❞❡ ❝❧❛ss❡ C é ♦r✐❡♥tá✈❡❧ s❡ ♦ ❝❛♠♣♦ ❞❡ ♣❧❛♥♦s t❛♥❣❡♥t❡s ❛ F é ♦r✐❡♥tá✈❡❧✳ ❙✐♠✐❧❛r ❛ F é tr❛♥s✈❡rs❛❧♠❡♥t❡ ♦r✐❡♥tá✈❡❧ s❡ ♦ ❝❛♠♣♦ ❞❡ ♣❧❛♥♦s t❛♥❣❡♥t❡s ❛ F é tr❛♥s✈❡rs❛❧♠❡♥t❡ ♦r✐❡♥tá✈❡❧✳ ❉❡♥♦t❛r❡♠♦s ♦ ❝❛♠♣♦ ❞❡ ♣❧❛♥♦s t❛♥❣❡♥t❡s ❛ F ♣♦r T F

  ✷✾ ✶✳✺✳ ❚❊❖❘❊▼❆ ❉❊ ❍❆❊❋▲■●❊❘

  ❍♦❧♦♥♦♠✐❛ r , r ≥ 1

  ◆❡st❛ s❡çã♦ F ❞❡♥♦t❛ ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❝♦❞✐♠❡♥sã♦ n ❡ ❝❧❛ss❡ C ✱ m ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M ✳ ❖ ♥♦ss♦ ♦❜❥❡t✐✈♦ ❡ ❡st✉❞❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s ❢♦❧❤❛s ✈✐③✐♥❤❛s ❛ ✉♠❛ ❢♦❧❤❛ ❝♦♠♣❛❝t❛ F ✜①❛❞❛ ❛ ♣r✐♦r✐✳ P❡❧❛ ✉♥✐❢♦r♠✐❞❛❞❡ tr❛♥s✈❡rs❛❧ ❞❡ F é s✉✜❝✐❡♥t❡ ❡st✉❞❛r ♦s r❡t♦r♥♦s ❞❛s ❢♦❧❤❛s ❛ ✉♠❛ ♣❡q✉❡♥❛ s❡çã♦ tr❛♥s✈❡rs❛❧ Σ

  ❞❡ ❞✐♠❡♥sã♦ n ♣❛ss❛♥❞♦ ♣♦r ✉♠ ♣♦♥t♦ p ∈ F ✳ P❛r❛ ❝❛❞❛ ❝❛♠✐♥❤♦ γ ❢❡❝❤❛❞♦ ❡♠ F ♣❛ss❛♥❞♦ ♣♦r p✱ ❡st❡s r❡t♦r♥♦s ♣♦❞❡♠ s❡ ❡①♣r❡ss❛r ♣♦r ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ r

  , f γ , f γ (p) = p ❧♦❝❛❧ ❞❡ Σ ❞❡ ❝❧❛ss❡ C ✱ ♦♥❞❡ ♣❛r❛ x ∈ Σ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣ró①✐♠♦ γ (x) ❞❡ p, f é ♦ ♣r✐♠❡✐r♦ r❡t♦r♥♦ ✧s♦❜r❡ γ✧❞❛ ❢♦❧❤❛ ❞❡ F q✉❡ ♣❛ss❛ ♣♦r x✳ ❘❡s✉❧t❛ q✉❡ s❡ γ é ♦✉tr♦ ❝❛♠✐♥❤♦ ❡♠ F ❢❡❝❤❛❞♦ ❡♠ p ❡ ❤♦♠♦tó♣✐❝♦ ❛ γ ✭❝♦♠ ❡①tr❡♠♦s γ e f γ

  ♠❛♥t✐❞♦s ✜①♦s✮ ❡♥tã♦ f ❝♦✐♥❝✐❞❡♠ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❝♦♠✉♠ ❞❡ p ∈ Σ✱ γ γ e f ✐st♦ é✱ f tê♠ ♦ ♠❡s♠♦ ❣❡r♠❡ ❡♠ p ∈ Σ✳ ❉❛í✱ s❡❣✉❡✲ s❡ q✉❡ ❛ ❛♣❧✐❝❛çã♦ γ ←→ f γ

  ✐♥❞✉③ ✉♠ ❤♦♠♦♠♦r✜s♠♦ π (F, p) → G(Σ, p) 1

  ❡♥tr❡ ♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ F em p ✭❣r✉♣♦ ❞❡ ❝❧❛ss❡s ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ r ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s✮ ❡ ♦ ❣r✉♣♦ ❞❡ ❣❡r♠❡s ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s C ❞❡ Σ em p✳ ❊st❛ r❡♣r❡s❡♥t❛çã♦ é ❝❤❛♠❛❞❛ ❤♦❧♦♥♦♠✐❛ ❞❡ F em p✳ ❊st❛ ♥♦çã♦✱ ❢♦✐ ✐♥tr♦❞✉③✐❞❛ ♣♦r ❊❤r❡s♠❛♥♥✳ 1 (F, p)

  ➱ ❝❧❛r♦ q✉❡ q✉❛♥t♦ ♠❡♥♦r ❢♦r π ♠❡♥♦r s❡rá ❛ r❡❝♦rrê♥❝✐❛ ❞❛s ❢♦❧❤❛s ✈✐③✐♥❤❛s ❛ F ✳ ❖ ❚❡♦r❡♠❛ ❞❛ ❊st❛❜✐❧✐❞❛❞❡ ▲♦❝❛❧ ❡①♣r❡ss❛ ✐ss♦ ❞❛ ♠❛♥❡✐r❛ s❡❣✉✐♥t❡✿ s❡ F é ✉♠❛ ❢♦❧❤❛ ❝♦♠♣❛❝t❛ ❞❡ F ❝♦♠ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ✜♥✐t♦✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ V ❞❡ F ✱ s❛t✉r❛❞❛ ♣❡❧❛s ❢♦❧❤❛s ❞❡ F✱ t❛❧ q✉❡ t♦❞❛ ❢♦❧❤❛ ❡♠

  V é ❝♦♠♣❛❝t❛ ❝♦♠ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ✜♥✐t♦✳

  ❖ ❚❡♦r❡♠❛ ❞❛ ❊st❛❜✐❧✐❞❛❞❡ ❈♦♠♣❧❡t❛ ♦❢❡r❡❝❡ ❛ s❡❣✉✐♥t❡ ✈❡rsã♦ ❣❧♦❜❛❧✿ ❙❡ codF = 1✱ M é ❝♦♠♣❛❝t❛ ❡ ❝♦♥❡①❛✱ ❡①✐st❡ ✉♠❛ ❢♦❧❤❛ ❝♦♠♣❛❝t❛ ❝♦♠ ❣r✉♣♦

  ❢✉♥❞❛♠❡♥t❛❧ ✜♥✐t♦ ❡♥tã♦ t♦❞❛s ❛s ❢♦❧❤❛s sã♦ ❝♦♠♣❛❝t❛s ❡ tê♠ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ✜♥✐t♦✳

  ❊st❡s t❡♦r❡♠❛s sã♦ ❞❡✈✐❞♦s à ●✳ ❘❡❡❜✳ ❍♦❧♦♥♦♠✐❛ ❞❡ ✉♠❛ ❋♦❧❤❛

  ❉✉r❛♥t❡ ❡st❛ s❡çã♦ F s❡rá ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❝♦❞✐♠❡♥sã♦ n ❡ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❛ s❡çã♦ é ❞❡✜♥✐r ❡ ❞❛r ✉♠❛ ♥♦çã♦ s♦❜r❡ ❤♦❧♦♥♦♠✐❛ ❞❡ ✉♠❛ ❢♦❧❤❛ F de F✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ✈❡❥❛ ♦ ❈❛♣ít✉❧♦ ■❱ ❞❡

  , Σ 1 ❙❡❥❛♠ γ : [0, 1] → F ✉♠ ❝❛♠✐♥❤♦ ❝♦♥tí♥✉♦ ❡ Σ ♣❡q✉❡♥❛s s❡çõ❡s

  = γ(0) = γ(1) 1 tr❛♥s✈❡rs❛✐s ❛ F ❞❡ ❞✐♠❡♥sã♦ n ♣❛ss❛♥❞♦ ♣♦r p ❡ p ✱ e Σ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡✜♥✐r❡♠♦s ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧♦❝❛❧ ❡♥tr❡ Σ 1 ✧❛♦ em p ) 1

  ❧♦♥❣♦✧❞❛s ❢♦❧❤❛s ❞❡ F✱ ✧s♦❜r❡✧♦ ❝❛♠✐♥❤♦ γ ❧❡✈❛♥❞♦ p ✳ i ) k ❙❡❣✉❡ ❞♦ ▲❡♠❛ q✉❡ ❡①✐st❡♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❝❛rt❛s ❧♦❝❛✐s (U i ❡ =0

  < ... < t k = 1 i ∩ U j 6= ∅ i ∪ U j +1 ✉♠❛ ♣❛rt✐çã♦ ❞❡ [0, 1], 0 = t t❛✐s q✉❡ s❡ U ❡♥tã♦ U

  ✸✵ ✶✳✺✳ ❚❊❖❘❊▼❆ ❉❊ ❍❆❊❋▲■●❊❘

  ❋✐❣✉r❛ ✶✳✶✶✿ ❍♦❧♦♥♦♠✐❛ i , t i ]) ⊂ U i +1 ❡stá ❝♦♥t✐❞♦ ♥✉♠❛ ❝❛rt❛ ❧♦❝❛❧ ❞❡ F ❡ γ([t ♣❛r❛ t♦❞♦ 0 ≤ i ≤ k✳ ❉✐③❡♠♦s ❡♥tã♦ q✉❡ ❡①✐st❡ ✉♠❛ ❝❛❞❡✐❛ s✉❜♦r❞✐♥❛❞❛ ❛ γ ♦✉✱ ♣♦r s✐♠♣❧✐❝✐❞❛❞❡✱ i ) i k q✉❡ (U =0 é ✉♠❛ ❝❛❞❡✐❛ s✉❜♦r❞✐♥❛❞❛ ❛ γ✳ i ) ⊂ U i ∩ U i

  P❛r❛ ❝❛❞❛ 0 < i < k + 1 ✜①❡♠♦s ✉♠❛ s❡çã♦ tr❛♥s✈❡rs❛❧ ❛ F, D(t −1 i ) ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠ ❞✐s❝♦ ❞❡ ❞✐♠❡♥sã♦ n ♣❛ss❛♥❞♦ ♣♦r γ(t ✳ ❈♦❧♦q✉❡♠♦s t❛♠❜é♠ D(0) = Σ e D(1) = Σ i ) 1 i ) i i ) i (x) ✳ ❊♥tã♦ ♣❛r❛ ❝❛❞❛ x ∈ D(t s✉✜❝✐❡♥t❡♠❡♥t❡ ♣ró①✐♠♦

  ❞❡ γ(t ❛ ♣❧❛❝❛ ❞❡ U q✉❡ ♣❛ss❛ ♣♦r x ❡ ✐♥t❡r❝❡♣t❛ D(t +1 ♥✉♠ ú♥✐❝♦ ♣♦♥t♦ f ✳ i ⊂ D(t i ) i ) i ❖ ❞♦♠í♥✐♦ ❞❛ ❛♣❧✐❝❛çã♦ f ❝♦♥té♠ ✉♠ ❞✐s❝♦ D ❝♦♥t❡♥❞♦ γ(t ✳ ❉❛í✱ é ❝❧❛r♦ q✉❡ ❛ ❝♦♠♣♦s✐çã♦ f γ = f k ◦ f k ◦ ... ◦ f −1

  ∈ Σ γ ❊stá ❜❡♠ ❞❡✜♥✐❞❛ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ p ✳ ❈❤❛♠❛r❡♠♦s f ❞❡ ❛♣❧✐❝❛çã♦

  ❞❡ ❤♦❧♦♥♦♠✐❛ ❛ss♦❝✐❛❞❛ ❛ γ✳ ❉❡✜♥✐çã♦ ✶✳✶✵✹✳ ❙❡❥❛♠ X, Y ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❡ x ∈ X✳ ◆♦ ❝♦♥❥✉♥t♦ ❞❛s ❛♣❧✐❝❛çõ❡s X ⊃ f : V → Y ✱ ♦♥❞❡ V é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x✱ ✐♥tr♦❞✉③✐♠♦s ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ R : fRg s❡ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ W ❞❡ x t❛❧ q✉❡ f | W = g| W

  ✳ ❆ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ f é ❞❡♥♦♠✐♥❛❞❛ ❣❡r♠❡ ❞❡ f ❡♠ x✳ ◗✉❛♥❞♦ X = Y ✱ ♦ ❝♦♥❥✉♥t♦ G(X, x) ❞❡ ❣❡r♠❡s ❞❡ ❤♦♠❡♦♠♦r✜s♠♦s ❧♦❝❛✐s q✉❡ ❞❡✐①❛♠ ✜①♦ x é ✉♠ ❣r✉♣♦ ❝♦♠ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❣❡r♠❡ (f) ◦ germe(g) = germe(f ◦ g) −1 ✱ s❡♥❞♦ q✉❡ ♦ ❞♦♠í♥✐♦ ❞❡ f ◦ g é ❛ ✐♥t❡rs❡çã♦ ❞♦ ❞♦♠í♥✐♦ ❞❡ g ❝♦♠ g

  ✭❞♦♠í♥✐♦ ❞❡ f✮✳ , γ : [0, 1] → M

  Pr♦♣♦s✐çã♦ ✶✳✶✵✺✳ ❙❡❥❛♠ γ 1 2 ❝❛♠✐♥❤♦s ❝♦♥t✐❞♦s ❡♠ ✉♠❛ ❢♦❧❤❛ F de F (0) = γ (0) = p e γ (1) = γ (1) = p , Σ 1 2 1 2 1 1 t❛✐s q✉❡ γ

  ✳ ❙❡❥❛♠ Σ ∈ Σ ∈ Σ ⊂ Σ → Σ e p 1 1 γ : D 1 1 s❡çõ❡s tr❛♥s✈❡rs❛✐s ❛ F ❝♦♠ p ✳ ❙❡❥❛♠ f γ : D ⊂ Σ → Σ 2 2 1 1 γ e f γ 1 2

  ✸✶ γ ) e germ(f γ ) def γ e f γ 1 2 ✶✳✺✳ ❚❊❖❘❊▼❆ ❉❊ ❍❆❊❋▲■●❊❘ 1 2 s❡❥❛♠ germ(f ♦s ❣❡r♠❡s ❞❡ ❤♦❧♦♥♦♠✐❛ ❡♠ p ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ss✐♠✱ eγ

  ✶✳ ❙❡ γ 1 sã♦ ❤♦♠♦tó♣✐❝♦s ❝♦♠ ❡①tr❡♠♦s ✜①♦s ❞❡♥tr♦ ❞❡ F ❡♥tã♦ germ(f γ ) = germ(f γ ) 1 2 ✳ −1 = p 1 e Σ = Σ 1 γ )

  ✷✳ ❙❡ p ✱ ❡♥tã♦ ❛ tr❛♥s❢♦r♠❛çã♦ γ ←→ germ(f ✐♥❞✉③ ✉♠ ❤♦♠♦♠♦r✜s♠♦

  Φ : φ (F, p ) → G(Σ , p ), Φ([γ]) = ϕ 1 γ −1 ❞♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ F ❡♠ p ♥♦ ❣r✉♣♦ ❞❡ ❣❡r♠❡s ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s r C de Σ q✉❡ ❞❡✐①❛♠ p ✜①♦✳

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ♣á❣✐♥❛ ✻✺✱ ❝❛♣✐t✉❧♦ ■❱✳ ) = Φ(φ (F, p )) de G(Σ , p ) 1

  ❉❡✜♥✐çã♦ ✶✳✶✵✻✳ ❖ s✉❜❣r✉♣♦ Hol(F, p é ∈ F

  , p 1 ❝❤❛♠❛❞♦ ❣r✉♣♦ ❞❡ ❤♦❧♦♥♦♠✐❛ ❞❡ F em p ✳ ❉❛❞♦s p ✱ q✉❛✐sq✉❡r e α(1) = p 1

  ❝❛♠✐♥❤♦ α : I → F ❝♦♠ α(0) = p ✐♥❞✉③ ✉♠ ✐s♦♠♦r✜s♠♦ ϑ : Hol(F, p ) → Hol(F, p ) 1 α α ◦ Φ[µ] ◦ ϕ

  ♣♦♥❞♦ ϑ(Φ[µ]) = ϕ −1 ✳ ❉❡st❛ ♠❛♥❡✐r❛ ♣♦❞❡♠♦s ❢❛❧❛r ❞♦ ❣r✉♣♦ ❞❡ ❤♦❧♦♥♦♠✐❛ ❞❡ F ❝♦♠♦ s❡♥❞♦

  ) q✉❛❧q✉❡r ❣r✉♣♦ ✐s♦♠♦r❢♦ ❛ Hol(F, p ✳ P❛r❛ s✐♠♣❧✐✜❝❛r ♥♦t❛çã♦✱ às ✈❡③❡s ❝♦♥❢✉♥❞✐r❡♠♦s ✉♠ ❡❧❡♠❡♥t♦ ❞❛ ❤♦❧♦♥♦♠✐❛ ❝♦♠ ✉♠ ❞♦s s❡✉s r❡♣r❡s❡♥t❛♥t❡s✳

  ❆ s❡❣✉✐r✱ ❞❡✜♥✐r❡♠♦s ❤♦❧♦♥♦♠✐❛ ♥❛ ❢♦❧❤❡❛çã♦ ❡stá✈❡❧ ❡ ✐♥stá✈❡❧ ❡♠ ✉♠ ✢✉①♦ ❆♥♦s♦✈✳ ❚❛❧ ❞❡✜♥✐çã♦ é ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ❡ s❡rá ✉s❛❞❛ ♥♦ ▲❡♠❛ ❉❡✜♥✐çã♦ ✶✳✶✵✼✳ ❙❡❥❛ F ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠ ♥❛ ✈❛r✐❡❞❛❞❡ M✳ ❉✐③❡♠♦s q✉❡✱ F ∈ F t❡♠ ❤♦❧♦♥♦♠✐❛ tr✐✈✐❛❧ s❡ Hol(F ) é ✉♠ ❣r✉♣♦ tr✐✈✐❛❧✱ ♣❛r❛ t♦❞❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ γ ∈ F ❝♦rr❡s♣♦♥❞❡ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ❤♦❧♦♥♦♠✐❛ q✉❡ é ❛ ✐❞❡♥t✐❞❛❞❡ γ = Id

  ❡♠ ❛❧❣✉♠❛ s❡çã♦ tr❛♥s✈❡rs❛❧ ❛ γ e F ✱ ✐st♦ é✱ f ✳ ❉✐③❡♠♦s q✉❡✱ γ é ✉♠❛ γ ❝✉r✈❛ ❢❡❝❤❛❞❛ ❡♠ F t❡♠ ❤♦❧♦♥♦♠✐❛ ❡①♣❛♥s✐✈❛ s❡ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ❤♦❧♦♥♦♠✐❛ f é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❧♦❝❛❧ ❡①♣❛♥s✐✈♦ ❡♠ ❛❧❣✉♠❛ s❡çã♦ tr❛♥s✈❡rs❛❧ J tr❛♥s✈❡rs❛❧ ❛ γ✳ ❯♠❛ ✈❡③ q✉❡ ❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❤♦❧♦♥♦♠✐❛ ❞❡✜♥✐❞❛s ❡♠ s❡çõ❡s tr❛♥s✈❡rs❛✐s ❞✐st✐♥t❛s sã♦ ❝♦♥❥✉❣❛❞❛s ♣♦r ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✱ ❛ss✐♠ ♣❛r❛ q✉❡ ✉♠❛ ❝✉r✈❛ γ ❢❡❝❤❛❞❛ t❡♥❤❛ ❤♦❧♦♥♦♠✐❛ ❡①♣❛♥s✐✈❛ ✐♥❞❡♣❡♥❞❡ ❞♦ s❡❣♠❡♥t♦ tr❛♥s✈❡rs❛❧ q✉❡ s❡ ❝♦♥s✐❞❡r❡✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❡✜♥✐♠♦s q✉❛♥❞♦ ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ t❡♠ ❤♦❧♦♥♦♠✐❛ ❝♦♥tr❛t✐✈❛✳ ❖❜s❡r✈❛çã♦ ✶✳✶✵✽✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❛ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❛s ❢♦❧❤❛s ❞❡ ✉♠❛ ❢♦❧❤❡❛çã♦ ✐♥stá✈❡❧ ✭♦✉ ❡stá✈❡❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✮ ❞❡ ✉♠ ✢✉①♦ ❆♥♦s♦✈ s♦❜r❡ ✉♠❛ u +1 ✈❛r✐❡❞❛❞❡ r✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛ M é ❤♦♠❡♦♠♦r❢♦ ❛ R ✱ ✭♦♥❞❡ u é ❛ ❞✐♠❡♥sã♦ u +1 ❞❛ ❢♦❧❤❡❛çã♦ ✐♥stá✈❡❧✮ ♦✉ q✉❡ ♥ã♦ s❡❥❛ ❤♦♠❡♦♠♦r❢❛ ❛ R ✳ ❉❡st❛ ❢♦r♠❛✱ ♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❢♦❧❤❛ ✜❝❛ ❣❡r❛❞♦ ♣♦r ✉♠❛ ú♥✐❝❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ ❞♦ ✢✉①♦✳ ◆♦t❡ q✉❡✱ ♦ ✢✉①♦ ✐♥❞✉③ ✉♠❛ ♦r✐❡♥t❛çã♦ ❛ t♦❞❛s ❝✉r✈❛s ♥ã♦ ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ♥✉❧❛s ❞❡♥tr♦ ❞❛ ❢♦❧❤❛✳

  ✸✷ ✶✳✺✳ ❚❊❖❘❊▼❆ ❉❊ ❍❆❊❋▲■●❊❘

  ❚❡♦r❡♠❛ ❞❡ ❍❛❡✢✐❣❡r ❊♥✉♥❝✐❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❍❛❡✢✐❣❡r ♥❡st❛ s❡çã♦ ♣❛r❛ ❢♦❧❤❡❛çõ❡s ❞❡

  ❝♦❞✐♠❡♥sã♦ ✉♠ ❡ ❢❛r❡♠♦s ♦ r♦t❡✐r♦ ❞❡st❛ ❞❡♠♦♥str❛çã♦✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ❚❡♦r❡♠❛ ✶✳✶✵✾ ✭❚❡♦r❡♠❛ ❞❡ ❍❛❡✢✐❣❡r✮✳ ❙❡❥❛ F ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❝♦❞✐♠❡♥sã♦ 2

  ✉♠ ❞❡ ❝❧❛ss❡ C ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ tr❛♥s✈❡rs❛❧ ❛ F ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦✳ ❊♥tã♦ ❡①✐st❡♠ ✉♠❛ ❢♦❧❤❛ F

  ❞❡ F ❡ ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ Γ ⊂ F ❝✉❥♦ ❣❡r♠❡ ❞❡ ❤♦❧♦♥♦♠✐❛ ♥✉♠ s❡❣♠❡♥t♦ J tr❛♥s✈❡rs❛❧ ❛ F✱ ❝♦♠ p = J ∩ Γ✱ é ❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ J − {p}

  ♠❛s ❞✐❢❡r❡ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ ❞❡ p em J✳ ❋❛r❡♠♦s ♦ ❡s❜♦ç♦ ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❍❛❡✢✐❣❡r ❜❛s❡❛❞♦s ❡♠ 1

  → M ❙❡❥❛ α : S ✉♠❛ ❝✉r✈❛ tr❛♥s✈❡rs❛❧ ❛ F ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦✳ ■st♦ 2

  → M s✐❣♥✐✜❝❛ q✉❡✱ ❡①✐st❡ ✉♠❛ ❡①t❡♥sã♦ ❞❡ α ❛ ✉♠❛ ❛♣❧✐❝❛çã♦ A : D ✳ ❊st❛ ❛♣❧✐❝❛çã♦ é ❝♦♥tí♥✉❛✱ ♠❛s ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❛♣r♦①✐♠❛çã♦ ❞❡ ❙t♦♥❡✲ ❲❡✐❡rstr❛ss ♣♦❞❡♠♦s s✉♣♦r q✉❡ A é ❞❡ ❝❧❛ss❡ C ✳ ❆ ❛♣❧✐❝❛çã♦ A é ❛ s❡❣✉✐r ❛♣r♦①✐♠❛❞❛ ♣♦r 2

  → M ✉♠❛ ❛♣❧✐❝❛çã♦ g : D tr❛♥s✈❡rs❛❧ ❛ F✱ ❡①❝❡t♦ ♥✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣♦♥t♦s {q 1 ♦♥❞❡ ❛ t❛♥❣ê♥❝✐❛ ❞❡ g ❝♦♠ ❛s ❢♦❧❤❛s ❞❡ F é ♥ã♦ ❞❡❣❡♥❡r❛❞❛✱ ♦✉ s❡❥❛✱ , ..., q l )} ❞❡ ✉♠ ❞♦s ❞♦✐s t✐♣♦s s❡❣✉✐♥t❡s✿

  ❋✐❣✉r❛ ✶✳✶✷✿ ❚❛♥❣ê♥❝✐❛ t✐♣♦ ❝❡♥tr♦ ❋✐❣✉r❛ ✶✳✶✸✿ ❚❛♥❣ê♥❝✐❛ t✐♣♦ ❙❡❧❛ 2 ∗ ∗

  , F = g (F) ❉❡st❛ ❢♦r♠❛ ♦❜t❡♠♦s ✉♠❛ ❢♦❧❤❡❛çã♦ ❝♦♠ s✐♥❣✉❧❛r✐❞❛❞❡s ❞❡ D ✱ 1 l , ..., q )} tr❛♥s✈❡rs❛❧ ❛♦ ❜♦r❞♦ ❡ ❝✉❥♦ ❞♦♠í♥✐♦ s✐♥❣✉❧❛r {q é ❝♦♥st✐t✉í❞♦ ❞❡ ❝❡♥tr♦s

  ❡ s❡❧❛s✳ i ◗✉❛♥❞♦ q é ✉♠❛ s❡❧❛ ❞❡ F ✱ s❡ ♥♦s r❡str✐♥❣✐r♠♦s ❛ ✉♠❛ ♣❡q✉❡♥❛ ✈✐③✐♥❤❛♥ç❛

  |V V i 1 , γ 2 , γ 3 , γ 4

  ❞❡ q ❝♦♠♦ ♥❛ ✜❣✉r❛ 1✱ F ♣♦ss✉✐ q✉❛tr♦ ✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s γ i i

  ✸✸ ✶✳✺✳ ❚❊❖❘❊▼❆ ❉❊ ❍❆❊❋▲■●❊❘ i

  ❙❡ γ é ✉♠❛ ❢♦❧❤❛ ❞❡ F t❛❧ γ ∩ V ❝♦♥té♠ ✉♠❛ s❡♣❛r❛tr✐③ ❧♦❝❛❧ ❞❡ q ✱ ❞✐③❡♠♦s i q✉❡ γ é ✉♠❛ s❡♣❛r❛tr✐③ ❞❡ ❞❡ q ✱ ❡ s❡ γ ∩ V ❝♦♥té♠ ❞✉❛s s❡♣❛r❛tr✐③❡s ❧♦❝❛✐s ❞❡ q i

  ✱ γ é ✉♠❛ ❛✉t♦✲❧✐❣❛çã♦ ❞❡ s❡❧❛s✳ ◗✉❛♥❞♦ γ é s❡♣❛r❛tr✐③ ❞❡ ❞✉❛s s❡❧❛s ❞✐st✐♥t❛s✱ ❞✐③❡♠♦s q✉❡ γ é ✉♠❛ ❧✐❣❛çã♦ ❞❡ s❡❧❛s ❡ q✉❡ s❡ ❞✉❛s s❡❧❛s sã♦ ❧✐❣❛❞❛s✳ ◆❛ ❢❛s❡ ✜♥❛❧ ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ✉s❛r❡♠♦s q✉❡ é ♣♦ssí✈❡❧ ♦❜t❡r g ❞❡ t❛❧ ❢♦r♠❛ i e q j q✉❡ ❞✉❛s s❡❧❛s ❞✐st✐♥t❛s ❞❡ F ♥ã♦ s❡❥❛♠ ❧✐❣❛❞❛s✳ ❖❜s❡r✈❡ q✉❡✱ s❡ q i ) e g(q j ) sã♦ ❞✉❛s s❡❧❛s ❧✐❣❛❞❛s ❞❡ F ✱ ❡♥tã♦ g(q ❡stã♦ ❝♦♥t✐❞❛s ♥✉♠❛ ♠❡s♠❛

  ❢♦❧❤❛ ❞❡ F✳ ◗✉❛♥❞♦ ✐ss♦ ♦❝♦rr❡✱ ♣♦❞❡♠♦s ♠♦❞✐✜❝❛r g✱ ♣❛r❛ ♦❜t❡r ✉♠❛ ❢✉♥çã♦ 2 g : D ˜ → M ) e g( ˜ ˜ q ) i j ✱ ♣ró①✐♠❛ ❞❡ g ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ˜g(˜q ❡stã♦ ❡♠ ❢♦❧❤❛s i 6= ˜ q j (F)

  ❞✐st✐♥t❛s ❞❡ F s❡♠♣r❡ q✉❡ ˜q sã♦ s❡❧❛s ❞❡ ˜g ✳ P♦❞❡♠♦s ❡♥tã♦ s✉♣♦r✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ F ♥ã♦ ♣♦ss✉✐ s❡❧❛s ❞✐st✐♥t❛s ❧✐❣❛❞❛s✱ ♣♦❞❡♥❞♦ ♣♦ss✉✐r ♥♦ ❡♥t❛♥t♦ ❛✉t♦✲ ❧✐❣❛çõ❡s ❞❡ s❡❧❛s ✭q✉❡ sã♦ ✐♥❞❡str✉tí✈❡✐s ♣♦r ♣❡q✉❡♥❛s ♣❡rt✉❜❛çõ❡s✮✳ ◆❛ ✜❣✉r❛ ✭✈♦✉ ❝♦❧♦❝❛r ❛ r❡❢❡r❡♥❝✐❛ ❞❛ ✜❣✉r❛ ❛q✉✐✱ ❛✐♥❞❛ ❡st♦✉ ❞❡s❡♥❤❛♥❞♦✮✱ ✐❧✉str❛♠♦s ✉♠ ❡①❡♠♣❧♦ ❞♦ q✉❡ ♣♦❞❡ ♦❝♦rr❡r✿

  ❋✐❣✉r❛ ✶✳✶✹✿ ❋♦❧❤❡❛çõ❡s ♦r✐❡♥tá✈❡✐s ♥♦ ❝ír❝✉❧♦ ❖❜s❡r✈❡ q✉❡ ❛ ❢♦❧❤❡❛çã♦ F é ❧♦❝❛❧♠❡♥t❡ ♦r✐❡♥tá✈❡❧ ✭♠❡s♠♦ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ 2

  ❞❡ ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡✮✳ ❈♦♠♦ D é s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦ ❡①✐st❡ ✉♠ ❝❛♠♣♦ ❞❡ 2 1 , ..., q l )} ✈❡t♦r❡s Y ❡♠ D ✱ ❝♦♠ s✐♥❣✉❧❛r✐❞❛❞❡s {q ✱ ❝✉❥❛s ór❜✐t❛s r❡❣✉❧❛r❡s sã♦ ❛s ❢♦❧❤❛s ❞❡ F ✳ ❆♣❧✐❝❛♥❞♦✲ s❡ ❛ t❡♦r✐❛ ❞❡ P♦✐♥❝❛ré✲ ❇❡♥❞✐①s♦♥ ❛♦ ❝❛♠♣♦ Y ✱ ♠♦str❛✲ s❡ q✉❡ ❡①✐st❡♠ ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ Γ✱ ✐♥✈❛r✐❛♥t❡ ♣♦r Y ✱ ❡ ✉♠ s❡❣♠❡♥t♦ J

  ✱ tr❛♥s✈❡rs❛❧ ❛ Y ❞❡ t❛❧ ❢♦r♠❛ q✉❡ é ♣♦ssí✈❡❧ ❞❡✜♥✐r✲ s❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ♣r✐♠❡✐r♦ r❡t♦r♥♦ f✱ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ p = Γ ∩ J ❡♠ J✱ s❡❣✉✐♥❞♦✲ s❡ ❛s ór❜✐t❛s ♣♦s✐t✐✈❛s ❞❡ Y ✱ ❛ q✉❛❧ é ❛ ✐❞❡♥t✐❞❛❞❡ ❡ ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ J −{p}✱ ♠❛s ♥ã♦ é ❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ ♥❡♥❤✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ p ❡♠ J✳ ❆ ✐♠❛❣❡♠ ❞❡ Γ ♣♦r g ❞❡✜♥❡ ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ g(Γ) ♥✉♠❛ ❢♦❧❤❛ ❞❡ F ❝✉❥❛ ❤♦❧♦♥♦♠✐❛ é ❝♦♥❥✉❣❛❞❛ ❛ f✳ ■st♦ ❝♦♥❝❧✉✐ ♦ ❡s❜♦ç♦ ❞❛ ❞❡♠♦♥str❛çã♦✳

  ❈❛♣ít✉❧♦ ✷ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧

  ◆❡st❡ ❝❛♣ít✉❧♦ t❡♠♦s ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛♣r❡s❡♥t❛r ❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ q✉❡ ♠♦t✐✈♦✉ ❡st❡ ❡st✉❞♦✳ P❛r❛ ✐st♦✱ ✉t✐❧✐③❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ❡ ❝♦♥❝❡✐t♦s ❛♣r❡s❡♥t❛❞♦s ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r

  ■♥✐❝✐❛❧♠❡♥t❡✱ ♣r♦✈❛r❡♠♦s q✉❡ ❛ ❢✉♥çã♦ Γ t❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ s❡✱ s♦♠❡♥t❡ s❡✱ ❛ ❢✉♥çã♦ P t❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧✳ }

  , ..., g m (M )

  ▲❡♠❛ ✷✳✶✳ ❙❡❥❛ G = {g 1 ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ❣❡r❛❞♦r❡s ❞♦ π 1 ✳ 1 , k 2 ❊♥tã♦✱ ❡①✐st❡ k t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ α 6= e✱ l (α) 2

  ≤ ≤ k k 1 2 l (α) 1 ε ε ε 1 2 n g ...g i = ±1 ❉❡♠♦♥str❛çã♦✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ s❡ α = g i i i n ✱ ♦♥❞❡ ε 1 2

  | + |ε | + ... + |ε | (α) = |ε n t❡♠♦s l 1 1 2 ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ❛ ❞❡✜♥✐çã♦✱ l (α) ≤ l (g i ) + l (g i ) + ... + l (g i ) 2 2 1 2 2 ✳ ❈♦♠♦ G é ✜♥✐t♦ t♦♠❡♠♦s 2 n k 2 = max{l 2 (g 1 ), l 2 (g 2 ), ..., l 2 (g m )}.

  ❉❛í✱ l (α) l (g i ) + l (g i ) + ... + l (g i ) k + k + ... + k nk 2 2 1 2 2 2 n 2 2 2 2 ≤ ≤ = ≤ k . 2 |ε | + |ε | + ... + |ε | |ε | + |ε | + ... + |ε | l 1 (α) 1 2 n 1 2 n n

  P❛r❛ ♦❜t❡r♠♦s k 1 ✱ ❝♦♥s✐❞❡r❛r❡♠♦s ♦ r❡❝♦❜r✐♠❡♥t♦ ✉♥✐✈❡rs❛❧ ❞❛ ✈❛r✐❡❞❛❞❡ M✳ M → M

  M ❙❡ p : f ❛ ❛♣❧✐❝❛çã♦ r❡❝♦❜r✐♠❡♥t♦✱ ✜①❡♠♦s ex ∈ f t❛❧ q✉❡ p(ex) = x ✳ ❙❡❥❛ K

  ✉♠ ❞♦♠í♥✐♦ ❢✉♥❞❛♠❡♥t❛❧ ❝♦♠♣❛❝t♦ ❝♦♠ ❞✐â♠❡tr♦ D ❡ d ♦ ❞✐â♠❡tr♦ ❞❡ M✳ M

  ❈♦♥s✐❞❡r❡ ❛ ❜♦❧❛ ❡♠ f ❞❡ ❝❡♥tr♦ ex ❡ r❛✐♦ 3d + D x, 3d + D) B(e 1 ( ˜ M )

  ❖❜s❡r✈❡ q✉❡✱ s❡ ♣r❛ ❛❧❣✉♠ γ ∈ π t❡♠✲s❡ q✉❡ γK ✐♥t❡rs❡❝t❛ ❛ ❜♦❧❛ ❞❡ ❝❡♥tr♦ x ˜ ❡ r❛✐♦ 3d B(ex, 3d) ❡♥tã♦ γK ❡stá ❝♦♥t✐❞❛ ♥❛ ❜♦❧❛ B(ex, 3d + D)

  ✸✺ ❋✐❣✉r❛ ✷✳✶✿ ❇✭ex, 3d + D✮ 1 (M ) : l (α) ≤ 3d} 2 −1 −1 ◆♦t❡♠♦s q✉❡ {α ∈ π é ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦✳ ❉❡ ❢❛t♦✱ ❝♦♠♦

  {x } {x p é ❞✐s❝r❡t♦ t❡♠♦s q✉❡ p }∩B(ex, 3d) é ✜♥✐t♦✱ ❧♦❣♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❧❛ss❡s

  ❤♦♠♦tó♣✐❝❛s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ 3d q✉❡ ❧✐❣❛♠ ˜x ❝♦♠ ❛❧❣✉♠ ♣♦♥t♦ −1 {x

  ❡♠ p } ∩ B(ex, 3d) é t❛♠❜é♠ ✜♥✐t♦✳ ❆ss✐♠ ❛s s✉❛s ♣r♦❥❡çõ❡s ♣♦r p ❢♦r♠❛♠ (M ) : l (α) ≤ 3d}

  ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❡♠ M✳ ■st♦ é✱ {α ∈ π 1 2 é ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦✳ 1 (α) : α ∈ {α ∈ π (M ) : l (α) ≤ 3d}} 1 2 ❙❡❥❛ k = max{l 1 ∗ γ ∗ ... ∗ γ 2 m

  ❉❛❞♦ ✉♠ ❝❛♠✐♥❤♦ α ♣♦❞❡♠♦s ♣❛rt✐❝✐♦♥á✲❧♦ ♥❛ ❢♦r♠❛ α = γ ♦♥❞❡ |γ i | = d, i = 1, ..., m − 1 m | < d

  ❡ 0 ≤ |γ ✱ ✈❡❥❛ ❛ ❋✐❣✉r❛ m | < d ❋✐❣✉r❛ ✷✳✷✿ 0 ≤ |γ 1 |+|γ |+|γ |+...+|γ m |+|γ m | ≥ |γ |+...+|γ m | = (m−1)·d 2 3 1

  ▲♦❣♦✱ |α| = |γ −1 −1 |α| ≥ m − 1

  P♦rt❛♥t♦✱ ♦✉ d |α| m ≤ + 1 (∗) d

  ✸✻ i i | ≤ d ❆❧é♠ ❞✐ss♦✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❝❛♠✐♥❤♦s β ❝♦♠ |β ✱ t❛❧ q✉❡ α = −1 −1 −1 (γ ∗ β ) ∗ (β ∗ γ ∗ β ) ∗ ... ∗ (β m ∗ γ m ∗ β ) ∗ (β m ∗ γ m ). 1 1 1 2 2 −2 −1 m −1 ❱❡❥❛ ❋✐❣✉r❛ −1

  

  i | ≤ 3d −1 −1 −1 ❋✐❣✉r❛ ✷✳✸✿ |α 1 = (γ 1 ), α ∗ β ∗ γ ∗ β ∗ γ ∗ β 2 = (β 1 2 ), α 3 = (β 2 3 ), ..., α m =

  ❆ss✐♠✱ s❡ α 1 2 3 (β m ∗ γ m ) −1 ✳

  ∗ ... ∗ α −1 −1 α = α 1 m 1 | = |γ |+|β | = 2d ≤ 3d, |α | = |β |+|γ |+|β | ≤ 3d, ..., |α | ≤ 1 2 1 2 m ❖❜s❡r✈❡ q✉❡✱ |α 1 2

  |β m | + |γ m | ≤ 3d (α i ) ≤ 3d −1 ✳ P♦rt❛♥t♦✱ l ✳ 2 ❆❣♦r❛✱ l (α) ≤ l (α ) + l (α ) + ... + l (α m ) 1 1 1 1 2 1 l (α) ≤ k + k + k + ... + k ≤ mk 1 P♦r (∗) t❡♠♦s

  |α| l 1 (α) ≤ ( + 1) · k (∗∗) d = inf{|β|; β 6≃ 0} > 0

  ❙❡❥❛ d ✱ ♦❜s❡r✈❡ q✉❡✱ d ✳ ❈❛s♦ ❝♦♥trár✐♦ s❡ ❡①✐st✐r✐❛ β 6≃ 0 ❡ |β| = 0 ❧♦❣♦ t❡rí❛♠♦s q✉❡ β ❡st❛r✐❛ ❝♦♥t✐❞♦ ♥✉♠❛ ❜♦❧❛✱ ❛ss✐♠ s❡r✐❛ ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦✳ ❆❜s✉r❞♦✳ d

  )|α| ❆✜r♠❛♠♦s q✉❡ |α| + d ≤ (1 + ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ α 6≃ 0✱ |α| ≤ d ✳ ❊♥tã♦✱ |α| d d ≤ d > 0 d ✱ ♣♦✐s d ≥ 0 ❡ d

  |α| |α| + d ≤ |α| + d d d |α| + d ≤ (1 + )|α|. d

  ✐st♦ ♣r♦✈❛ ❛ ❛✜r♠❛çã♦✳

  ✸✼ ✷✳✶✳ P❘❖❱❆ ❉❖ ❚❊❖❘❊▼❆

  P♦r (∗∗)✱ t❡♠♦s✿ |α| l (α) ≤ ( + 1)k 1 d

  |α| + d = ( ) d d

  (1 + )|α|k d ≤ d

  (d + d)|α|k = dd

  ♣❛r❛ q✉❛❧q✉❡r α 6= 0✱ d + d l (α) ≤ inf { k|α|} 1 α ∈α dd d + d

  ≤ k. inf {α} α ∈α dd dd 1 = ❉❡✜♥❛ k ✱ k (d+d ) ❧♦❣♦✱ l (α) dd 2

  ≥ ≥ k 1 l (α) k(d + d ) 1

  ✷✳✶ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛

  ❈♦♠♦ ♦ ✢✉①♦ é ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠✱ ♣♦❞❡♠♦s s✉♣♦r✱ s❡♠ ♣❡r❝❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ s ) = 1 q✉❡ dim(E ✭❝❛s♦ ❝♦♥trár✐♦✱ ✐♥✈❡rt❡♠♦s ♦ s❡♥t✐❞♦ ❞♦ ✢✉①♦✮✳ ▲❡♠❜r❡♠♦s q✉❡ ss s u

  (x) (x) s❡ x ∈ M ❡♥tã♦ W é ❛ ✈❛r✐❡❞❛❞❡ t❛♥❣❡♥t❡ à E ♥♦ ♣♦♥t♦ x ❡ W ❞❡♥♦t❛

  ❛ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧ ❢r❛❝❛ ❞❡ x ❈♦♠♦ ❛ ❢♦❧❤❡❛çã♦ ❡stá✈❡❧ ♣♦❞❡ ❞❡✜♥✐r ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ♥ã♦ ♥✉❧♦✱ ♣♦❞❡♠♦s t

  ❞❡✜♥✐r ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ✉♥✐tár✐♦ ❡♠ M✱ ✈❡❥❛ ❆ss✐♠ s❡❥❛ η ♦ ✢✉①♦ ❝♦♥tí♥✉♦ ❛ss♦❝✐❛❞♦ ❛ ❡ss❡ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❡♠ M✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ ♦r❜✐t❛ é ❞❛❞♦ ♣❡❧♦ t❡♠♣♦✳ ❈♦♥s✐❞❡r❡ N ⊂ M ✉♠ ❝♦♥❥✉♥t♦ ♠✐♥✐♠❛❧ u t ∈ N (x )

  ♣❛r❛ η ❡ t♦♠❡ x ✳ ❙❡❥❛ D ⊂ W ✉♠ ❞✐s❝♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❢❡❝❤❛❞♦ ❡ ♠❡r❣✉❧❤❛❞♦ ❝♦♠ x ♣♦♥t♦ ✐♥t❡r✐♦r à D✳ t

  ◆♦t❡ q✉❡✱ η t❡♠ ✈❡❧♦❝✐❞❛❞❡ ✉♠ ❡♠ t♦❞♦ ♣♦♥t♦✳ ❙❡❥❛ V ✉♠ ❛❜❡rt♦ t❛❧ q✉❡ [

  V = η t (D) −δ≤t≤δ ♣❛r❛ δ > 0 ♣❡q✉❡♥♦✳ t ❖❜s❡r✈❡ q✉❡✱ ❛ ór❜✐t❛ ❢✉t✉r❛ ❞❡ x ♣♦r η ❞á ✈ár✐❛s ✈♦❧t❛s✱ ♣♦ré♠ s❡♠♣r❡

  ∈ N r❡t♦r♥❛ ❞❡♥tr♦ ❞❡ V ✱ ✐ss♦ s❡ ❞❡✈❡ ❛♦ ❢❛t♦ ❞❡ x s❡r ✉♠ ♣♦♥t♦ r❡❝♦rr❡♥t❡ ❞❡ η t t

  ✱ ❝♦♠♦ V ❡stá ❞❡✜♥✐❞♦ ♣♦r ✉♠❛ ✧❝❛✐①❛ ❞❡ ✢✉①♦s✧❞❡ η ✱ ✈❡♠♦s q✉❡ ❡ss❛s ór❜✐t❛s

  ✸✽ ✷✳✶✳ P❘❖❱❆ ❉❖ ❚❊❖❘❊▼❆

  ❋✐❣✉r❛ ✷✳✹✿ ❞✐s❝♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❢❡❝❤❛❞♦ ❡ ♠❡r❣✉❧❤❛❞♦ ❡♠ M ❋✐❣✉r❛ ✷✳✺✿ ❈❛✐①❛ ❞❡ ❋❧✉①♦ V t❛♠❜é♠ ❝♦rt❛♠ D✱ ✐♥✜♥✐t❛s ✈❡③❡s✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❞❡✜♥✐r t❡♠♣♦s s✉❝❡ss✐✈♦s ♥❛

  ór❜✐t❛ ❞❡ x q✉❡ ❡stá ❛✐♥❞❛ ✈♦❧t❛ ❛ ❝♦rt❛r D✳ ❊①✐st❡ ✉♠ ♥ú♠❡r♦ ♠❡♥♦r ♣♦s✐t✐✈♦ t t (x ) ∈ D n n n 1 t❛❧ q✉❡ η 1 ✳ ■♥❞✉t✐✈❛♠❡♥t❡✱ ❞❛❞♦ t −1 ✱ ❞❡✜♥❛ t ❝♦♠♦ ♦ ♠❡♥♦r t > t −1 t n (x ) ∈ D ❞❡ t❛❧ ♠♦❞♦ q✉❡ η ✳ t n − t n t (x ) e η t (x ) −1 é ♦ t❡♠♣♦ ❡♥tr❡ η n−1 ✳ n + n n − t < L

  ▲❡♠❛ ✷✳✷✳ ❊①✐st❡ L > 0 t❛❧ q✉❡ t −1 ♣❛r❛ t♦❞♦ n ∈ Z ✳

  ✸✾ ✷✳✶✳ P❘❖❱❆ ❉❖ ❚❊❖❘❊▼❆

  • + ❉❡♠♦♥str❛çã♦✳ ❋❛r❡♠♦s ❛ ♣r♦✈❛ ♣♦r ❛❜s✉r❞♦✳ ❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ t♦❞♦ L > 0 L ∈ Z n − t n ≥ L (∗) L L +

  ❡①✐st❡ n t❛❧ q✉❡ t −1 ✳ + ∈ Z − t n n > 1

  ❆ss✐♠✱ ♣❛r❛ L = 1 ❡①✐st❡ n 1 t❛❧ q✉❡ t 1 1 ✳ P❛r❛ L = 2 −1 + 2 > n ∈ Z n − t n > 2 1 ❛✜r♠❛♠♦s q✉❡ ❡①✐st❡ n t❛❧ q✉❡ t 2 2 −1 ✳ ❈❛s♦ ❝♦♥trár✐♦✱ ♣❛r❛ 2 ∈ Z 2 > n 1 t♦❞♦ n t❛❧ q✉❡ n ✱ t n − t n ≤ 2 2 2 −1 i i ; i = 1, ..., n − t } 1

  ❙❡ t♦♠❛r♠♦s L = max{t −1 ❡ eL = max{L, 2}✱ t❡rí❛♠♦s ♣❛r❛ i −t i ≤ e L t♦❞♦ i ∈ Z✱ t −1 ✉♠❛ ❝♦♥tr❛❞✐çã♦ ❛ ✭✯✮✳ ❉❡ss❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r✱ k k > n k > ... > n 2 > n 1 ♥ú♠❡r♦s ✐♥t❡✐r♦s n t❛✐s q✉❡ n −1 ✳ ❊✱ t (x )} ∈ D ∩ N t n − t n > k (∗∗) k k −1 t (x )} ◆♦t❡ q✉❡✱{η ✳ ❊①✐st❡ ✉♠ ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦ y ❞❡ {η ✳ nk t nk

  ❈♦♠♦ D∩N é ❢❡❝❤❛❞♦✱ t❡♠♦s q✉❡✱ y ∈ D∩N✳ P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ η ✱ ❞❛❞♦ ǫ > 0✱ 1 t (x) : 0 ≤ t ≤ L } ∩ V 6= ∅ 1 ❡①✐st❡ L t❛❧ q✉❡ ♣❛r❛ t♦❞♦ x ∈ B(y, ǫ)✱ t❡♠♦s q✉❡✱ {η ✱

  [ η t (D)

  ♦♥❞❡ V = ✳ −δ≤t≤δ L (x)t ∈ [0, 2δ]} ∩ D 6= ∅ 1 +t ❋✐❣✉r❛ ✷✳✻✿ ❈❛✐①❛ ❞❡ ❋❧✉①♦s ❆ss✐♠✱ {η ♣❛r❛ t♦❞♦ x ∈ B(y, ǫ)✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ x ∈ B(y, ǫ) ∩ D 1 + 2δ

  ♦ t❡♠♣♦ ❞❡ r❡t♦r♥♦ ❞❡ x é ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ L ✳ ❈♦♠♦ y é t η t (x ) = y + 2δ 1 ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦ ❡♥tã♦ lim →∞ ✳ P♦rt❛♥t♦✱ ❡①✐st❡ k > L t❛❧ t (x ) ∈ B(y, ǫ) ∩ D t (x ) ≤ L + 2δ nk 1 q✉❡ η ✳ ▲♦❣♦✱ ♦ t❡♠♣♦ ❞❡ r❡t♦r♥♦ ❞❡ η kn−1 ✳ nk−1 k n − t n ≤ L + 2δ

  ■st♦ é✱ t k −1 1 ✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦ à ✭✯✯✮✳ t : M → M ▲❡♠❛ ✷✳✸✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❡ ❝♦♥s✐❞❡r❡ ϕ ✉♠ ✢✉①♦ ❆♥♦s♦✈✳ ❊♥tã♦✱ ♥ã♦ ♣♦❞❡ ❡①✐st✐r ✉♠❛ ❝✉r✈❛ γ ❝♦♥tí♥✉❛✱ ❢❡❝❤❛❞❛✱ ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ♥✉❧❛ ❡ tr❛♥s✈❡rs❛❧ ♣❛r❛ t♦❞♦ ♣♦♥t♦ ❞❛ ❢♦❧❤❡❛çã♦ ✐♥stá✈❡❧ ♦✉ ❡stá✈❡❧✳ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s ♣♦r ❝♦♥tr❛❞✐çã♦ q✉❡ ❡①✐st❡ γ ❝♦♥tí♥✉❛✱ ❢❡❝❤❛❞❛✱ ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ♥✉❧❛ ❡ tr❛♥s✈❡rs❛❧ ❛ ❢♦❧❤❡❛çã♦ ✐♥stá✈❡❧✳ P❡❧♦ t❡♦r❡♠❛ ❞❡ ❍❛❡✢✐❣❡r ❡①✐st❡ ✉♠❛ ❢♦❧❤❛ F ❞❛ ❢♦❧❤❡❛çã♦ ✐♥stá✈❡❧ ❡ ✉♠❛ ❝✉r✈❛ γ ❡♠ F

  ✹✵ ✷✳✶✳ P❘❖❱❆ ❉❖ ❚❊❖❘❊▼❆

  ❝✉❥♦ ❣❡r♠❡ ❞❡ ❤♦❧♦♥♦♠✐❛ ❡♠ ✉♠ s❡❣♠❡♥t♦ J é tr❛♥s✈❡rs❛❧ ❛ F✱ ❝♦♠ p = J ∩ γ✱ é ❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ J − {p} ♠❛s ❞✐❢❡r❡ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ ❞❡ p em J✳ ❆ss✐♠✱ t❡♠♦s ❞♦✐s ❝❛s♦s ❛ ❝♦♥s✐❞❡r❛r✿ u +1

  ❙✉♣♦♥❤❛♠♦s q✉❡ F s❡❥❛ ❤♦♠❡♦♠♦r❢❛ ❛ R ✱ ♦♥❞❡ u é ❛ ❞✐♠❡♥sã♦ ❞❛ ❢♦❧❤❡❛çã♦ ✐♥stá✈❡❧✳ ❊♥tã♦✱ t♦❞❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ γ ⊂ F é ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ♥✉❧❛ ❡♠ F ❡ ❛ γ = Id ❛♣❧✐❝❛çã♦ ❞❡ ❤♦❧♦♥♦♠✐❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ γ é f ✱ ♠❛s ✐ss♦ ❝♦♥tr❛❞✐③ ♦ ❚❡♦r❡♠❛ ❞❡ ❍❛❡✢✐❣❡r✱ ♣♦✐s ❡①✐st❡ ✉♠❛ ❢♦❧❤❛ F ❞❛ ❢♦❧❤❡❛çã♦ ✐♥stá✈❡❧ ❡ ✉♠❛ ❝✉r✈❛ γ

  ❡♠ F ❝✉❥♦ ❣❡r♠❡ ❞❡ ❤♦❧♦♥♦♠✐❛ ❡♠ ✉♠ s❡❣♠❡♥t♦ J é tr❛♥s✈❡rs❛❧ ❛ F✱ ❝♦♠ p = J ∩ γ ✱ é ❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ J − {p} ♠❛s ❞✐❢❡r❡

  ❞❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ ❞❡ p ❡♠ J✳ ❆❣♦r❛ s✉♣♦♥❤❛♠♦s q✉❡ F ♥ã♦ u +1 s❡❥❛ ❤♦♠❡♦♠♦r❢❛ ❛ R ✳ P♦rt❛♥t♦✱ γ ❝♦♥té♠ ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ γ ❛ q✉❛❧ é ♥ã♦ ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ♥✉❧❛ ❡♠ F ✳ ❚♦♠❛♥❞♦ J ✉♠ s❡❣♠❡♥t♦ tr❛♥s✈❡rs❛❧ ❛ ❢♦❧❤❡❛çã♦ F

  ❡♠ p ∈ J ∩ γ ✱ ♥❡st❡ ❝❛s♦ t❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ❤♦❧♦♥♦♠✐❛ s❡rá ❡①♣❛♥s✐✈❛ ❡♠ ❛♠❜❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ❞❡ J − {p}✱ ♠❛s ✐ss♦ ♥ã♦ ♣♦❞❡ ❛❝♦♥t❡❝❡r ♣♦✐s ♣❡❧♦ ❚❡♦r❡♠❛✱ ❡①✐st❡ ✉♠❛ ❢♦❧❤❛ F ❞❛ ❢♦❧❤❡❛çã♦ ✐♥stá✈❡❧ ❡ ✉♠❛ ❝✉r✈❛ γ ❡♠ F ❝✉❥♦ ❣❡r♠❡ ❞❡ ❤♦❧♦♥♦♠✐❛ ❡♠ ✉♠ s❡❣♠❡♥t♦ J é tr❛♥s✈❡rs❛❧ ❛ F✱ ❝♦♠ p = J ∩ γ✱ é ❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ J − {p} ♠❛s ❞✐❢❡r❡ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ ❞❡ p em J✳ ▲♦❣♦✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳ n = α n ∗ β n n t (x ) n

  ❉❡✜♥❛ ♦ ❝❛♠✐♥❤♦ γ ✱ ♦♥❞❡ α ✈❛✐ ❞❡ x ❛ η ❛♦ ❧♦♥❣♦ ❞❛ ór❜✐t❛ n t n (x ) ❡ ✐♠❛❣❡♠ ❞❡ β ❡♥❝♦♥tr❛✲ s❡ ❡♠ D ❡ ✈❛✐ ❞❡ η ❛té x ✳ n m

  = γ ▲❡♠❛ ✷✳✹✳ γ s❡✱ s♦♠❡♥t❡ s❡✱ n = m✳ n m

  = γ ❉❡♠♦♥str❛çã♦✳ ❙❡ n = m✱ s❡❣✉❡ q✉❡✱ γ ✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛♠♦s ♣♦r

  ∼ n = γ m n n m m ∗ β ≃ α ∗ β ❛❜s✉r❞♦ q✉❡ n > m✳ P♦r ❤✐♣ót❡s❡✱ γ ✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡✱ α ✳ m m −1 −1 ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣♦r α ❡ ♣♦r β ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❡♠♦s✿ −1 −1 −1 −1 −1 −1 α ∗ α n ∗ β n ≃ α ∗ α m ∗ β m m m

  ∗ α ∗ α ∗ β ≃ ∗ β

  ✹✶ ✷✳✶✳ P❘❖❱❆ ❉❖ ❚❊❖❘❊▼❆

  ■st♦ é✱ −1 −1 β ∗ α ∗ α n ∗ β n ≃ 0 (∗) m m −1

  ∗α n ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ α m ✭❝♦♠ ❡①tr❡♠♦s ♠❛♥t✐❞♦s ✜①♦s✮ é ❤♦♠♦tó♣✐❝♦ n,m t t (x ) m

  à ❝✉r✈❛ α ❛ q✉❛❧ é ✉♠ s❡❣♠❡♥t♦ ❞❡ ♦r❜✐t❛ ❞❡ η q✉❡ ❝♦♠❡ç❛ ❡♠ η ✈❛✐ ❛té η t n (x )

  ✳ α n,m (0) = η t n (x ) n,m (1) = η t m (x )

  ✱ α ✳ ❆ss✐♠✱ −1 −1 α ∗ α n ≃ α ∗ α n ∗ α n,m m m

  ❱❡❥❛ ✜❣✉r❛

  

−1

  ∗ α n ≃ α n,m ❋✐❣✉r❛ ✷✳✼✿ α m

  −1

  ∗ α ∗ β ∼ 0 n,m n ❋✐❣✉r❛ ✷✳✽✿ β m

  ✹✷ −1 ✷✳✶✳ P❘❖❱❆ ❉❖ ❚❊❖❘❊▼❆ m ∗ α n ≃ α n,m

  ❙❡❣✉❡ q✉❡✱ α ✳ ❘❡❡s❝r❡✈❡♥❞♦ (∗)✱ ♦❜t❡♠♦s✿ −1 −1 −1 −1 −1 0 ≃ β ∗ α ∗ α n ∗ β n ≃ β ∗ α n,m ∗ β n m m m m m ∗ α ∗ α n ∗ β n P♦rt❛♥t♦✱ β ✱ ✈❡❥❛ ❛ ✜❣✉r❛ é ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ♥✉❧❛ ❡

  ♣♦❞❡♠♦s ♣❡rt✉❜á✲❧❛ ❧✐❣❡✐r❛♠❡♥t❡ ❡♠ t♦r♥♦ ❞❡ x ♣❛r❛ q✉❡ ❡stá s❡❥❛ tr❛♥s✈❡rs❛❧ ❡♠ t♦❞♦ ♣♦♥t♦ ❞❛ ❢♦❧❤❡❛çã♦ ✐♥stá✈❡❧✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♥str✉í♠♦s ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ♥✉❧❛ ❡ tr❛♥s✈❡rs❛❧ ❛ ❢♦❧❤❡❛çã♦ ✐♥stá✈❡❧✱ ♦ q✉❛❧ ❝♦♥tr❛❞✐③ ♦ ▲❡♠❛

  ❋✐❣✉r❛ ✷✳✾✿ ❈✉r✈❛ tr❛♥s✈❡rs❛❧ ❡♠ t♦❞♦ ♣♦♥t♦ à ❢♦❧❤❡❛çã♦ ✐♥stá✈❡❧ t ▲❡♠❛ ✷✳✺✳ ❙❡ α : [0, 1] −→ M r❡♣r❡s❡♥t❛ q✉❛❧q✉❡r s❡❣♠❡♥t♦ ❞❛ ór❜✐t❛ η ✳ ❊♥tã♦ α

  é ❤♦♠♦tó♣✐❝♦ ✭❝♦♠ ❡①tr❡♠♦s ♠❛♥t✐❞♦s ✜①♦s✮ ❛ ✉♠ ❝❛♠✐♥❤♦ α t❛❧ q✉❡ |α | ≤ C + 2(σ/λ)ln|α|

  ❉❡♠♦♥str❛çã♦✳ P❛r❛ ❝❛❞❛ x ∈ M e t 6= 0✱ ❝♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ❝❛♠✐♥❤♦✿ δ x,t : [0, 1] −→ M x,t (s) = ϕ st (x)

  ❞❛❞♦ ♣♦r s 7−→ δ ✳ ❖❜s❡r✈❡ q✉❡✱

  δ x,t (0) = ϕ (x) = ϕ (x) = x 0.t δ x,t (1) = ϕ t (x)

  δ α ,t (0) = ϕ (α ) = ϕ t α(1) 0.t 1 δ α ,t (1) = ϕ 1.t (α 1 1 ) = ϕ t (α(0))

  ❈♦♥s✐❞❡r❡ ♦ ❝❛♠✐♥❤♦ α ✱ ❞❛❞♦ ♣♦r✿ ′ −1 α = δ α ,t ∗ ϕ t ◦ α ∗ δ α ,t 1

  = α ◆♦t❡ q✉❡✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ α ∼ ✳

  ❖ ❝♦♠♣r✐♠❡♥t♦ ❞❡ α é ❞❛❞♦ ♣♦r✿ |α | = |δ | + |ϕ ◦ α| + |δ | α ,t t α ,t 1 Z 1 Z 1 Z 1

  ∂δ α ,t ∂ϕ t ◦ α(s) ∂δ α ,t 1 = | (s)|ds + | |ds + | (s)|ds (∗)

  ∂s ∂s ∂s

  ✹✸ ✷✳✶✳ P❘❖❱❆ ❉❖ ❚❊❖❘❊▼❆

  t

  ❈♦♠♦ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ❡ Dϕ é ❝♦♥tí♥✉❛✱ ✜①❛♠♦s σ = sup x |Dϕ t (p)| ∈M ✳ ∂δ α ,t (s) ∂ϕ s,t (α )

  | | = | | ∂s ∂s

  ∂ϕ(st, α ) |

  = | ∂s

  = |Dϕ(st, α ).t| ≤ |Dϕ(st, α )||t| ≤ |Dϕ (α )||t| st ∂δ α1,t (s) ≤ σ|t|

  | ≤ σ|t| ❆♥❛❧♦❣❛♠❡♥t❡✱ | ✳ ❊✱ ∂ϕ t ∂ϕ ∂α (α(s)) (t,α(s)) (s) ∂s | | = | | = |Dϕ st (α )|.| | ∂s ∂s ∂s ss s ∂α (s) ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ α é ✉♠

  (x ), s❡❣♠❡♥t♦ ❞❡ W é ✉♠ ✈❡t♦r ❡♠ E ✳ P❡❧❛ ✐♥❡q✉❛çã♦✱ ∂s −λt |Dϕ | ≤ Ce |v | t (p).v p p

  ❙❡q✉❡ q✉❡✱ (α ) ∂α (s) (s) −λt

  |∂ϕ st | ≤ Ce | | ∂s ∂s

  P♦rt❛♥t♦✱ −λt ∂α (s) |Dϕ st (α(s))| ≤ Ce | |

  ∂s ❙✉❜st✐t✉✐♥❞♦ ❡♠ ✭✯✮✱ t❡♠♦s✿

  Z 1 Z 1 Z 1 ′ −λt ∂(α (s) ) |α | ≤ σ|t|ds + Ce | |ds + σ|t|ds

  ∂s −λt = 2σ|t| + Ce |α|

  ✹✹ 1 ✷✳✶✳ P❘❖❱❆ ❉❖ ❚❊❖❘❊▼❆ ln|α|

  ❙❡ ❞❡✜♥✐♠♦s t = λ ✱ t❡♠♦s✱ ′ −ln|α| σ |α | ≤ 2 ln|α| + Ce |α|

  λ 2σ |α | ≤ ln|α| + c.

  λ

  n

  Pr♦✈❛ ❞♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧✿ ❈♦♠♦ ❝❛❞❛ α ❡stá ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ❡ ♣❡❧♦ ▲❡♠❛ |α | ≤ n.L n

  ≃ α n P❡❧♦ ▲❡♠❛ ❡①✐st❡ α n ✭❝♦♠ ❡①tr❡♠♦s ♠❛♥t✐❞♦s ✜①♦s✮ t❛❧ q✉❡✿

  |α | ≤ | + C ln|α n λ

  |α | ≤ ln|n.L| + C λ

  ❉❛í✱ s❡ 2σ z = ln|n.L| + C z ln λ (n.L) ) ( 2σ λ e = A.e z ( ) λ e = B.nL ( ) λz n | ≤ C n | = |α n | + C n = B.e (∗)

  ❈♦♠♦ ♦ ❞✐s❝♦ é ❝♦♠♣❛❝t♦✱ |β ✱ ♣❛r❛ t♦❞♦ n✳ ❆ss✐♠✱ |γ ✳ ❆❧é♠ n | = nL ❞✐ss♦✱ |α ✳ ❉❛í✱ l (γ ) ≤ C + |α | 2 n n

  2σ ≤ C ln|n.L| + C +

  λ 2σ

  ≤ C + ln|n.L| λ ln|n.L| (γ n ) ≤ r (γ ) ≤ l (γ ) ≤ ... ≤ 2 2 1 2 2 ❙❡ r = C + ✱ t❡♠♦s q✉❡✱ l ✳ ❈♦♠♦ l λ l 2 (γ n ) ≤ r 1 (M )

  ✳ ❙❡❣✉❡ q✉❡✱ ❡①✐st❡♠ ♣❡❧♦ ♠❡♥♦s n ❡❧❡♠❡♥t♦s ❞❡ π ✳ P♦rt❛♥t♦✱ #{α ∈ π (M ); l (α) ≤ r} ≥ n 1 2

  ■st♦ é✱ P (r) ≥ n✱ ♣♦r (∗)✱ ( ) λz P (r) ≥ B.e

  ❆❣♦r❛✱ ❛❥✉st❛♥❞♦ ❛ ❝♦♥st❛♥t❡ B✱ ❢❛③❡♠♦s ❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣❛r❛ t♦❞♦ r✳ ■ss♦

  ✹✺ ✷✳✶✳ P❘❖❱❆ ❉❖ ❚❊❖❘❊▼❆

  ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛ ♣❛r❛ t♦❞♦ ✢✉①♦ ❆♥♦s♦✈ ❞❡ ❝♦♦❞✐♠❡♥sã♦ ✉♠✳

  ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

  ❬✶❪ ❆♥♦s♦✈✱ ❉✳ ❱✳ ●❡♦❞❡s✐❝ ✢♦✇s ♦♥ ❝❧♦s❡❞ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ ♥❡❣❛t✐✈❡ ❝✉r✈❛t✉r❡✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❙t❡❦❧♦✈ ■♥st✐t✉t❡ ♦❢ ▼❛t❤♠❡t✐❝s✱ ✈♦❧✳ ✾✵ ✭✶✾✻✼✮ ✭❆✳ ▼✳ ❙✳ tr❛♥s❧❛t✐♦♥✱ ✶✾✻✾✮

  ❬✷❪ ❆♥♦s♦✈✱ ❉✳ ❱✳ ❡ ❙✐♥❛✐✱ ●✳✱ ❙♦♠❡ s♠♦t❤ ❡r❣♦❞✐❝ s②st❡♠s✳✳ ❯s♣❡❦❤✐ ▼❛t❤✳ ◆❛✉❦ ✷✷ ✭✶✾✻✼✮✳ ✶✵✼ ✲ ✶✼✷✱ ❘✉ss✐❛♥ ▼❛t❤✱ ❙✉r✈❡②s ✷✷ ✭✶✾✻✼✮✱ ✶✵✸ ✲ ✶✻✽✳

  ❬✸❪ ❇❛✉t✐st❛✱ ❙✳❀ ▼♦r❛❧❡s ❈✳ ▲❡❝t✉r❡s ♦♥ s❡❝t✐♦♥❛❧ ❆♥♦s♦✈✳ ❉✐s♣♦♥í✈❡❧ ❡♠✿ ❤tt♣✿✴✴✇✇✇✳♣r❡♣r✐♥t✳✐♠♣❛✳❜r✴❙❤❛❞♦✇s✴❙❊❘■❊❴❉✴✷✵✶✶✴✽✻✳❤t♠❧✳ Ú❧t✐♠♦ ❛❝❡ss♦ ❡♠✿ ✺ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✼✳

  ❬✹❪ ❈❛♠❛❝❤♦✱ ❈✳✱ ◆❡t♦✱ ❆✳✱ ❚❡♦r✐❛ ●❡♦♠étr✐❝❛ ❞❛s ❋♦❧❤❡❛çõ❡s✳✳ Pr✐♠❡✐r❛ ❡❞✐çã♦✳ ❘✐♦ ❞❡ ❥❛♥❡✐r♦✿ Pr♦❥❡t♦ ❊✉❝❧✐❞❡s✱ ■▼P❆✱ ✶✾✼✾✳

  ❬✺❪ ●❛❧❧♦t✱ ❙②❧✈❡str❡❀ ❍✉❧✐♥✱ ❉♦♠✐♥✐q✉❡ ❡ ▲❛❢♦♥t❛✐♥❡✱ ❏❛❝q✉❡s✱ ❘✐❡♠❛♥♥✐❛♥ ●❡♦♠❡tr②✳ ❚❡r❝❡✐r❛ ❡❞✐çã♦✳ ❙♣r✐♥❣❡r✳

  ❬✻❪ ❍❛❡✢✐❣❡r✱ ❆✳ ❱❛r✐❡té❡s ❢❡✉✐❧❧❡t❡❡s✱ ❆♥♥✳ ❙❝✉♦❧❛ ◆♦r♠✳ ❙✉♣✳ P✐s❛✱ ✈♦❧✳ ✭✸✮✶✻ ✭✶✾✻✷✮✱ ♣♣✳ ✸✻✼✲✸✾✼✳

  ❬✼❪ ❍❛t❝t❤❡r✱ ❆✳ ❇❛s✐❝ ❚♦♣♦❧♦❣② ♦❢ ✸✲▼❛♥✐❢♦❧❞s✳ Pr❡♣r✐♥t ❛✈❛✐❧❛❜❧❡ ❛t ❤tt♣✿✴✴✇✇✇✳♠❛t❤✳❝♦r♥❡❧❧✳❡❞✉✴∼❤❛t❝❤❡r✳

  ❬✽❪ ❍✐rs❝❤✱ ▼✳ ❲✳❀ P✉❣❤✱ ❈✳ ❈✳❀ ❙❤✉❜✱ ▼✳ ■♥✈❛r✐❛♥t ▼❛♥✐❢♦❧❞s✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✺✽✸✳ ❇❡r❧✐♥✲◆❡✇ ❨♦r❦✿ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✶✾✼✼✳

  ❬✾❪ ❑❛t♦❦✱ ❆✳❀ ❍❛ss❡❧❜❧❛tt ❇✳ ❆ ▼♦❞❡r♥❛ ❚❡♦r✐❛ ❞❡ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s ✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✭✶✾✾✾✮✳

  ❬✶✵❪ ▲✐♠❛✱ ❊✳ ▲✳ ●r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ❡s♣❛ç♦s ❞❡ r❡❝♦❜r✐♠❡♥t♦✳ ❘✐♦ ❞❡ ❥❛♥❡✐r♦✿ Pr♦❥❡t♦ ❊✉❝❧✐❞❡s✱ ■▼P❆✱ ✶✾✾✸✳

  ❬✶✶❪ ▲✐♠❛✱ ❊✳ ▲✳ ❱❛r✐❡❞❛❞❡s ❉✐❢❡r❡♥❝✐á✈❡✐s✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ■▼P❆✱ ✶✾✼✼✳ ❬✶✷❪ ▼❡❧♦✱ ❲✳ ❚♦♣♦❧♦❣✐❛ ❞❛s ❱❛r✐❡❞❛❞❡s✳ ❉✐s♣♦♥í✈❡❧ ❡♠✿ ✇✸✳✐♠♣❛✳❜r✴∼

  ❞❡♠❡❧♦✴t♦♣♦❧♦❣✐❛❞✐❢❡r❡♥❝✐❛❧✷✵✶✶✳ Ú❧t✐♠♦ ❛❝❡ss♦ ❡♠✿ ✶✵ ❞❡ ❥❛♥❡✐r♦ ❞❡ ✷✵✶✼✳

  ❬✶✸❪ ❆♥♦s♦✈✱ ❉✳ ❱✳ ●❡♦❞❡s✐❝ ✢♦✇s ♦♥ ❝❧♦s❡❞ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ ♥❡❣❛t✐✈❡ ❝✉r✈❛t✉r❡✱ ✶✾✻✼✳

  ✹✼ ❘❊❋❊❘✃◆❈■❆❙ ❇■❇▲■❖●❘➪❋■❈❆❙

  ❬✶✹❪ ◆♦✈✐❦♦✈✱ ❙✳ P✳✱ ❚♦♣♦❧♦❣② ♦❢ ❋♦❧✐❛t✐♦♥s✳ ❚r❛♥s ▼♦s❝♦✇ ▼❛t❤✳ ❙♦❝ ✶✹ ✭✶✾✻✸✮ ✷✻✽✲ ✸✵✺✳

  ❬✶✺❪ P❛❧✐s✱ ❏✳ ❡ ▼❡❧♦✱ ❲✳ ■♥tr♦❞✉çã♦ ❛♦s ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ Pr♦❥❡t♦ ❊✉❝❧✐❞❡s✱ ■▼P❆✱ ✶✾✼✽✳

  ❬✶✻❪ P❧❛♥t❡✱ ❏✳ ❋✳❀ ❚❤✉rst♦♥✱ ❲✳ P✳ ❆♥♦s♦✈ ✢♦✇s ❛♥❞ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣✳ ❚♦♣♦❧♦❣② ✶✶ ✭✶✾✼✷✮✱ ✶✹✼✲✶✺✵✳

  ❬✶✼❪ P❧❛♥t❡✱ ❏✳✱ ❆♥♦s♦✈ ❋❧♦✇s✳ ❆♠✳ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✾✹✭✶✾✼✷✮✱ ✸✻✶ ✲ ✸✼✷✳

  ❬✶✽❪ ❘♦❜✐♥s♦♥✱ ❈✳ ❉②♥❛♠✐❝❛❧ ❙②st❡♠✳ ❙t❛❜✐❧✐t②✱ ❙②♠❜♦❧✐❝ ❉②♥❛♠✐❝s✱ ❛♥❞ ❝❤❛♦s✳ ❙❡❝♦♥❞ ❡❞✐t✐♦♥✳ ❙t✉❞✐❡s ✐♥ ❆❞✈❛♥❝❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❈❘❈ Pr❡ss✱ ❇♦❝❛ ❘❛t♦♥✱ ❋▲✱ ✶✾✾✾✳

  ❬✶✾❪ ❙♠❛❧❡✱ ❙✳ ❉✐✛❡r❡♥t✐❛❜❧❡ ❞②♥❛♠✐❝❛❧ s②st❡♠s✳ ❇✉❧❧✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✼✸ ✶✾✻✼ ✼✹✼✕✽✶✼✳

  ❬✷✵❪ ❙♦t♦♠❛②♦r✱ ❏✳✱ ▲✐çõ❡s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ Pr♦❥❡t♦ ❊✉❝❧✐❞❡s✱ ■▼P❆ ✭✶✾✼✾✮✳

  ❬✷✶❪ ❱✐❧❝❤❡s✱ ▼✳ ❆✳ ■♥tr♦❞✉çã♦ à ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛✱ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❆♥á❧✐s❡ ✲ ■▼❊✱ ❯❊❘❏✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✳

Novo documento