Incompressobilidade de Toro Transversal a Campos de Vetores.

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  ❘❖❙➶◆●❊▲❆ ❆❙❙■❙ P■❘❊❙ ■◆❈❖▼P❘❊❙❙■❇■▲■❉❆❉❊ ❉❊ ❚❖❘❖ ❚❘❆◆❙❱❊❘❙❆▲ ❆

  ❈❆▼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✐❛❡✳

  ❱■➬❖❙❆ ▼■◆❆❙ ●❊❘❆■❙ ✲ ❇❘❆❙■▲

  ✷✵✶✼ Ficha catalográfica preparada pela Biblioteca Central da Universidade Federal de Viçosa - Câmpus Viçosa

  T Pires, Rosângela Assis, 1993-

P667i Incompressibilidade de toro transversal a campos de vetores

2017 / Rosângela Assis Pires. – Viçosa, MG, 2017. v, 73f. : il. (algumas color.) ; 29 cm. Orientador: Enoch Humberto Apaza Calla. Dissertação (mestrado) - Universidade Federal de Viçosa. Referências bibliográficas: f.72-73.

  1. Toro (Geometria). 2. Campos vetoriais. I. Universidade Federal de Viçosa. Departamento de Matemática. Programa de Pós-graduação em Matemática. II. Título.

  

CDD 22 ed. 512.2

  ❘❖❙➶◆●❊▲❆ ❆❙❙■❙ P■❘❊❙ ■◆❈❖▼P❘❊❙❙■❇■▲■❉❆❉❊ ❉❊ ❚❖❘❖ ❚❘❆◆❙❱❊❘❙❆▲ ❆

  ❈❆▼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❘❖❱❆❉❆✿ ✷✵ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✼✳ ❈❛r❧♦s ❆r♥♦❧❞♦ ▼♦r❛❧❡s ❘♦❥❛s

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

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

  ✭❖r✐❡♥t❛❞♦r✮

  ❆❣r❛❞❡❝✐♠❡♥t♦s

  ❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛ ❉❡✉s✳ ❙♦✉ ♠✉✐tíss✐♠♦ ❣r❛t❛ ❛♦s ♠❡✉s ♣❛✐s✱ ♣❡❧♦ ❡①❡♠♣❧♦✱ ❝❛r✐♥❤♦ ❡ ♠♦t✐✈❛çã♦✳ ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❊♥♦❝❤ ❡ ❛♦ ♠❡✉ ❝♦♦r✐❡♥t❛❞♦r ❇✉❧♠❡r✱ ♣❡❧❛

  ♣❛❝✐ê♥❝✐❛✱ ❛♣r❡♥❞✐③❛❞♦✱ ♣❡❧❛s s✉❛s ❝♦rr❡çõ❡s ❡ ✐♥❝❡♥t✐✈♦✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❞❡ ❞❡❢❡s❛ ♣❡❧❛s s✉❣❡stõ❡s ❡ ❝♦rr❡çõ❡s✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ ♣❡❧❛ ❛♠✐③❛❞❡✱ ♣❡❧♦s ♠♦♠❡♥t♦s

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

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

  ✐✐

  ❙✉♠ár✐♦

   ✐✈

   ✈

   ✶

   ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✸

   ✸✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

   ✺✼

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾

   ✼✶

   ✼✷

  ✐✐✐

  ❘❡s✉♠♦

  P■❘❊❙✱ ❘♦sâ♥❣❡❧❛ ❆ss✐s✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✼✳ ■♥❝♦♠♣r❡ss✐❜✐❧✐❞❛❞❡ ❞❡ ❚♦r♦ ❚r❛♥s✈❡rs❛❧ ❛ ❈❛♠♣♦s ❞❡ ❱❡t♦r❡s✳ ❖r✐❡♥t❛❞♦r✿ ❊♥♦❝❤ ❍✉♠❜❡rt♦ ❆♣❛③❛ ❈❛❧❧❛✳ ❈♦♦r✐❡♥t❛❞♦r✿ ❇✉❧♠❡r ▼❡❥í❛ ●❛r❝í❛✳ ◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ ♥♦ss♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ é ❞❛r ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ♣❛r❛ ✉♠ t♦r♦ T ♠❡r❣✉❧❤❛❞♦ ♥✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ ♦r✐❡♥tá✈❡❧ M s❡r ✐♥❝♦♠♣r❡ssí✈❡❧✱ 1 (T ) → π (M ) 1

  ✐st♦ é✱ ♦ ❤♦♠♦♠♦r✜s♠♦ π ✐♥❞✉③✐❞♦ ♣❡❧❛ ❛♣❧✐❝❛çã♦ ✐♥❝❧✉sã♦ é ✐♥❥❡t♦r✳ ◆ós ❛ss✉♠✐♠♦s q✉❡ T é tr❛♥s✈❡rs❛❧ ❛ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X✱ ❡①✐❜✐♥❞♦ ✉♠❛ ú♥✐❝❛ ór❜✐t❛ O q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛ T ✳ ❙❡✱ ❛❧é♠ ❞✐ss♦✱ O é ❤✐♣❡r❜ó❧✐❝❛ ❡ ♥ã♦ ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦ ❡♠ M ❡♥tã♦ T é ✐♥❝♦♠♣r❡ssí✈❡❧ ❡ M é ✐rr❡❞✉tí✈❡❧ ✭t♦❞❛ ❡s❢❡r❛ ♠❡r❣✉❧❤❛❞❛ ❡♠ M ❜♦r❞❛ ✉♠❛ ❜♦❧❛✮✳

  ✐✈

  ❆❜str❛❝t

  P■❘❊❙✱ ❘♦sâ♥❣❡❧❛ ❆ss✐s✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❋❡❜r✉❛r②✱ ✷✵✶✼✳ ■♥❝♦♠♣r❡ss✐❜✐❧✐t② ♦❢ ❚♦r✉s ❚r❛♥s✈❡rs❡ t♦ ❱❡❝t♦r ❋✐❡❧❞s✳ ❆❞✈✐s❡r✿ ❊♥♦❝❤ ❍✉♠❜❡rt♦ ❆♣❛③❛ ❈❛❧❧❛✳ ❈♦✲❛❞✈✐s❡r✿ ❇✉❧♠❡r ▼❡❥í❛ ●❛r❝í❛✳ ■♥ t❤✐s ♣❛♣❡r✱ ♦✉r ♠❛✐♥ ❣♦❛❧ ✐s t♦ ❣✐✈❡ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r ❛ t♦r✉s T ❡♠❜❡❞❞❡❞ ✐♥ ❛ ❝❧♦s❡❞ ♦r✐❡♥t❛❜❧❡ ✸✲♠❛♥✐❢♦❧❞ M t♦ ❜❡ ✐♥❝♦♠♣r❡ss✐❜❧❡✱ t❤✐s ✐s✱ t❤❡ 1 (T ) → π 1 (M ) ❤♦♠♦♠♦r♣❤✐s♠ π ✐♥❞✉❝❡❞ ❜② t❤❡ ✐♥❝❧✉s✐♦♥ ♠❛♣ ✐s ✐♥❥❡❝t✐✈❡✳ ❲❡ ❛ss✉♠❡ t❤❛t T ✐s tr❛♥s✈❡rs❡ t♦ ❛ ✈❡❝t♦r ✜❡❧❞ X✱ ❡①❤✐❜✐t✐♥❣ ❛ ✉♥✐q✉❡ ♦r❜✐t O ✇❤✐❝❤ ❞♦❡s ♥♦t ✐♥t❡rs❡❝t T ✳ ■❢✱ ✐♥ ❛❞❞✐t✐♦♥✱ O ✐s ❤②♣❡r❜♦❧✐❝ ❛♥❞ ♥♦t ♥✉❧❧ ❤♦♠♦t♦♣✐❝ ✐♥ M t❤❡♥ T ✐s ✐♥❝♦♠♣r❡ss✐❜❧❡ ❛♥❞ M ✐s ✐rr❡❞✉❝✐❜❧❡ ✭❡✈❡r② ❡♠❜❡❞❞❡❞ ✷✲s♣❤❡r❡ ✐♥ M ❜♦✉♥❞s ❛ ✸✲❜❛❧❧✮✳

  ✈

  ■♥tr♦❞✉çã♦

  ❖ ❡st✉❞♦ ❞❛s ór❜✐t❛s ❞♦s ♣♦♥t♦s ❞❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ♦✉ ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ♥✉♠❛ ✈❛r✐❡❞❛❞❡ r❡✈❡❧❛ ♦✉ ♣r❡❞✐③ ♦ s❡✉ ❝♦♠♣♦rt❛♠❡♥t♦ ❢✉t✉r♦ ❡ ♣❛ss❛❞♦✱ ❡st❡ é ♦ ♣r✐♥❝í♣✐♦ ❜ás✐❝♦ ❞❡ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s✳ ❊st❛ ❧✐♥❤❛ ❞❡ ♣❡sq✉✐s❛ ❛❜r❛♥❣❡✿ ●❡♦♠❡tr✐❛✱ ❚♦♣♦❧♦❣✐❛✱ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ✭❊❉❖✬s✮✱ ❞❡♥tr❡ ♦✉tr❛s ár❡❛s ❡ s✉❜ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛✳ ❚❛❧ é ❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ q✉❡ s✉r❣✐r❛♠ t❡♦r✐❛s ♣❡rt✐♥❡♥t❡s ❛ ❝❛❞❛ ♣r♦❜❧❡♠❛ ❡♥❢r❡♥t❛❞♦✱ ❝♦♠♦ ❤✐♣❡r❜♦❧✐❝✐❞❛❞❡✱ ❤✐♣❡r❜♦❧✐❝✐❞❛❞❡ ♣❛r❝✐❛❧ ❡✱ r❡❝❡♥t❡♠❡♥t❡✱ ❤✐♣❡r❜♦❧✐❝✐❞❛❞❡ s❡❝❝✐♦♥❛❧✳

  ❆s ♣r✐♠❡✐r❛s ✐❞❡✐❛s ❞ã♦ ♦r✐❣❡♠ ❛♦s ✢✉①♦s ❆♥♦s♦✈ ❡✱ ❛s ú❧t✐♠❛s ❛♦s ✢✉①♦s ❙❡❝❝✐♦♥❛❧✲❆♥♦s♦✈✱ ♦s q✉❛✐s✱ ❡♠ ❧✐♥❤❛s ❣❡r❛✐s✱ sã♦ ✢✉①♦s tr❛♥s✈❡rs❛✐s ❛ ❢r♦♥t❡✐r❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛♠❜✐❡♥t❡✱ ❝✉❥❛ ❞❡❝♦♠♣♦s✐çã♦ ❞♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❡♠ ✉♠ s✉❜✜❜r❛❞♦ ❡stá✈❡❧ ❡ ✉♠ ❝❡♥tr❛❧ ❡♠ ❝❛❞❛ ♣♦♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ✐♥✈❛r✐❛♥t❡ ♠❛①✐♠❛❧ é ❞♦♠✐♥❛❞❛ ❡ s❡❝❝✐♦♥❛❧♠❡♥t❡ ❡①♣❛♥s♦r ♥♦ s✉❜✜❜r❛❞♦ ❝❡♥tr❛❧✳

  ❉❛❞♦ ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ ✉♠ ❞♦s q✉❡st✐♦♥❛♠❡♥t♦s ❣❡r❛❞♦s sã♦ ❛s r❡❧❛çõ❡s ❡♥tr❡ ❛ ❞✐♥â♠✐❝❛ ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ❞❛s ✈❛r✐❡❞❛❞❡s q✉❡ s✉♣♦rt❛ t❛❧ ❞✐♥â♠✐❝❛✱ t❛✐s ❝♦♠♦ ✐♥❝♦♠♣r❡ss✐❜✐❧✐❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡s ❡ ✐rr❡❞✉t✐❜✐❧✐❞❛❞❡ ❞❛ ✈❛r✐❡❞❛❞❡✳

  ❯♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ q✉❡ ❛❞♠✐t❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❆♥♦s♦✈ ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠✱ ✐st♦ é✱ ❛ ❞✐♠❡♥sã♦ ❞♦ s✉❜✜❜r❛❞♦ ❡stá✈❡❧ ♦✉ ❞♦ s✉❜✜❜r❛❞♦ ✐♥stá✈❡❧ é ✐❣✉❛❧ ❛ ✉♠✱ é ✐rr❡❞✉tí✈❡❧✳ ❖✉tr♦ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦ é q✉❡ s❡ X é ✉♠ ❝❛♠♣♦ s❡❝❝✐♦♥❛❧ ❆♥♦s♦✈ ♥✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ❡ ✐rr❡❞✉tí✈❡❧ ♦♥❞❡ t♦❞❛s ❛s s✐♥❣✉❧❛r✐❞❛❞❡s sã♦ ❞♦ t✐♣♦ ▲♦r❡♥③ ❡ ♥ã♦ ❛♣r❡s❡♥t❛ ór❜✐t❛s ♣❡r✐ó❞✐❝❛s ❤♦♠♦tó♣✐❝❛s ❛ ✉♠ ♣♦♥t♦ ❡♥tã♦ t♦❞♦ t♦r♦ T tr❛♥s✈❡rs❛❧ ❛♦ ❝❛♠♣♦ X é ✐♥❝♦♠♣r❡ssí✈❡❧✱

  ❊♠ ✢✉①♦s ❆♥♦s♦✈ ♥ã♦ tr❛♥s✐t✐✈♦s ♥✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ M ✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s r ♥ã♦ ❡rr❛♥t❡s✱ ❞❡♥♦t❛❞♦ ♣♦r Ω(X)✱ ♦♥❞❡ X é ♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s C ✭r ≥ 1✮✱ ❛❞♠✐t❡ n

  [ Ω j j

  ✉♠❛ ❞❡❝♦♠♣♦s✐çã♦ ❡s♣❡❝tr❛❧✿ Ω(X) = ♦♥❞❡ ❝❛❞❛ Ω ✭❞❡♥♦♠✐♥❛❞♦ ❝♦♥❥✉♥t♦ j =1 ❜ás✐❝♦✮ é ❢❡❝❤❛❞♦✱ ❞✐s❥✉♥t♦s ❞♦✐s ❛ ❞♦✐s✱ ✐♥✈❛r✐❛♥t❡ ♣❡❧♦ ✢✉①♦ ❡ tr❛♥s✐t✐✈♦ j ❊♠ ♣r♦✈❛✲s❡ q✉❡ ♦s ❝♦♥❥✉♥t♦s Ω ✭j = 1, ..., n✮ ♣♦❞❡♠ s❡r s❡♣❛r❛❞♦s ♣♦r t♦r♦s ❞♦✐s ❛ ❞♦✐s ❞✐s❥✉♥t♦s✱ ♥ã♦ ✐s♦tó♣✐❝♦s ❡ ✐♥❝♦♠♣r❡ssí✈❡✐s✳ ❚❡♠♦s t❛♠❜é♠ q✉❡ ✉♠ t♦r♦ T é ✐♥❝♦♠♣r❡ssí✈❡❧ ♥✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ✐rr❡❞✉tí✈❡❧ s❡ T é tr❛♥s✈❡rs❛❧ ❛ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❆♥♦s♦✈

  ❆ ❡①✐stê♥❝✐❛ ❞❡ s✉♣❡r❢í❝✐❡s ✐♥❝♦♠♣r❡ssí✈❡✐s ❞✐s❥✉♥t❛s ❞♦✐s ❛ ❞♦✐s ❡♠ ✸✲ ✈❛r✐❡❞❛❞❡s ♦r✐❡♥tá✈❡✐s✱ ❢❡❝❤❛❞❛s ❡ ✐rr❡❞✉tí✈❡✐s ♣♦❞❡♠ ❛❥✉❞❛r ♥❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ✈❛r✐❡❞❛❞❡s ♦❜t✐❞❛s ♣♦r ✉♠ ❝♦rt❡ ❛♦ ❧♦♥❣♦ ❞❡ss❛s s✉♣❡r❢í❝✐❡s ✐♥❝♦♠♣r❡ssí✈❡✐s✱ ♣♦r

  ✶

  ✷ ❙❯▼➪❘■❖

  ❡①❡♠♣❧♦✱ s❡r ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛tr❛t♦r✐❛❧ ♦✉ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❙❡✐❢❡rt ✭✈❡r ❊st❛ ❞✐ss❡rt❛çã♦ t❡♠ ❝♦♠♦ ❜❛s❡ ♦ ❛rt✐❣♦ ■♥❝♦♠♣r❡ss✐❜✐❧✐t② ♦❢ t♦r✉s tr❛♥✈❡rs❡ t♦

  ✈❡❝t♦r ✜❡❧❞s✱ ❞♦ ♣r♦❢❡ss♦r ❈❛r❧♦s ❆✳ ▼♦r❛❧❡s✱ ♣✉❜❧✐❝❛❞♦ ♥♦ ❚♦♣♦❧♦❣② Pr♦❝❡❞✐♥❣s✱ ✈♦❧✉♠❡ ✷✽✱ ◆♦✳ ✶✱ ♥♦ ❛♥♦ ❞❡ ✷✵✵✹✱ ❝✉❥♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ é ❞❛r ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ✉♠ t♦r♦ ♠❡r❣✉❧❤❛❞♦ ♥✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ M s❡❥❛ ✐♥❝♦♠♣r❡ssí✈❡❧✳

  ❆ss✉♠✐♠♦s q✉❡ T é tr❛♥s✈❡rs❛❧ ❛♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X✱ ♠❛s ♥ã♦ s✉♣♦♠♦s q✉❡

  X é ❆♥♦s♦✈✳ ❊♠ ✈❡③ ❞✐ss♦✱ ❛ss✉♠✐♠♦s q✉❡ X ❡①✐❜❡ ✉♠❛ ú♥✐❝❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛

  O ❛ q✉❛❧ ♥ã♦ ✐♥t❡rs❡❝t❛ T ✳ ❙❡ ❛❧é♠ ❞✐ss♦ O é ❤✐♣❡r❜ó❧✐❝♦ ❡ ♥ã♦ ❤♦♠♦tó♣✐❝♦ ❛ ✉♠

  ♣♦♥t♦ ❡♠ M✱ ❡♥tã♦ T é ✐♥❝♦♠♣r❡ssí✈❡❧ ❡ M é ✐rr❡❞✉tí✈❡❧✳ ❊♠ r❡s✉♠♦✱ ❡st❡ é ♦ ❝♦♥t❡ú❞♦ ❞♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❞♦ ❛rt✐❣♦✳

  ◆♦ ❝❛♣ít✉❧♦ ✶ ♥♦s ❞❡❞✐❝❛♠♦s ❛ ❛♣r❡s❡♥t❛r ❛♦ ❧❡✐t♦r ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✱ ♦ q✉❛❧ ❡stá ❡str✉t✉r❛❞♦ ❡♠ q✉❛tr♦ s❡❝çõ❡s q✉❡✱ ❡♠ ❧✐♥❤❛s ❣❡r❛✐s✱ ❛❜♦r❞❛♥❞♦ ♦s s❡❣✉✐♥t❡s t❡♠❛s✿ ♥♦çõ❡s ❞❡ ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡ ❝♦♥❝❡✐t♦s ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s✳

  ◆❛ s❡çã♦ ✶✱ ❛♣r❡s❡♥t❛r❡♠♦s r❡s✉❧t❛❞♦s s♦❜r❡ ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✱ t❛✐s ❝♦♠♦ ❞❡✜♥✐çõ❡s ❞❡ ✈❛r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛ ❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❞❡ ❢✉♥çõ❡s ❡♥tr❡ ✈❛r✐❡❞❛❞❡s✱ ❝♦♥❝❡✐t♦ ❞❡ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❡ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡ p M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ✐s♠♦r✜s♠♦ ❡♥tr❡ ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ T ❡ ♦ ❡s♣❛ç♦ n ❡✉❝❧✐❞✐❛♥♦ R ✱ ♦♥❞❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦ n ❡ p ✉♠ ♣♦♥t♦ ❞❡ M✱ ❞❡✜♥✐çã♦ ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ ✈❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s✱ ❞❡♥t❡ ♦✉tr♦s✳ ❊st❛ s❡çã♦ t❡♠♦ ❝♦♠♦ ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s

  P❛r❛ ❛ s❡çã♦ ✷ ✉t✐❧✐③❛♠♦s ❝♦♠♦ ♣r✐♥❝✐♣❛✐s ❢♦♥t❡s ❛s r❡❢❡rê♥❝✐❛s ❡ ❊st❛ s❡çã♦ ♣♦ss✉✐ ❝♦♥❝❡✐t♦s s♦❜r❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ✢✉①♦ ❛ss♦❝✐❛❞♦ ❛♦ ❝❛♠♣♦✱ ❤✐♣❡r❜♦❧✐❝✐❞❛❞❡✱ r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ♥❛ t❡♦r✐❛ ❞❡ s✐st❡♠❛s ❞✐♠â♠✐❝♦s ❝♦♠♦ ❚❡♦r❡♠❛ ❞♦ ❋❧✉①♦ ❚✉❜✉❧❛r✱ ❚❡♦r❡♠❛ ❞❛ ❱❛r✐❡❞❛❞❡ ❊stá✈❡❧ r❡s✉❧t❛❞♦s s♦❜r❡ ♦s ❝♦♥❥✉♥t♦s ω− ❧✐♠✐t❡ ❡ α−❧✐♠✐t❡ ❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ♥ã♦ ✲❡rr❛♥t❡s ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s✳ ❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠✱ ✉♠ ♣♦✉❝♦ ❞❛ t❡♦r✐❛ ❞❛ ❢✉♥çã♦ ❞❡ Pr✐♠❡✐r♦ ❘❡t♦r♥♦ ❞❡ P♦✐♥❝❛ré✳

  ❖ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ é ❞❡❞✐❝❛❞♦ ❛ ♥♦çõ❡s ❞❡ t♦♣♦❧♦❣✐❛ ❛❧❣é❜r✐❝❛ ❡ ❛s♣❡❝t♦s t♦♣♦❧ó❣✐❝♦s✱ ❡st❡ ❝❛♣ít✉❧♦ ❡stá ❡str✉t✉r❛❞♦ ❡♠ ❞✉❛s s❡çõ❡s s♦❜r❡ ♦s t❡♠❛s r❡❢❡r✐❞♦s✳ ◆❛ ♣r✐♠❡✐r❛ s❡çã♦✱ ❛❜♦r❞❛♠♦s ♥♦çõ❡s ❞❡ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ ❝❛♠✐♥❤♦s ❡♠ ✈❛r✐❡❞❛❞❡s✱ t✐♣♦s ❞❡ ❤♦♠♦t♦♣✐❛s✱ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡✱ ❛❧é♠ ❞❡ ❝♦♥❝❡✐t♦s s♦❜r❡ ❤♦♠♦♠♦r✜s♠♦ ✐♥❞✉③✐❞♦✱ ♦ q✉❛❧ t❡♠ ❣r❛♥❞❡ r❡❧❛çã♦ ❝♦♠ ❛ ✐♥❝♦♠♣r❡ss✐❜✐❧✐❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡s ♥✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❆ s❡❣✉♥❞❛ s❡çã♦ é ❞❡st✐♥❛❞❛ ❛♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s t♦♣♦❧ó❣✐❝♦s ✉t✐❧✐③❛❞♦s ♥❡ss❡ tr❛❜❛❧❤♦✱ t❛✐s ❝♦♠♦ ✐♥❝♦♠♣r❡ss✐❜✐❧✐❞❛❞❡ ❞❡ s✉♣❡r❢í❝✐❡s ❡ ✐rr❡❞✉t✐❜✐❧✐❞❛❞❡ ❞❡ ✈❛r✐❡❞❛❞❡s✱ ❛❧é♠ ❞❡ ♦✉tr♦s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s✳ ❚❡♥❞♦ ❝♦♠♦ ♣r✐♥❝✐♣❛❧ r❡❢❡rê♥❝✐❛

  ◆♦ ❝❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s ❧❡♠❛s ❡ t❡♦r❡♠❛s ❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ❞❡♠♦♥str❛r ♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❞♦ ❛rt✐❣♦ ❞❡ r❡❢❡rê♥❝✐❛✳ ❯♠ ❡①❡♠♣❧♦ ❞❡ ✉♠ t♦r♦ s❛t✐s❢❛③❡♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠

  ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧✳ ❙❡❥❛ T ✉♠ t♦r♦ ♠❡r❣✉❧❤❛❞♦ ❡♠ ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ M✳ ❙✉♣♦♥❤❛ q✉❡

  ✸ 1 ❙❯▼➪❘■❖

  ✶✳ T é tr❛♥s✈❡rs❛❧ ❛ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X✱ C ✱ ❡♠ M✳ ✷✳ ❊①✐st❡ ✉♠❛ ú♥✐❝❛ ór❜✐t❛ O ❞❡ X q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛ T ✳ ✸✳ O é ❤✐♣❡r❜ó❧✐❝♦ ❡ ♥ã♦ é ❤♦♠♦tó♣✐❝♦ ❛ ✉♠ ♣♦♥t♦ ❡♠ M✳

  ❊♥tã♦✱ T é ✐♥❝♦♠♣r❡ssí✈❡❧ ❡ M é ✐rr❡❞✉tí✈❡❧✳

  ❈❛♣ít✉❧♦ ✶ ❈♦♥❝❡✐t♦s ❞❡ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦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✐❡❞❛❞❡✳ ❆❧é♠ ❞❛ ✜①❛çã♦ ❞❛ ♥♦t❛çã♦ q✉❡ s❡rá ✉t✐❧✐③❛❞❛ ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✳ ❚❡♠✲s❡ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❛❥✉❞❛r ♦ ❧❡✐t♦r ❛ s❡ ❢❛♠✐❧✐❛r✐③❛r ❝♦♠ ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s q✉❡ sã♦ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ♥♦ss♦ 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♦ ✶✳✶✳✶✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛ ❞❡ ❞✐♠❡♥sã♦ n✱ é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ M ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

  • M é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❞❡ ❍❛✉s❞♦r✛✳
  • M t❡♠ ✉♠❛ ❜❛s❡ ❡♥✉♠❡rá✈❡❧ ❞❡ ❛❜❡rt♦s✳ n

  P❛r❛ q✉❛❧q✉❡r ♣♦♥t♦ p ∈ M ❡①✐st❡♠ ❛❜❡rt♦s U ⊂ M ❝♦♥t❡♥❞♦ p ❡ A ❡♠ R

  • ❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ φ : U → A✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❝❛❞❛ ♣♦♥t♦ p ∈ M n

  ♣♦ss✉✐ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠ ❛❜❡rt♦ ❞❡ R ✳ ❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❙❡❥❛♠ M ✉♠❛ ✈❛r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛ ❡ U ⊂ M ❛❜❡rt♦ t❛❧ q✉❡ n p ∈ U (p ∈ M )

  ✱ A ⊂ R ❛❜❡rt♦ ❡ φ : U → A ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✳ ❖ ♣❛r (U, φ)

  é ❞❡♥♦♠✐♥❛❞♦ ❝❛rt❛ ❧♦❝❛❧ ♦✉ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛❧ ❞❡ M ❡♠ p✳ U é ❞❡♥♦♠✐♥❛❞♦ ✈✐③✐♥❤❛♥ç❛ ❝♦♦r❞❡♥❛❞❛✳ n n

  ➚s ✈❡③❡s✱ ❞✐r❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ φ : U → A ⊂ R ✱ ❝♦♠ U ⊂ M✱ A ⊂ R ❛❜❡rt♦s ❡ p ∈ M✱ é ✉♠❛ ❝❛rt❛ ♦✉ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ❡♠ ✈❡③ ❞❡ (U, φ) é ✉♠❛ ❝❛rt❛ ❧♦❝❛❧ ♦✉ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛❧ ❞❡ M ❡♠ p✳

  ✹

  ✺ ✶✳✶✳ ◆❖➬Õ❊❙ ❙❖❇❘❊ ❱❆❘■❊❉❆❉❊❙ ❉■❋❊❘❊◆❈■➪❱❊■❙ i i : U → ❉❡✜♥✐çã♦ ✶✳✶✳✸✳ ❯♠ ❛t❧❛s ❞❡ ❞✐♠❡♥sã♦ n ❞❡ M é ✉♠❛ ❝♦❧❡çã♦ U = {φ n A i } i i ⊂ M i ⊂ R i U i = M ∈I ∈I

  ❞❡ ❤♦♠❡♠♦r✜s♠♦s ♦♥❞❡ U ❛❜❡rt♦✱ A ❛❜❡rt♦ ❡ ∪ ✳ ❖s ❤♦♠❡♦♠♦r✜s♠♦s✿ −1

  ◦ φ ∩ U → φ ∩ U φ j : φ i (U i j ) ⊂ A i j (U i j ) ⊂ A j i sã♦ ❝❤❛♠❛❞♦s ♠✉❞❛♥ç❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ r

  ❯♠ ❛t❧❛s U é ❞✐t♦ ❞❡ ❝❧❛ss❡ C ✱ 1 ≤ r ≤ ∞✱ s❡ t♦❞❛s ❛s ♠✉❞❛♥ç❛s ❞❡ r ❝♦♦r❞❡♥❛❞❛s ❞♦ ❛t❧❛s U sã♦ ❞❡ ❝❧❛ss❡ C ✳

  M Ai Aj

  Φ i Φ j Rm

  Rm -1 o Φ j Φ i U i Uj

  ❋✐❣✉r❛ ✶✳✶ m ❯♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ψ : W → R ❞❡ M ❞✐③✲s❡ ❛❞♠✐ssí✈❡❧ r r❡❧❛t✐✈❛♠❡♥t❡ ❛ ✉♠ ❛t❧❛s U ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C ✱ r > 0✱ ❞❡ M s❡ ♣❛r❛ m t♦❞♦ φ ∈ U ❝♦♠ U ∩ W 6= ∅✱ ♦♥❞❡ φ : U → A ⊂ R t❡♠✲s❡ q✉❡ ❛s ♠✉❞❛♥ç❛s ❞❡ −1 −1 r

  ❝♦♦r❞❡♥❛❞❛s φ ◦ ψ ❡ ψ ◦ φ sã♦ ❞❡ ❝❧❛ss❡ C ✳ ❖✉ s❡❥❛✱ U ∪ {ψ} é t❛♠❜é♠ ✉♠ r ❛t❧❛s ❞❡ ❝❧❛ss❡ C ✳ r

  ❯♠ ❛t❧❛s U ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C ✱ r > 0✱ ❞❡ M é ❝❤❛♠❛❞♦ ♠á①✐♠♦ U q✉❛♥❞♦ ❝♦♥té♠ t♦❞♦s ♦s s✐st❡♠❛s ❝♦♦r❞❡♥❛❞❛s q✉❡ sã♦ ❛❞♠✐ssí✈❡✐s ❡♠ r❡❧❛çã♦ ❛ ✳ r

  ❱❛❧❡ r❡ss❛❧t❛r q✉❡ t♦❞♦ ❛t❧❛s ❞❡ ❞✐♠❡♥sã♦ m ❡ ❞❡ ❝❧❛ss❡ C ✱ r > 0✱ ❞❡ M✱ r ♣♦❞❡ s❡r ❛♠♣❧✐❛❞♦ ❛té s❡ t♦r♥❛r ✉♠ ❛t❧❛s ♠á①✐♠♦ ❞❡ ❝❧❛ss❡ C ✱ ♣❛r❛ ✐ss♦ ❜❛st❛ ❛❝r❡s❝❡♥t❛r✲❧❤❡ t♦❞♦s ♦s s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛❞♠✐ssí✈❡✐s✳ r ❉❡✜♥✐çã♦ ✶✳✶✳✹✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C ✱ r > 0

  ✱ é ✉♠ ♣❛r (M, U)✱ ♦♥❞❡ M é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❞❡ ❍❛✉s❞♦r✛✱ ❝♦♠ ❜❛s❡ r ❡♥✉♠❡rá✈❡❧ ❡ U é ✉♠ ❛t❧❛s ♠á①✐♠♦ ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C ✳

  ❆ ❡①✐❣ê♥❝✐❛ ❞❡ ♦ ❛t❧❛s U s❡r ♠á①✐♠♦ ♥ã♦ é ♥❡❝❡ssár✐❛✱ ♠❛s é ❝♦♥✈❡♥✐❡♥t❡ ♣♦✐s ♠✉✐t♦s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s✱ ❝♦♠♦ ❛❧❣✉♥s t❡♦r❡♠❛s ❞❡ ❲❤✐t♥❡② ♥❡❝❡ss✐t❛♠ ❞❡ss❛ ♣r♦♣r✐❡❞❛❞❡✱ ❛ss✐♠ ❝♦♠♦ ❢❛t♦ ❞❡ M s❡r ❞❡ ❍❛✉s❞♦r✛ ❡ t❡r ❜❛s❡ ❡♥✉♠❡rá✈❡❧✳

  ✻ ✶✳✶✳ ◆❖➬Õ❊❙ ❙❖❇❘❊ ❱❆❘■❊❉❆❉❊❙ ❉■❋❊❘❊◆❈■➪❱❊■❙ m ❖❜s❡r✈❛çã♦ ✶✳✶✳✺✳ ➚s ✈❡③❡s✱ ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦ M ♣❛r❛ ✐♥❞✐❝❛r q✉❡ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M é ❞✐♠❡♥sã♦ m✱ ✐st♦ é✱ ♦ ❛t❧❛s ♠á①✐♠♦ U t❡♠ ❞✐♠❡♥sã♦ m✳ ❖❜s❡r✈❛çã♦ ✶✳✶✳✻✳ ❆s s✉♣❡r❢í❝✐❡s sã♦ ❛s ❝❤❛♠❛❞❛s ✈❛r✐❡❞❛❞❡s ❞❡ ❞✐♠❡♥sã♦ ❞♦✐s✳ ❊♥q✉❛♥t♦ ✉♠❛ ❝✉r✈❛ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦ ✉♠✳

  ✶✳✶✳✶ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❡♥tr❡ ✈❛r✐❡❞❛❞❡s m n r

  ❉❡✜♥✐çã♦ ✶✳✶✳✼✳ ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡ ❝❧❛ss❡ C ✱ r > 0✳ ❯♠❛ ❛♣❧✐❝❛çã♦ f : M → N é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ p ∈ M s❡ ❡①✐st❡♠ ❝❛rt❛s m n x : U → x(U ) ⊂ R xy = y ◦ f ◦ x : x(U ) ⊂ R → y(V ) ⊂ R ❡♠ M✱ y : V → y(V ) ⊂ R ❡♠ N✱ ❝♦♠ p ∈ U ❡ f(U) ⊂ V −1 m n t❛✐s q✉❡ f

  é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ x(p) ✳ xy

  ●❡r❛❧♠❡♥t❡ ❝❤❛♠❛♠♦s f ❝♦♠♦ ❛ ❡①♣r❡ssã♦ ❞❛ ❛♣❧✐❝❛çã♦ f ♥❛s ❝♦♦r❞❡♥❛❞❛s ♦✉ ❝❛rt❛s x ❡ y✳ 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❛ ✶✳✷ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ❛ ♥♦çã♦ ❞❡ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❡♥tr❡ ✈❛r✐❡❞❛❞❡s ✐♥❞❡♣❡♥❞❡ ′ ′ ′ ′ m

  : U → x (U ) ⊂ R ❞❛ ❡s❝♦❧❤❛ ❞❛s ❝❛rt❛s x ❡ y✳ ❈♦♥s✐❞❡r❡ ❛s ❝❛rt❛s x ❡♠ M ❡ ′ ′ ′ ′ n ′ ′ ′

  → y y : V (V ) ⊂ R ) ⊂ V ′ ′ −1 ′ −1 ′ −1 −1 ❡♠ N t❛✐s q✉❡ p ∈ U ❡ f(U ✳ ❊♥tã♦✱ y ◦ f ◦ (x ) = y ◦ y ◦ y ◦ f ◦ (x ◦ x ◦ x) ′ −1 −1 ′ −1 = y ◦ y ◦ y ◦ f ◦ x ◦ x ◦ (x ) ′ −1 ′ −1

  ◦ y ◦ f ◦ x ◦ (x = y xy ) ′ −1 ′ −1 −1 ′ −1 ′ −1 = y ◦ y ◦ f xy ◦ ((x ) ◦ x ) .

  ◦ y ◦ x xy ❈♦♠♦ y ❡ x sã♦ ❞✐❢❡♦♠♦r✜s♠♦s ❡ f é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ x(p) s❡❣✉❡ x y = y ◦ f ◦ (x ) (p) ′ ′ ′ ′ −1 ′ q✉❡ f é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ x ✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ ❛ ♥♦çã♦ ❞❡

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

  ✼ ✶✳✶✳ ◆❖➬Õ❊❙ ❙❖❇❘❊ ❱❆❘■❊❉❆❉❊❙ ❉■❋❊❘❊◆❈■➪❱❊■❙ m n r ❉❡✜♥✐çã♦ ✶✳✶✳✾✳ ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡ ❝❧❛ss❡ C ✱ r > 0✳ r ❯♠❛ ❛♣❧✐❝❛çã♦ f : M → N é ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❝❧❛ss❡ C ✱ ❝♦♠ k ≤ r✱ ♣❛r❛ ❝❛❞❛ m n ♣♦♥t♦ p ∈ M s❡ ❡①✐st❡♠ ❝❛rt❛s x : U → x(U) ⊂ R ❡♠ M✱ y : V → y(V ) ⊂ R −1 m n

  : x(U ) ⊂ R → y(V ) ⊂ R ❡♠ N✱ ❝♦♠ p ∈ U ❡ f(U) ⊂ V t❛✐s q✉❡ y ◦ f ◦ x é k ❞❡ ❝❧❛ss❡ C ✳ r ❈♦♠♦ ❛s ♠✉❞❛♥ç❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡♠ M ❡ N sã♦ ❞✐❢❡♦♠♦r✜♠♦s ❞❡ ❝❧❛ss❡ C t❡♠✲s❡ q✉❡ ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❛s ❝❛rt❛s x ❡ y✳

  ✶✳✶✳✷ ❊s♣❛ç♦ ❚❛♥❣❡♥t❡ m r

  ❈♦♥s✐❞❡r❡ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❝❧❛ss❡ C ✱ r > 0 ❡ x ∈ M ✳ x ❉❡♥♦t❛r❡♠♦s ♣♦r C ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❝✉r✈❛s α : (−ǫ, ǫ) → M✱ ǫ ≥ 0✱ t❛✐s x q✉❡ α(0) = x ❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ 0✳ ❙❡❥❛♠ α, β ∈ C ✱ ❞✐③❡♠♦s q✉❡ ❡❧❛s t❡♠ ♦ i : U i → e U i ⊂ R i m

  ♠❡s♠♦ ✈❡t♦r t❛♥❣❡♥t❡ ❡♠ x s❡ ♣❛r❛ ❛❧❣✉♠❛ ❝❛rt❛ φ ✱ x ∈ U i ◦ α) (0) = (φ i ◦ β) (0) ′ ′ t❡♠✲s❡ q✉❡ (φ ✳ ❈❛s♦ s❡❥❛ ♥❡❝❡ssár✐♦ ❝♦♥s✐❞❡r❡ ǫ ≥ 0 ❞❡ t❛❧ i ❢♦r♠❛ q✉❡ α((−ǫ, ǫ)) ⊂ U ✳ j : U j → e U j ⊂ R m

  ❊st❛ ♣r♦♣r✐❡❞❛❞❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❛ ❝❛rt❛✳ ❙❡❥❛ φ ✱ x ∈ U j ✱ ✉♠❛ ♦✉tr❛ ❝❛rt❛✳ ❆ss✐♠✱ ′ −1 ′

  (φ j ◦ α) (0) = D(φ j ◦ φ )(φ i (x))(φ i ◦ α) (0) ′ −1 ′ i ◦ β) ◦ φ ◦ β) i ◦ α) (0) = (φ i ◦ β) (0) j ◦ α) (0) = (φ j ◦ β) (0) ′ ′ ′ ′ (φ j (0) = D(φ j )(φ i (x))(φ i (0). i

  ❈♦♠♦ (φ s❡❣✉❡ q✉❡ (φ ✳ ◆❡st❡ ❝❛s♦ ❞✐③❡♠♦s q✉❡ t❛✐s ❝✉r✈❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s✳ ❊st❛ r❡❧❛çã♦ é ✉♠❛ r❡❧❛çã♦ ❞❡ x ❡q✉✐✈❛❧ê♥❝✐❛ ♥♦ ❝♦♥❥✉♥t♦ C ❡ ❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ α s❡rá ❞❡♥♦t❛❞❛ ♣♦r [α]✱ é ❝❤❛♠❛❞❛ ♦ ✈❡t♦r t❛♥❣❡♥t❡ ❛ α ♥♦ ♣♦♥t♦ x ∈ M✳ ❖ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❛ M ♥♦ x M ♣♦♥t♦ x✱ ❞❡♥♦t❛❞♦ ♣♦r T ✱ é ♦ ❝♦♥❥✉♥t♦ ❞❡ t❛✐s ✈❡t♦r❡s t❛♥❣❡♥t❡s✳ m i : U i U i i x M → e

  ❯♠❛ ❝❛rt❛ φ ✱ x ∈ U ✱ ❡st❛❜❡❧❡❝❡ ✉♠❛ ❜✐❥❡çã♦ ❡♥tr❡ T ❡ R ✳ ❊st❛ x M i ◦ α) (0) ∈ R m ❜✐❥❡çã♦ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ [α] ∈ T ♦ ✈❡t♦r (φ ✱ m i x : T M → R ♦✉ s❡❥❛✱ φ ❞❛❞❛ ♣♦r✿

  φ i ([α]) = (φ i ◦ α) (0) é ✉♠❛ ❜✐❥❡çã♦✳ ❉❡ ❢❛t♦✱ 1. φ i

  é ✐♥❥❡t✐✈❛✳ ′ ′ φ i ([α]) = φ i ([β]) ⇒ (φ i ◦ α) (0) = (φ ◦ β) (0). i ❊♥tã♦ α ❡ β sã♦ ❡q✉✐✈❛❧❡♥t❡s✱ ❛ss✐♠ [α] = [β]✳ ▲♦❣♦✱ φ é ✐♥❥❡t✐✈❛✳ 2. φ i

  é s♦❜r❡❥❡t✐✈❛✳ m −1 x (φ i (x) + tv) ❉❛❞♦ v ∈ R ✱ s❡❥❛ α ∈ C ❞❛❞❛ ♣♦r α(t) = φ i t❡♠♦s q✉❡

  φ i ([α]) = (φ i ◦ α) (0) = v ✳

  ✽ ✶✳✶✳ ◆❖➬Õ❊❙ ❙❖❇❘❊ ❱❆❘■❊❉❆❉❊❙ ❉■❋❊❘❊◆❈■➪❱❊■❙ i P♦rt❛♥t♦✱ φ é ✉♠❛ ❜✐❥❡çã♦✳ x M ❆ss✐♠✱ ♣♦❞❡♠♦s ♠✉♥✐r T ❝♦♠ ❛ ❡str✉t✉r❛ ❞❡ ✉♠❛ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧✱ ❞❡ i

  ♠♦❞♦ q✉❡ φ s❡❥❛ ✉♠ ✐s♦♠♦r✜s♠♦✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❛s ♦♣❡r❛çõ❡s ❞❡ s♦♠❛ ❡ ♣r♦❞✉t♦ ♣♦r ✉♠ ♥ú♠❡r♦ r❡❛❧ s❡rã♦ ❞❡✜♥✐❞❛s ♣♦r✿ −1

  [α] + [β] = (φ i ) (φ i ([α]) + φ i ([β])) −1 c · [α] = (φ i ) (c · φ i ([α])). i j : U j → f U j ❖❜s❡r✈❡ q✉❡ ❡st❛ ❞❡✜♥✐çã♦ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ φ ✳ ❙❡❥❛ φ ✱ x ∈ U j

  ✱ ❡♥tã♦✿ (φ j )([α]) = (φ j ◦ α) (0) −1 ′

  ◦ (φ ◦ α)) = ((φ j i ) ) ◦ (φ i (0) −1

  ◦ (φ j ◦ (φ i ) )(φ i (x)) j ◦ (φ i ) −1 = D(φ j i ) )(φ i (x)) · φ i ([α]). −1 ❈♦♠♦ D(φ é ✉♠ ✐s♦♠♦r✜s♠♦ ✭♣♦✐s φ é ✉♠ i j

  ❞✐❢❡♦♠♦r✜s♠♦✮✱ ❡ φ é ✉♠ ✐s♦♠♦r✜s♠♦ t❡♠♦s q✉❡ φ t❛♠❜é♠ é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❆ss✐♠✱ ❛ ❡str✉t✉r❛ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ✐♥❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❛ ❝❛rt❛✳ m ∂ ∂ ❉❛❞♦s ✉♠❛ ❝❛rt❛ φ : U → V ⊂ R ❡♠ M ❡ ✉♠ ♣♦♥t♦ x ∈ U ✐♥❞✐❝❛♠♦s ♣♦r m { 1 (x), ..., (x)} x M x M → R m ∂x ∂x ❛ ❜❛s❡ ❞❡ T q✉❡ é ❧❡✈❛❞❛ ♣❡❧♦ ✐s♦♠♦r✜s♠♦ φ : T

  , ..., e } i (p) ∈ T x M s♦❜r❡ ❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ {e 1 x (0) = e i 2 ✳ ❖ ✈❡t♦r é ❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ∂x ❞❡ q✉❛❧q✉❡r ❝❛♠✐♥❤♦ α ∈ C t❛❧ q✉❡ (φ ◦ α) ✳ x = R R m m ❊①❡♠♣❧♦ ✶✳✶✳✶✵✳ T m m

  → R ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✿ φ = id : R ✳ ❊♥tã♦ ❛

  ❛♣❧✐❝❛çã♦✿ R m m φ : T x → R

  ❞❛❞❛ ♣♦r ′ ′ φ([α]) = (id ◦ α) (0) = α (0)

  ❈♦♠♦ ❥á ❢♦✐ ✈✐st♦✱ φ é ✉♠ ✐s♦♠♦r✜s♠♦✳ x | α (0) = ❈♦♠ ❡st❡ ✐s♦♠♦r✜s♠♦ ❡st❛♠♦s ✐❞❡♥t✐✜❝❛♥❞♦ ❛ ❝♦❧❡çã♦ [α] = {β ∈ C ′ ′ ′ m ′

  (id ◦ α) (0) = (id ◦ β) (0) = β (0)} (0) = v ❝♦♠ ♦ ✈❡t♦r v ∈ R t❛❧ q✉❡ α ✳ v m n ❋✐❣✉r❛ ✶✳✸

  Pr♦♣♦s✐çã♦ ✶✳✶✳✶✶✳ ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡ f : M → N p M → T f N (p) ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ p ∈ M✳ ❆ ❛♣❧✐❝❛çã♦ Df(p) : T p M f (p) N q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ✈❡t♦r t❛♥❣❡♥t❡ [α] ∈ T ♦ ❡❧❡♠❡♥t♦ [f ◦ α] ∈ T é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r q✉❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ α✳

  ✾ ✶✳✶✳ ◆❖➬Õ❊❙ ❙❖❇❘❊ ❱❆❘■❊❉❆❉❊❙ ❉■❋❊❘❊◆❈■➪❱❊■❙ M N f

  ε α f o

  α - ε ❋✐❣✉r❛ ✶✳✹

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ❉❡✜♥✐çã♦ ✶✳✶✳✶✷✳ ❆ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r Df(p) ❞❛ Pr♦♣♦s✐çã♦ é ❝❤❛♠❛❞❛ ❞❡r✐✈❛❞❛ ❞❡ f ♥♦ ♣♦♥t♦ p✳ m n k Pr♦♣♦s✐çã♦ ✶✳✶✳✶✸✳ ✭❘❡❣r❛ ❞❛ ❈❛❞❡✐❛✮ ❙❡❥❛♠ M ✱ N ✱ P ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✱ f : M → N ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ ✉♠ ♣♦♥t♦ p ∈ M ❡ g : N → P

  ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ f(p) ∈ N✳ ❊♥tã♦ g ◦f : M → P é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ p ∈ M ❡ D(g ◦ f )(p) = Dg(f (p)) ◦ Df (p) : T p M → T g (f (p)) P.

  ❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ ❛s s❡❣✉✐♥t❡s ❝❛rt❛s x : U → x(U) ❡♠ M ❡ y : V → y(V ) ❡♠ N ❡ z : W → z(W ) ❡♠ P ✱ t❛✐s q✉❡ p ∈ U✱ f(U) ∈ V ❡ g(V ) ∈ W ✳ ❚❡♠♦s xy = y ◦ f ◦ y : x(U ) ⊂ R → y(V ) ⊂ R −1 m n q✉❡✱ f −1 n p é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ x(p) ❡ g yz = z ◦ g ◦ y : y(V ) ⊂ R → z(W ) ⊂ R

  é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ y(f(p))✳ P❡❧❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛ ❡♠ ❡s♣❛ç♦s ✈❡t♦r✐❛s r❡❛✐s✱ s❡❣✉❡ q✉❡✿ −1 m p ◦ f → z(W ) ⊂ R g yz xy = z ◦ (g ◦ f ) ◦ x : x(U ) ⊂ R

  é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ x(p)✳ ❆ss✐♠✱ g ◦ f : M → P é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ p ∈ M ✳ p M

  ❆❣♦r❛✱ ❞❛❞♦ [α] ∈ T t❡♠♦s q✉❡ D(g ◦ f )(p)[α] = [g ◦ f ◦ α]

  = [g ◦ (f ◦ α)] = Dg(f (p))[f ◦ α] = Dg(f (p)) ◦ Df (p)[α].

  P♦rt❛♥t♦✱ g ◦ f : M → P é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ p ∈ M ❡ D(g ◦ f)(p) = Dg(f (p)) ◦ Df (p) : T p M → T g P m (f (p)) ✳ r ❊①❡♠♣❧♦ ✶✳✶✳✶✹✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❝❧❛ss❡ C ✱ r > 0✱ ❡ m φ : U ⊂ M → A ⊂ R

  ✉♠❛ ❝❛rt❛ ♣❛r❛ ♦ ♣♦♥t♦ x ∈ M✳ ❊♥tã♦✱ R m Dφ(x) : T x M → T φ (x)

  [α] 7→ [φ ◦ α]

  ✶✵ ✶✳✶✳ ◆❖➬Õ❊❙ ❙❖❇❘❊ ❱❆❘■❊❉❆❉❊❙ ❉■❋❊❘❊◆❈■➪❱❊■❙ P❡❧♦ ❡①❡♠♣❧♦ ♣♦❞❡ s❡r ❞❡✜♥✐❞❛ ❝♦♠♦✿ m

  Dφ(x) : T x M → R t❛❧ q✉❡ Dφ(x)[α] = (φ ◦ α) (0).

  ❆ss✐♠✱ Dφ(x) é ✉♠ ✐s♦♠♦r✜s♠♦✳ m n ❉❡✜♥✐çã♦ ✶✳✶✳✶✺✳ ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❉✐③❡♠♦s q✉❡ f : M → N

  é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ s❡ f é ❜✐❥❡t♦r❛✱ ❞✐❢❡r❡♥❝✐❛❧ ❡ ❝♦♠ ✐♥✈❡rs❛ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❊ f é ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧ ❡♠ p ∈ M s❡ ❡①✐st❡♠ ✈✐③✐♥❤❛♥ç❛s U ❞❡ p U : U → V

  ❡♠ M ❡ V ❞❡ f(p) t❛❧ q✉❡ f| é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✳ ❆ ♥♦çã♦ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ é ❛ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✱ m

  ♦✉ s❡❥❛✱ s❡ ❡①✐st❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❡♥tr❡ ❞✉❛s ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s M ❡ n N

  ❞✐③❡♠♦s q✉❡ ❡❧❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s✳ ❆ss✐♠ ❝♦♠♦ ❛ ♥♦çã♦ ❞❡ ❤♦♠❡♦♠♦r✜s♠♦ ❡♥tr❡ ✈❛r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s é ❛ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♠ ✈❛r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s✳

  ❯♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ❞❛ ❋✉♥çã♦ ❈♦♠♣♦st❛ é q✉❡ s❡ f : M → N é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❡♥tr❡ ❞✉❛s ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s M ❡ N ❡♥tã♦ Df(p) : T p M → T f N (p) é ✉♠ ✐s♦♠♦r✜s♠♦ ♣❛r❛ t♦❞♦ p ∈ M✱ ❡♠ ♣❛rt✐❝✉❧❛r ❛s ❞✐♠❡♥sõ❡s ❞❡ M ❡ N sã♦ ✐❣✉❛✐s✳ ❯♠❛ r❡❝í♣r♦❝❛ ❞❡st❡ ❢❛t♦ é ♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✳ n n

  → N ❚❡♦r❡♠❛ ✶✳✶✳✶✻✳ ❙❡❥❛ f : M ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♥tr❡ ❞✉❛s p M → T f N (p) ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s M ❡ N✱ t❛❧ q✉❡ Df(p) : T é ✉♠ ✐s♦♠♦r✜s♠♦✱ ❝♦♠ p ∈ M✳ ❊♥tã♦✱ f é ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧ ❡♠ p✳

  ✶✳✶✳✸ ■♠❡rsõ❡s ❡ ▼❡r❣✉❧❤♦s m n

  ❉❡✜♥✐çã♦ ✶✳✶✳✶✼✳ ❙❡❥❛ M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❯♠❛ ❛♣❧✐❝❛çã♦ p M → T f N ❞✐❢❡r❡♥❝✐á✈❡❧ f : M → N é ✉♠❛ s✉❜♠❡rsã♦ s❡ Df(p) : T (p) é s♦❜r❡❥❡t✐✈❛ ♣❛r❛ t♦❞♦ p ∈ M✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ s❡ f é ✉♠❛ s✉❜♠❡rsã♦ ❡♥tã♦ m ≥ n

  ✳ m n ❉❡✜♥✐çã♦ ✶✳✶✳✶✽✳ ❙❡❥❛ M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❯♠❛ ❛♣❧✐❝❛çã♦ p f (p) M → T N ❞✐❢❡r❡♥❝✐á✈❡❧ f : M → N é ✉♠❛ ✐♠❡rsã♦ s❡ Df(p) : T é ✐♥❥❡t✐✈❛ ♣❛r❛ t♦❞♦ p ∈ M✳ ❙❡ ❛❧é♠ ❞✐ss♦✱ f ❢♦r ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ s♦❜r❡ ❛ s✉❛ ✐♠❛❣❡♠ f (M ) ⊂ N

  ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ♣♦r N✱ ❞✐③❡♠♦s q✉❡ f é ✉♠ ♠❡r❣✉❧❤♦✳ ❙❡ M ⊂ N

  ❡ ❛ ✐♥❝❧✉sã♦ i : M ֒→ N é ✉♠ ♠❡r❣✉❧❤♦ ❡♥tã♦ M é ❝❤❛♠❛❞❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ N✳

  ❖❜s❡r✈❡ q✉❡ ♣❛r❛ f : M → N s❡r ✉♠❛ ✐♠❡rsã♦ é ♥❡❝❡ssár✐♦ q✉❡ m ≤ n✱ ❛ ❞✐❢❡r❡♥ç❛ n − m é ❝❤❛♠❛❞❛ ❞❡ ❝♦❞✐♠❡♥sã♦ ❞❛ ✐♠❡rsã♦ f✳ ❖s t❡♦r❡♠❛s ✐♥t✐t✉❧❛❞♦s ❚❡♦r❡♠❛s ❞❡ ❲❤✐t♥❡② ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❚❡♦r❡♠❛ ✶✳✶✳✶✾✳ ✭❲❤✐t♥❡②✮ ❙❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦ 2m+1 m

  ❡♥tã♦ ❡①✐st❡ ✉♠ ♠❡r❣✉❧❤♦ ❛❞❡q✉❛❞♦ f : M → R ✳

  ✶✶ ✶✳✶✳ ◆❖➬Õ❊❙ ❙❖❇❘❊ ❱❆❘■❊❉❆❉❊❙ ❉■❋❊❘❊◆❈■➪❱❊■❙ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ S ⊂ M ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡✳ ❯♠❛

  ✈✐③✐♥❤❛♥ç❛ t✉❜✉❧❛r ❞❡ S é ♦ ♣❛r (V, S) ♦♥❞❡ V é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ S ❡♠ M ❡ π : V → S é ✉♠❛ s✉❜♠❡rsã♦ ❞❡ ❝❧❛ss❡ C t❛❧ q✉❡ π(p) = p✱ ∀p ∈ S✳ ❚❡♦r❡♠❛ ✶✳✶✳✷✵✳ ❙❡❥❛ S ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ M✱ ❝♦♠♣❛❝t❛ ❡ s❡♠ ❜♦r❞♦✱ ❞❡ ❝❧❛ss❡ C ❡♥tã♦ S ♣♦ss✉✐ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ t✉❜✉❧❛r✳ r

  ❚♦❞❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M ❞❡ ❝❧❛ss❡ C ✱ r > 0✱ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♦ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❝❧❛ss❡ C ✳ m ❚❡♦r❡♠❛ ✶✳✶✳✷✶✳ ✭❲❤✐t♥❡②✮ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❝❧❛ss❡ r r 2m+1 C f : M → R

  ✱ r > 0✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ♠❡r❣✉❧❤♦ C t❛❧ q✉❡ f(M) é ✉♠❛ ∞ 2m+1 ✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ ❞❡ ❝❧❛ss❡ C ❡♠ R ✳ m ❉❡✜♥✐çã♦ ✶✳✶✳✷✷✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❉❡✜♥✐♠♦s ♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡ M ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦✿

  T M = {(x, v); x ∈ M, v ∈ T x M } ✳

  ❙❡❥❛ π : T M → M ❛ ♣r♦❥❡çã♦ (x, v) 7→ x✳ P♦❞❡✲s❡ ❞❡✜♥✐r ✉♠❛ t♦♣♦❧♦❣✐❛ ❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ ✈❛r✐❡❞❛❞❡ ❡♠ T M✳ m i : U i U i , i ∈ I} ⊂ M → e ⊂ R

  ❙❡❥❛ U = {φ ✉♠ ❛t❧❛s ♠❛①✐♠❛❧ ❞❡ M✳ ❉❡✜♥❛✿ −1 m Φ i : π (U i ) ⊂ T M → U i × R t❛❧ q✉❡

  Φ i (x, v) = (x, Dφ i (x)v). 1. Φ i

  é ✐♥❥❡t✐✈❛✳ Φ i (x, v) = Φ i (y, w) ⇒ (x, Dφ i (x)v) = (y, Dφ i (y)w) ⇒ i (x) ⇒ x = y, Dφ i (x)v = Dφ i (y)w.

  ❈♦♠♦ x = y ❡ Dφ é ✉♠ ✐s♦♠♦r✜s♠♦✱ ✈❡r ❡①❡♠♣❧♦✱ s❡❣✉❡ q✉❡ v = w ❡✱ Φ é ✐♥❥❡t✐✈❛✳ m −1 x (φ i (x) + tv)

  ❉❛❞♦ v ∈ R ✱ s❡❥❛ α ∈ C ❞❛❞❛ ♣♦r α(t) = φ i t❡♠♦s q✉❡ φ i ([α]) = (φ i ◦ α) (0) = v

  ✳ 2. Φ i

  é s♦❜r❡❥❡t✐✈❛✳ i × R x (φ i (x) + tv) m −1 i ❉❛❞♦ (x, v) ∈ U s❡❥❛ α ∈ C ❞❛❞❛ ♣♦r α(t) = φ t❡♠♦s q✉❡

  Φ i (x, [α]) = (x, v) i P♦rt❛♥t♦✱ Φ é ✉♠❛ ❜✐❥❡çã♦✳ ❊ t❛♠❜é♠✱ −1 m m

  Φ j ◦ Φ : (U i ∩ U j ) × R → (U i ∩ U j ) × R 1

  ✶✷ ✶✳✶✳ ◆❖➬Õ❊❙ ❙❖❇❘❊ ❱❆❘■❊❉❆❉❊❙ ❉■❋❊❘❊◆❈■➪❱❊■❙ ❞❛❞❛ ♣♦r

  M ♣♦❞❡ s❡r

  ∂x i (p),

  ✱ 1 ≤ i, j ≤ m ❞❡✜♥✐❞❛s ♣♦r✿ g x ij (p) = g x (x(p); e i , e j ) = h ∂

  ❈♦♥s✐❞❡r❡ ❛s ❢✉♥çõ❡s g x ij : U → R

  ♣♦r✿ (a, b) 7→ g x (x(p); a, b).

  a, (Dx p ) −1 bi p ◆♦t❡ q✉❡✱ ♣❛r❛ ❝❛❞❛ p ∈ U✱ t❡♠✲s❡ ❞❡✜♥✐❞♦ ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♠ R m ❞❛❞♦

  × R m → R ❞❡✜♥✐❞❛ ♣♦r✱ g x (x(p); a, b) = h(Dx p ) −1

  ❞❡♥♦♠✐♥❛❞❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✳ ❆ ❝❛❞❛ ❝❛rt❛ ❡♠ M m ✱ x : U ⊂ M → x(U) ⊂ R m ❛ss♦❝✐❡ ❛ ❢✉♥çã♦✿ g x : X(U ) × R m

  ❊st❛ ♥♦r♠❛ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ♥♦r♠❛ ❘✐❡♠❛♥♥✐❛♥❛✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ♦♥❞❡ ❡stá ❞❡✜♥✐❞❛ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ é

  ❞❡✜♥✐❞♦ ❝♦♠♦✿ | v | p = q hv, vi 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 p M ✳ ❉❡✜♥✐çã♦ ✶✳✶✳✷✹✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ♦✉ ♥♦r♠❛ ❞❡ ✉♠ ✈❡t♦r v ∈ T p

  Φ j ◦ Φ −1 1 (x, w) = (x, D(φ j ◦ φ −1 i )(φ i (x))w).

  ❡♥tã♦ ❡ss❡ ❝♦♥❥✉♥t♦ ❞❡ ❛♣❧✐❝❛çõ❡s ❢♦r♠❛♠ ✉♠ ❛t❧❛s ❡♠ T M✳ ❉❡✜♥✐çã♦ ✶✳✶✳✷✸✳ ❙❡❥❛ M m ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❯♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ✭♦✉ ❡str✉t✉r❛ ❘✐❡♠❛♥♥✐❛♥❛✮ é ✉♠❛ ❧❡✐ q✉❡ ❢❛③ ❝♦rr❡s♣♦♥❞❡r ❛ ❝❛❞❛ ♣♦♥t♦ p ∈ M ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♠ T p M

  Φ i (π −1 (U i )) = T M ✱

  = R m t❡♠♦s q✉❡ [ i ∈I e

  φ i (U i ) = M ❡ Dφ i (x)R m

  ❈♦♠♦ [ i ∈I

  ❞❡✜♥✐❞❛ ♣♦r e Φ i (x, v) = (φ i (x), Dφ i (x)v).

  Φ i : π −1 (U i ) → e U i × R m

  ❛♣❧✐❝❛çõ❡s Φ i sã♦ ❤♦♠❡♦♠♦r✜s♠♦s✳ ❈♦♥s✐❞❡r❡ ❛s s❡❣✉✐♥t❡s ❛♣❧✐❝❛çõ❡s ♣❛r❛ ❝❛❞❛ i ∈ I ✿ e

  (U i )) é ❛❜❡rt♦ ♣❛r❛ t♦❞♦ i ∈ I✳ ❉❡ss❡ ♠♦❞♦✱ ❛s

  ❈♦♠♦ φ j ◦ φ −1 i é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✱ ❧♦❣♦ Φ j ◦ Φ −1 1 é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✳ ❉❡✜♥✐♠♦s ✉♠❛ t♦♣♦❧♦❣✐❛ ❡♠ T M ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ W ⊂ T M é ❛❜❡rt♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ Φ i (W ∩ π −1

  ∂ ∂x j (p)i p .

  ✶✸ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ 1 m m m 1 , ..., a ) , ..., b )

  ❙❡❥❛♠ a = (a ❡ b = (b ✈❡t♦r❡s ❞♦ R ❡♥tã♦✱ m

  X −1 i ∂ (Dx p ) a = a (p) i i =1 m ∂x

  X −1 j ∂ (Dx p ) b = b (p). j m j =1 ∂x x x i j

  X (x(p); a, b) = g (p)a b

  ▲♦❣♦✱ g ij ✳ i,j =1 ❉❡✜♥✐çã♦ ✶✳✶✳✷✺✳ ❉✐③✲s❡ q✉❡ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ g ♥✉♠❛ ✈❛r✐❡❞❛❞❡ m k ❞✐❢❡r❡♥❝✐á✈❡❧ M é ❞❡ ❝❧❛ss❡ C ✱ k > 0✱ s❡✱ ♣❛r❛ ❝❛❞❛ ❝❛rt❛ x : U ⊂ m x m m k

  × R → R M → x(U ) ⊂ R : x(U ) × R

  ❛ ❢✉♥çã♦ g é ❞❡ ❝❧❛ss❡ C ✱ ♦✉ x k ij : U → R ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ s❡ ❛s ❢✉♥çõ❡s g sã♦ ❞❡ ❝❧❛ss❡ C ✳

  ❈♦♠♦ ❛s ♠✉❞❛♥ç❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s sã♦ ❞✐❢❡♦♠♦r✜s♠♦✱ ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❝❛rt❛ x✳ k

  Pr♦♣♦s✐çã♦ ✶✳✶✳✷✻✳ ❚♦❞❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M ❞❡ ❝❧❛ss❡ C ✱ k > 0✱ ❛❞♠✐t❡ k −1 ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡ ❝❧❛ss❡ C ✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ♣á❣✐♥❛ ✷✶✵✳ ❉❡✜♥✐çã♦ ✶✳✶✳✷✼✳ ❉✉❛s ✈❛r✐❡❞❛❞❡s V ❡ W sã♦ tr❛♥s✈❡rs❛✐s ❡♠ M s❡ ♣❛r❛ q

  V q W q✉❛❧q✉❡r ♣♦♥t♦ q ∈ V ∩ W t❡♠♦s q✉❡ ♦s ❡s♣❛ç♦s t❛♥❣❡♥t❡s ❞❡ T ❡ T ❣❡r❛♠ T q M

  ✳

  ✶✳✷ ◆♦çõ❡s ❞❡ ❙✐st❡♠❛s ❞✐♥â♠✐❝♦s

  ◆❡st❛ s❡çã♦ ✈❛♠♦s ❞❡✜♥✐r ❡ ❛♣r❡s❡♥t❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s s♦❜r❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❡ ✢✉①♦s ♥✉♠❛ ✈❛r✐❡❞❛❞❡ ♦r✐❡♥tá✈❡❧ ❡ ❝♦♠♣❛❝t❛✳ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ A ♥✉♠❛ ✈❛r✐❡❞❛❞❡ M ❞❡♥♦t❛r❡♠♦s ♣♦r intA✱ A ❡ frA✱ ♦ ✐♥t❡r✐♦r✱ ♦ ❢❡❝❤♦ ❡ ❛ ❢r♦♥t❡✐r❛ ❞❡ A✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ♣❛r❛ ❡st❛ s❡çã♦ ❢♦r❛♠ r ❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❯♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❞❡ ❝❧❛ss❡ C ✱ r > 0✱ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ n M

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

  ✶✹ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ T x M M

x

❋✐❣✉r❛ ✶✳✺ r

  (M ) ❆♣r❡s❡♥t❛r❡♠♦s ✉♠ t♦♣♦❧♦❣✐❛ ♥❛t✉r❛❧ ♥♦ ❡s♣❛ç♦ X ✱ r > 0✱ ❞❡ ❝❛♠♣♦s r

  ❞❡ ✈❡t♦r❡s ❞❡ ❝❧❛ss❡ C ♥✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛✳ ◆❡st❛ t♦♣♦❧♦❣✐❛ ❞♦✐s ❝❛♠♣♦s r X, Y ∈ X (M )

  ❡st❛rã♦ ♣ró①✐♠♦s s❡ ♦s ❝❛♠♣♦s ❡ s✉❛s ❞❡r✐✈❛❞❛s ❛té ❛ ♦r❞❡♠ r ❡st✐✈❡r❡♠ ♣ró①✐♠♦s ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ M✳ r s r

  (M, R ) ❈♦♥s✐❞❡r❡ ✐♥✐❝✐❛❧♠❡♥t❡ ♦ ❡s♣❛ç♦ C ❞❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❝❧❛ss❡ C ✱ 0 ≤ r < ∞

  ✱ ❞❡✜♥✐❞❛s ♥❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ M✳ ❚❡♠♦s ✉♠❛ ❡str✉t✉r❛ ♥❛t✉r❛❧ r s (M, R )

  ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡♠ C ✿ (f + g)(p) = f(p) + g(p) ❡ (λf)(p) = λf(p) r s (M, R )

  ♣❛r❛ f, g ∈ C ❡ λ ∈ R✳ ❚♦♠❡ ❡♠ M ✉♠❛ ❝♦❜❡rt✉r❛ ✜♥✐t❛ ♣♦r ❛❜❡rt♦s V , ..., V k i 1 i , U i ) t❛❧ q✉❡ ❝❛❞❛ V ❡st❡❥❛ ❝♦♥t✐❞♦ ♥♦ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❝❛rt❛ ❧♦❝❛❧ (x ❝♦♠ x (U ) = B(2) (V ) = B(1) i i i i

  ❡ x ✱ ♦♥❞❡ B(1) ❡ B(2) sã♦ ❜♦❧❛s ❞❡ r❛✐♦s ✶ ❡ ✷ ❡ ❝❡♥tr♦ n r s i −1 s (M, R ) = f ◦ x : B(2) → R i

  ♥❛ ♦r✐❣❡♠ ❞❡ R ✳ P❛r❛ f ∈ C ❞❡♥♦t❛♠♦s ♣♦r f ✳ ❉❡✜♥✐♠♦s✿ i i r i

  || f || {|| f r = max sup (u) ||, || Df (u) ||, ..., || D f (u) ||}. i u ∈B(1) r (M, R ) r s Pr♦♣♦s✐çã♦ ✶✳✷✳✷✳ || . || é ✉♠❛ ♥♦r♠❛ ❝♦♠♣❧❡t❛ ❡♠ C ✳ r

  (M ) ❉❡✜♥✐çã♦ ✶✳✷✳✸✳ ❯♠❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ ❞❡ ✉♠ ❝❛♠♣♦ X ∈ X ♣❛ss❛♥❞♦ ♣♦r r +1 p ∈ M α : I → M

  é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ❝❧❛ss❡ C ✱ ♦♥❞❡ I é ✉♠ ✐♥t❡r✈❛❧♦ ❞❛ r❡t❛ (t) = X(α(t)) r❡❛❧ ❝♦♥t❡♥❞♦ ♦ ③❡r♦✱ t❛❧ q✉❡ α(0) = p ❡ α ✱ ∀t ∈ I✳ dx

  = X(x) ❉✐③❡♠♦s q✉❡ α é ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❝♦♠ ❝♦♥❞✐çã♦ dt

  ✐♥✐❝✐❛❧ x(0) = p✳ ❆ ✐♠❛❣❡♠ ❞❡ ✉♠❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ é ❝❤❛♠❛❞❛ ór❜✐t❛ ♦✉ tr❛❥❡tór✐❛✳ r (M )

  ❉❡✜♥✐çã♦ ✶✳✷✳✹✳ ❯♠ ✢✉①♦ ❧♦❝❛❧ ❞❡ X ∈ X ❡♠ p ∈ M é ✉♠❛ ❛♣❧✐❝❛çã♦✿ ϕ : (−ǫ, ǫ) × V p → U

  V p p q : (−ǫ, ǫ) → U é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ p t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ q ∈ V ❛ ❛♣❧✐❝❛çã♦ ϕ q (t) = ϕ(t, q) q (0) = ϕ(0, q) = q

  ❞❡✜♥✐❞❛ ♣♦r ϕ é ✉♠❛ ❝✉r✈❛ ✐♥t❡❣r❛❧✱ ✐st♦ é✱ ϕ ❡✿ d ∂ϕ ϕ q (t) = ϕ q (t) = (t, q) = X(ϕ(t, q)) dt ∂t

  ∀(t, q) ∈ (−ǫ, ǫ) × V p ✳ r

  (M ) ❉❡✜♥✐çã♦ ✶✳✷✳✺✳ ❯♠ ✢✉①♦ ❣❧♦❜❛❧ ❞❡ X ∈ X é ✉♠❛ ❛♣❧✐❝❛çã♦ ϕ : R×M → M t❛❧ q✉❡ ϕ(0, p) = p ❡✱

  ∂ϕ (t, p) = X(ϕ(t, p)).

  ∂t

  ✶✺ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ p : M → M p (t) = ϕ(t, p) ▲♦❣♦✱ ♣❛r❛ ❝❛❞❛ p ∈ M ❛ ❛♣❧✐❝❛çã♦ ϕ ❞❡✜♥✐❞❛ ♣♦r ϕ p (0) = ϕ(0, p) = p

  é ✉♠❛ ❝✉r✈❛ ✐♥t❡❣r❛❧✱ ♦✉ s❡❥❛✱ ϕ ❡✱ d ∂ϕ ϕ p (t) = ϕ p (t) = (t, p) = X(ϕ(t, p)). dt ∂t P❛r❛ q✉❛✐sq✉❡r (t, p) ∈ R × M. r

  (M ) Pr♦♣♦s✐çã♦ ✶✳✷✳✻✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ❡ X ∈ X ✳ ❊①✐st❡ ❡♠ r M

  ✉♠ ✢✉①♦ ❣❧♦❜❛❧ ❞❡ ❝❧❛ss❡ C ♣❛r❛ X✳ ■st♦ é✱ ✉♠❛ ❛♣❧✐❝❛çã♦ ϕ : R × M → M t❛❧ q✉❡ ϕ(0, p) = p ❡ (∂/∂t)ϕ(t, p) = X(ϕ(t, p))✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r r

  (M ) ❈♦r♦❧ár✐♦ ✶✳✷✳✼✳ ❙❡❥❛♠ X ∈ X ❡ ϕ : R×M → M ♦ ✢✉①♦ ❞❡ X✳ P❛r❛ ❝❛❞❛ t ∈ R t : M → M t (p) = ϕ(t, p) r ✱ ❛ ❛♣❧✐❝❛çã♦ X ✱ X ✱ é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❝❧❛ss❡

  ◦ X C = identidade t +s = X t s

  ✳ ❆❧é♠ ❞✐ss♦✱ X ✱ X ✱ ∀t, s ∈ R✳ ❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ♠♦str❛r q✉❡ ϕ(t + s, p) = ϕ(t, ϕ(s, p))✱ ♣❛r❛ s, t ∈ R ❡ p ∈ M

  ✳ ❙❡❥❛♠ α(t) = ϕ(t + s, p) ❡ β(t) = ϕ(t, ϕ(s, p))✳ ❚❡♠♦s q✉❡ α ❡ β sã♦ ❝✉r✈❛s

  ✐♥t❡❣r❛✐s ❞♦ ❝❛♠♣♦ X ❡ α(0) = β(0) = ϕ(s, p)✱ ❡♥tã♦ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞❛s ❝✉r✈❛s ✐♥t❡❣r❛✐s s❡❣✉❡ q✉❡ α(t) = β(t)✱ ✐st♦ é✱ ϕ(t + s, p) = ϕ(t, ϕ(s, p))✳

  (p) = ϕ(0, p) = p P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ✢✉①♦s s❡❣✉❡ q✉❡ X ✱ ∀p ∈ M✱ ❛ss✐♠ X t : M → M

  é ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ M✳ ❊✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡ X é ✉♠❛ r ❛♣❧✐❝❛çã♦ ❞❡ ❝❧❛ss❡ C ♣❛r❛ t♦❞♦ t ∈ R ✜①❛❞♦✳ ❈♦♠♦✱ X t ◦ X (p) = ϕ(t, ϕ(−t, p)) = ϕ(t − t, p) = ϕ(0, p) = X (p) = p. −t

  ❊✱ ◦ X X −t t (p) = ϕ(−t, ϕ(t, p)) = ϕ(−t + t, p) = ϕ(0, p) = X (p) = p. r t −t = X −t t = identidade t ◦ X ◦ X

  ❊♥tã♦✱ X ✱ ❛ss✐♠ X é ❞❡ ❝❧❛ss❡ C ❡ t❡♠ r −t t : M → M ✐♥✈❡rs❛ ❞❡ ❝❧❛ss❡ C ❞❛❞❛ ♣♦r X ✳ P♦rt❛♥t♦✱ ❛ ❛♣❧✐❝❛çã♦ X é ✉♠ r ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❝❧❛ss❡ C ✱ ♣❛r❛ ❝❛❞❛ t ∈ R✳ r

  (M ) t ❙❡❥❛♠ X ∈ X ❡ X ♦ ✢✉①♦ ❞♦ ❝❛♠♣♦ X✳ t

  ❉❡✜♥✐çã♦ ✶✳✷✳✽✳ ❆ ór❜✐t❛ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M ♣❛r❛ ✉♠ ✢✉①♦ X é ♦ ❝♦♥❥✉♥t♦ O X (p) = {X t (p); t ∈ R}

  ✳ t + ❉❡✜♥✐çã♦ ✶✳✷✳✾✳ ❆ ór❜✐t❛ ♣♦s✐t✐✈❛ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M ♣❛r❛ ✉♠ ✢✉①♦ X é

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

  ✢✉①♦ X é ♦ ❝♦♥❥✉♥t♦ O ✳ t ❉❡✜♥✐çã♦ ✶✳✷✳✶✵✳ ❯♠ ♣♦♥t♦ p ∈ M é ❞✐t♦ ♣♦♥t♦ ✜①♦ ♣❛r❛ ♦ ❞✐❢❡♦♠♦r✜s♠♦ X t (p) = p s❡ X ✱ ♣❛r❛ ❝❛❞❛ t ∈ R✳

  ✶✻ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ ❉❡✜♥✐çã♦ ✶✳✷✳✶✶✳ ❙❡ X(p) = 0 ❡♥tã♦ ❞✐③❡♠♦s q✉❡ p ∈ M é ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ ❞♦ ❝❛♠♣♦ X✳ t

  ❖❜s❡r✈❛çã♦ ✶✳✷✳✶✷✳ ❙❡ p ∈ M é ✉♠ ♣♦♥t♦ ✜①♦ ♣❛r❛ ❝❛❞❛ X ✱ ❝♦♠ t ∈ R ❡♥tã♦ p é ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ ♣❛r❛ ♦ ❝❛♠♣♦ X✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ p ∈ M t ♣❛r❛ ♦ ❝❛♠♣♦ X é ✉♠ ♣♦♥t♦ ✜①♦ ♣❛r❛ X ✱ ♣❛r❛ ❝❛❞❛ t ∈ R ✜①❛❞♦✳ t (p) = p

  ❉❡ ❢❛t♦✱ ❝♦♠♦ X ✱ ∀t ∈ R ❡✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ t❡♠♦s q✉❡ X (p) = ϕ(t, p) t

  ✱ ❧♦❣♦ (∂/∂t)ϕ(t, p) = 0✱ ❛ss✐♠ (∂/∂t)ϕ(t, p) = X(ϕ(t, p)) = 0 ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ X(ϕ(t, p)) = X(p) = 0✳

  P♦rt❛♥t♦ X(p) = 0 ❡ p é ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ ♣❛r❛ ♦ ❝❛♠♣♦ X✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ ❊❉❖ ❝♦♠ ✈❛❧♦r ✐♥✐❝✐❛❧✿ dα

  = X(α(t)) dt α(0) = p. t

  ▲♦❣♦✱ α(t) = p s❛t✐s❢❛③ ❛ ❊❉❖ ❛❝✐♠❛✳ ❆ss✐♠✱ p é ♣♦♥t♦ ✜①♦ ♣❛r❛ ❝❛❞❛ X ✳ t ❉❡✜♥✐çã♦ ✶✳✷✳✶✸✳ ❯♠ ♣♦♥t♦ p ∈ M é ❝❤❛♠❛❞♦ ♣❡r✐ó❞✐❝♦ ♣❛r❛ ✉♠ ✢✉①♦ X ✱ s❡ T (p) = p t (p) 6= p ❡①✐st❡ T > 0 t❛❧ q✉❡ X ❡ X ✱ ∀t < T ✳

  ❆ ór❜✐t❛ ❞❡ ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ é ❝❤❛♠❛❞❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛✳ ❯♠❛ ór❜✐t❛ ❢❡❝❤❛❞❛ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M é q✉❛♥❞♦ O(p) é ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦✱ ❛s ór❜✐t❛s q✉❡ ♥ã♦ sã♦ s✐♥❣✉❧❛r✐❞❛❞❡s sã♦ ❞✐t❛s ór❜✐t❛s r❡❣✉❧❛r❡s✳ ❖s ♣♦♥t♦s r❡❣✉❧❛r❡s sã♦ ♦s ♣♦♥t♦s ♣❛r❛ ♦s q✉❛✐s ♦ ❝❛♠♣♦ ♥ã♦ s❡ ❛♥✉❧❛✳ 2 3

  ⊂ R ❊①❡♠♣❧♦ ✶✳✷✳✶✹✳ ❙❡❥❛ S ❛ ❡s❢❡r❛ ✉♥✐tár✐❛✳ ❈♦♥s✐❞❡r❡ ♦ ✢✉①♦ ❞❡✜♥✐❞♦ ♣❡❧❛ r♦t❛çã♦ ❞❡ ❝❛❞❛ ♣♦♥t♦ ❞❛ ❡s❢❡r❛ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ NS✱ ✈❡r ❋✐❣✉r❛ ❚❡♠♦s q✉❡ ♦ ♣♦❧♦ ♥♦rt❡ ❡ ♦ ♣♦❧♦ s✉❧ t❡♠ ❝♦♠♦ ór❜✐t❛ ♦ ♣ró♣r✐♦ ♣♦♥t♦✳ ❊✱ ❛s ❞❡♠❛✐s ór❜✐t❛s 2 sã♦ ♦s ♣❛r❛❧❡❧♦s ❞❛ ❡s❢❡r❛ S ✳ ❊♥tã♦✱ ❛s s✐♥❣✉❧❛r✐❞❛❞❡s sã♦ ♦s ♣♦❧♦s ♥♦rt❡ ❡ s✉❧ ❡

  ❛s ♦✉tr❛s ór❜✐t❛s sã♦ ór❜✐t❛s r❡❣✉❧❛r❡s ❡ ♣❡r✐ó❞✐❝❛s✳ N

  

S

2

  ❋✐❣✉r❛ ✶✳✻✳ ❋❧✉①♦ ❡♠ S ❉❡✜♥✐çã♦ ✶✳✷✳✶✺✳ ❖ ❝♦♥❥✉♥t♦ ω✲❧✐♠✐t❡ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M é ❞❛❞♦ ♣♦r✿

  → ∞ quando n → ∞ tal que lim ω(p) = {x ∈ M ; ∃ t n n →∞ X t (p) = x}. n

  ✶✼ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ ❆ss✐♠✱ ω(p) é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❡ ❛❝✉♠✉❧❛çã♦ ❞❛ ór❜✐t❛ ♣♦s✐t✐✈❛ ❞❡ p✳ ❉❡✜♥✐♠♦s t❛♠❜é♠ ♦ ❝♦♥❥✉♥t♦ α✲❧✐♠✐t❡ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M ❞❛❞♦ ♣♦r

  → −∞ quando n → ∞ tal que lim α(p) = {x ∈ M ; ∃ t n n →∞ X t (p) = x}. n

  ❊♥tã♦✱ ♦ ❝♦♥❥✉♥t♦ α(p) é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❡ ❛❝✉♠✉❧❛çã♦ ❞❛ ór❜✐t❛ ♥❡❣❛t✐✈❛ ❞❡ p✳

  ❖❜s❡r✈❛♠♦s q✉❡ ♦ α✲❧✐♠✐t❡ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M é ♦ ω✲❧✐♠✐t❡ ❞❡ p ♣❛r❛ ♦ ❝❛♠♣♦ −X

  ✳ ■♥t✉✐t✐✈❛♠❡♥t❡✱ α(p) é ♦♥❞❡ ❛ ór❜✐t❛ ❞❡ p ✧♥❛s❝❡✧❡ ω(p) é ♦♥❞❡ ❡❧❛ ✧♠♦rr❡✧✳ r (M )

  ❖❜s❡r✈❛çã♦ ✶✳✷✳✶✻✳ ❙❡ p é ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ ❞♦ ❝❛♠♣♦ X ∈ X ❡♥tã♦ ω(p) = α(p) = {p}

  ✳ ❉❡ ❢❛t♦✱ X(p) = 0 ⇔ X t (p) = p, ∀t ∈ R ⇒ ∀t n → ∞ ( n → −∞) ⇒ X t (p) → p. n

  ♦✉ t P♦rt❛♥t♦✱ ω(p) = α(p) = {p}✳ ◆♦ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r ✈❛♠♦s ❡①✐❜✐r ♦s ❝♦♥❥✉♥t♦s ω✲❧✐♠✐t❡ ❡ α✲❧✐♠✐t❡ ❞❡ ✉♠

  ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❝♦♠ s✐♥❣✉❧❛r✐❞❛❞❡s✳ 2 3 ⊂ R

  ❊①❡♠♣❧♦ ✶✳✷✳✶✼✳ ❈♦♥s✐❞❡r❡ ❛ ❡s❢❡r❛ ✉♥✐tár✐❛ S ❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠ ❡ 3 N = (0, 0, 1) s❡❥❛♠ (x, y, z) ❛s ❝♦♦r❞❡♥❛❞❛s ❝❛♥ô♥✐❝❛s ❡♠ R ✳ ❈❤❛♠❛♠♦s p ❞❡ S = (0, 0, −1) 2 ♣♦❧♦ ♥♦rt❡ ❡ p ❞❡ ♣♦❧♦ s✉❧ ❞❡ S ✳ ❉❡✜♥✐♠♦s ♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X 2

  ❞❡ ❝❧❛ss❡ C ❡♠ S ♣♦r✿ 2 2 X(x, y, z) = (−xz, −yz, x + y ). 2 ❆s s✐♥❣✉❧❛r✐❞❛❞❡s ❞♦ ❝❛♠♣♦ X sã♦ ♦s ♣♦♥t♦s (x, y, z) ∈ S t❛✐s q✉❡✿ 2 2 X(x, y, z) = (0, 0, 0) ⇒ (−xz, −yz, x + y ) = (0, 0, 0).

  ❊♥tã♦✱ x = y = 0.

  ❈♦♠♦ 2 2 2 x + y + z = 1 N S t❡♠♦s q✉❡ ❛s s✐♥❣✉❧❛r✐❞❛❞❡s sã♦ p ❡ p ✳ ❯♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❛ ❡s❢❡r❛ ♣♦❞❡ s❡r ❡①♣r❡ss❛❞❛ ❝♦♠♦✿ Y (θ, ϕ) = (cosθsenϕ, cosθϕ, senθ). 2 P❛r❛ ❝❛❞❛ ϕ ✜①❛❞♦ t❡♠♦s ❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❡ ✉♠ ♠❡r✐❞✐❛♥♦ ❞❡ S ✳ ❚❡♠♦s 2 q✉❡ ♦ ❝❛♠♣♦ X é t❛♥❣❡♥t❡ ❛♦s ♠❡r✐❞✐❛♥♦s ❞❡ S ✳

  ✶✽ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡ ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❡ ✉♠ ♠❡r✐❞✐❛♥♦ ❞❛❞❛ ♣♦r✿ α(θ) = (cosθsenϕ, cosθϕ, senθ).

  ❆ss✐♠✱ α (θ) = (−senθsenϕ, −senθϕ, cosθ).

  ❊✱ X(α(θ)) = cosθ(−senθsenϕ, −senθϕ, cosθ) = cosθα (θ).

  P♦rt❛♥t♦✱ ♦ ❝❛♠♣♦ é t❛♥❣❡♥t❡ ❛♦s ♠❡r✐❞✐❛♥♦s ❞❡ S 2 ❛♣♦♥t❛♥❞♦ ♣❛r❛ ❝✐♠❛ s❡❣✉❡ q✉❡ ω(p) = p N ❡ α(p) = p S ✱ ♣❛r❛ t♦❞♦ p ∈ S 2 − {p N , p S }

  ✳ ❊✱ ω(p N ) = α(p N ) = {p N }

  ✱ ω(p S ) = α(p S ) = {p S }p N

  

p

S

  ❋✐❣✉r❛ ✶✳✼ ❖❜s❡r✈❛çã♦ ✶✳✷✳✶✽✳ ❙❡❥❛♠ p, q ∈ M s❡ q ♣❡rt❡♥❝❡ ❛ ór❜✐t❛ ❞❡ p ♣❡❧♦ ❝❛♠♣♦ X ❡♥tã♦ ω(p) = ω(q)✳

  ❉❡ ❢❛t♦✱ ❝♦♠♦ q ♣❡rt❡♥❝❡ ❛ ór❜✐t❛ ♣♦s✐t✐✈❛ ❞❡ p✱ s❡❣✉❡ q✉❡ ❡①✐st❡ t ∈ R t❛❧ q✉❡ X t (p) = q ✳ ❊✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ t❡♠♦s q✉❡ X −t (q) = p ✳ ❱❛♠♦s ♠♦str❛r q✉❡ ω(p) ⊂ ω(q)✳

  ❙❡❥❛ x ∈ ω(p)✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ t n → ∞ t❛❧ q✉❡ X t n (p) → x

  ✳ P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦ ✢✉①♦ X t ❡ ♣❡❧♦ ❈♦r♦❧ár✐♦ t❡♠✲s❡ q✉❡✿

  X t (X t n (p)) → X t (x) ⇒ X t +t n (p) → X t (x) ⇒ ⇒ X t n

  (X t (p)) → X t (x) ⇒ X t n (q) → X t (x) ⇒ ⇒ X −t (X t n

  (q)) → X −t (X t (x)) ⇒ ⇒ X t n −t (q) → x.

  ❋❛③❡♥❞♦ s n = t n − t ✱ s❡❣✉❡ q✉❡ s n → ∞ ♣♦✐s t n → ∞ ✳ ❆ss✐♠✱ X s n (q) → x.

  P♦rt❛♥t♦✱ x ∈ ω(q) ❡✱ ω(p) ⊂ ω(q)✳

  ✶✾ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ ❆❣♦r❛ ✈❛♠♦s ♠♦str❛r q✉❡ ω(q) ⊂ ω(p)✳ ❙❡❥❛ x ∈ ω(q)✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ t n → ∞ t❛❧ q✉❡ X t n

  ✳ P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦ ✢✉①♦ ❡ ♣❡❧♦ ❈♦r♦❧ár✐♦ t❡♠♦s q✉❡✿ X t +t n (p) → X t (x) ∀t ∈ R.

  Λ = \ t ≥0 X t (U ).

  X t é ❝❤❛♠❛❞♦ s✉♠✐❞♦✉r♦ s❡✿

  Λ ♥ã♦ é ✉♠❛ ú♥✐❝❛ ór❜✐t❛✳ ❊ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ✐♥✈❛r✐❛♥t❡ Λ ♣❛r❛ ♦ ✢✉①♦

  (M ) ✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ Λ ⊂ M é s✐♥❣✉❧❛r s❡ ❡st❡ ♣♦ss✉✐ ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡❀ ♥ã♦ tr✐✈✐❛❧ s❡

  ❉❡✜♥✐çã♦ ✶✳✷✳✷✵✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ Λ ⊂ M é ❞✐t♦ ✐♥✈❛r✐❛♥t❡ ♣❛r❛ ✉♠ ✢✉①♦ X t s❡ X t (Λ) = Λ ✱ ∀t ∈ R✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ Λ ⊂ M é ❞✐t♦ ✐♥✈❛r✐❛♥t❡ ♣♦s✐t✐✈❛♠❡♥t❡ ✭ ♥❡❣❛t✐✈❛♠❡♥t❡✮ ♣❛r❛ ✉♠ ✢✉①♦ X t s❡ X t (Λ) ⊂ Λ ✱ ∀t ≥ 0 ✭∀t ≤ 0✮✳ ❉❡✜♥✐çã♦ ✶✳✷✳✷✶✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ❡ X ∈ X r

  P♦rt❛♥t♦✱ X t (x) ∈ ω(p) ✱ ♣❛r❛ t♦❞♦ t ∈ R✱ ♦✉ s❡❥❛✱ O X (x) ⊂ ω(p) ✳ ❘❡♣❡t✐♥❞♦ ♦ ♣r♦❝❡ss♦ ♣❛r❛ y ∈ α(p) t❡♠♦s q✉❡ O X (y) ⊂ α(p)

  ❋❛③❡♥❞♦ k n = t + t n t❡♠♦s q✉❡ k n → ∞ ✱ ♣♦✐s t n → ∞ ✱ ❡♥tã♦✿ X k n (p) → X t (x).

  → ∞ t❛❧ q✉❡ X t n (p) → x

  (q) → x ✳ P❡❧❛

  ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ y ∈ α(p)✱ ❡♥tã♦ O X (y) ⊂ α(p) ✳ ❉❡ ❢❛t♦✱ s❡ x ∈ ω(p)✱ t❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ t n

  ❖❜s❡r✈❛çã♦ ✶✳✷✳✶✾✳ ❙❡❥❛ p ∈ M✱ X t ✉♠ ✢✉①♦✳ ❙❡ x ∈ ω(p) t❡♠♦s q✉❡ O X (x) ⊂ ω(p)

  P♦rt❛♥t♦✱ ω(q) ⊂ ω(p) ❡✱ ❛ss✐♠ ω(p) = ω(q)✱ ❝♦♠♦ q✉❡rí❛♠♦s✳ ❯♠ r❡s✉❧t❛❞♦ ❛♥á❧♦❣♦ ❛ ❡st❡ s❡ ✈❡r✐✜❝❛ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ α✲❧✐♠✐t❡✳

  ❋❛③❡♥❞♦ k n = t n + t ✱ s❡❣✉❡ q✉❡ k n → ∞ ♣♦✐s t n → ∞ ✳ ❆ss✐♠✱ X k n (p) → x.

  (X −t (q)) → X −t (x) ⇒ X t n (p) → X −t (x) ⇒ ⇒ X t (X t n (p)) → X t (X −t (x)) ⇒ X t n +t (p) → x.

  X −t (X t n (q)) → X −t (x) ⇒ X t n −t (q) → X −t (x) ⇒ ⇒ X t n

  ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦ ✢✉①♦ X t ❡ ♣❡❧♦ ❈♦r♦❧ár✐♦ t❡♠✲s❡ q✉❡✿

  P❛r❛ ❛❧❣✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ Λ q✉❡ s❛t✐s❢❛③ X t (U ) ⊂ U ✱ ∀t ≥ 0✳

  ✷✵ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ ❉❡✜♥✐çã♦ ✶✳✷✳✷✷✳ ❯♠ ♣♦♥t♦ p ∈ M é ❞❡♥♦♠✐♥❛❞♦ ♣♦♥t♦ ♥ã♦✲❡rr❛♥t❡ ♣❛r❛ r

  (M ) ✉♠ ❝❛♠♣♦ X ∈ X s❡✱ ♣❛r❛ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ V ❞❡ p ❡ q✉❛❧q✉❡r ♥ú♠❡r♦ t (V ) ∩ V 6= ∅ r❡❛❧ T > 0✱ ❡①✐st❡ | t |> T t❛❧ q✉❡ X ✳ ❙❡ p ♥ã♦ é ✉♠ ♣♦♥t♦ ♥ã♦✲ ❡rr❛♥t❡✱ ❡♥tã♦ p é ❝❤❛♠❛❞♦ ♣♦♥t♦ ❡rr❛♥t❡✳ ❉❡♥♦t❛r❡♠♦s ♣♦r Ω(X) ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ♥ã♦ ❡rr❛♥t❡s ❞❡ X✳

  ❉❡♥♦t❛r❡♠♦s ♣♦r P er(X) ❡ Sing(X) ♦s ❝♦♥❥✉♥t♦s ❞♦s ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s ❡ s✐♥❣✉❧❛r❡s ❞♦ ❝❛♠♣♦ X✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ Pr♦♣♦s✐çã♦ ✶✳✷✳✷✸✳ ❖ ❝♦♥❥✉♥t♦ Ω(X) é ✐♥✈❛r✐❛♥t❡✱ ❢❡❝❤❛❞♦ ❡ P er(X) ∪ Sing(X) ⊂ Ω(X)

  ✳ t (Ω(X)) = Ω(X) ❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ♠♦str❛r q✉❡ Ω(X) é ✐♥✈❛r✐❛♥t❡✱ ✐st♦ é✱ X ✳ t (Ω(X t )) ⊂ Ω(X) Pr♦✈❛r❡♠♦s q✉❡ X ✱ ∀t ∈ R✳ t (Ω(X t )) −t (y) ∈ Ω(X) t )

  ❙❡❥❛ y ∈ X ✱ ❝♦♠ t ∈ R✱ ❛ss✐♠ x = X ✳ ❈♦♠♦ x ∈ Ω(X ❡♥tã♦ ♣❛r❛ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ V ❞❡ x ❡ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ T > 0✱ ❡①✐st❡ | t |> T t❛❧ q✉❡✿ t (x) X t (V ) ∩ V 6= ∅.

  (y) ∈ ❊✱ y = X ✳ ❙❡❥❛ U ✉♠❛ ✈✐③✐♥❤❛♥ç❛ q✉❛❧q✉❡r ❞❡ y✱ t❡♠♦s q✉❡ x = X −t

  X (U ) (U ) −t −t ✳ ▲♦❣♦✱ X é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x✱ ❡♥tã♦✿ X −t (U ) ∩ X t (X −t (U )) 6= ∅.

  ❆ss✐♠✱ X (X (U )) ∩ X (X (X (U ))) 6= ∅. t −t t t −t P❡❧♦ ❈♦r♦❧ár✐♦ t❡♠♦s q✉❡✱ U ∩ X t (U ) 6= ∅. t (Ω(X)) ⊂ Ω(X) P♦rt❛♥t♦✱ y ∈ Ω(X) ❡✱ ❛ss✐♠ X ✱ ∀t ∈ R✳ t (Ω(X)). ❆❣♦r❛ ♣r♦✈❛r❡♠♦s q✉❡ Ω(X) ⊂ X t (Ω(X)) ⊂ Ω(X) (x) ∈ Ω(X) −t ❙❡❥❛ x ∈ Ω(X)✱ ❝♦♠♦ X ✱ ∀t ∈ R s❡❣✉❡ q✉❡ X ✱ t (X (x)) ∈ X t (Ω(X)) −t

  ❛ss✐♠ x = X ♣❛r❛ t♦❞♦ t ∈ R✳ t (Ω(X)) ❊♥tã♦✱ Ω(X) ⊂ X ✱ ∀t ∈ R✳ t P♦rt❛♥t♦✱ Ω(X) é ✐♥✈❛r✐❛♥t❡ ♣❡❧♦ ✢✉①♦ X ✳ ❋✐♥❛❧♠❡♥t❡ ✈❛♠♦s ♠♦str❛r q✉❡ Ω(X) é ❢❡❝❤❛❞♦✳ ❙❡❥❛ p ∈ Ω(X)✱ ❡♥tã♦ ♣❛r❛ t♦❞❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ p t❡♠♦s q✉❡✱ U ∩ Ω(X) 6= ∅.

  ✷✶ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ ❆ss✐♠✱ ❡①✐st❡ x ∈ U t❛❧ q✉❡ x ∈ Ω(X)✳ ❈♦♠♦ x ∈ Ω(X) s❡❣✉❡ q✉❡ ♣❛r❛ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ V ❞❡ x ❡ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ T > 0✱ ❡①✐st❡ | t |> T t❛❧ q✉❡

  X t (V ) ∩ V 6= ∅ ✳ ❚❡♠♦s q✉❡✱ U é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x✱ ❡♥tã♦✿ U ∩ X t (U ) 6= ∅.

  ▲♦❣♦✱ p ∈ Ω(X) ❡✱ ♣♦rt❛♥t♦ Ω(X) é ❢❡❝❤❛❞♦✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ Ω(X) ❝♦♥té♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s✳

  > 0 t (p) = p ❉❡ ❢❛t♦✱ s❡ p é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❡♥tã♦ ❡①✐st❡ t t❛❧ q✉❡ X ✳

  ❆ss✐♠✱ ♣❛r❛ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ V ❞❡ p ❡ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ T > 0✱ ❡①✐st❡ n ∈ N > T nt (p) = p t❛❧ q✉❡ nt ❡ X ❡♥tã♦✱ X nt (V ) ∩ V 6= ∅.

  ▲♦❣♦✱ p ∈ Ω(X)✳ P♦r ú❧t✐♠♦✱ s❡❥❛ p ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ ♣❛r❛ ♦ ❝❛♠♣♦ X✱ ❧♦❣♦ s✉❛ ór❜✐t❛ s❡ r❡❞✉③ s♦♠❡♥t❡ ❛♦ ♣♦♥t♦ p✱ ❛ss✐♠ ❞❛❞♦ V ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ p ❡ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧

  T > 0 t (V ) ✱ t❡♠♦s q✉❡ p ∈ X ✱ ∀t > 0✱ ✐st♦ é✱ X t (V ) ∩ V 6= ∅.

  ❊♥tã♦✱ p ∈ Ω(X)✳ P♦rt❛♥t♦✱ Ω(X) é ✐♥✈❛r✐❛♥t❡✱ ❢❡❝❤❛❞♦ ❡ P er(X)∪Sing(X) ⊂ Ω(X)

  ✳ ❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r s❡rá ♠✉✐t♦ ✉t✐❧✐③❛❞❛ ♥♦ t❡①t♦ ❡ ♣❛r❛ ❡✈✐t❛r s✉❛

  ❝♦♥❣❧♦♠❡r❛çã♦ ❞❡ r❡❢❡rê♥❝✐❛s ♥❛ ❞✐ss❡rt❛çã♦✱ ❛ss✉♠✐r❡♠♦s q✉❡ é ❝♦♥❤❡❝✐❞❛✳ r (M )

  Pr♦♣♦s✐çã♦ ✶✳✷✳✷✹✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛✱ X ∈ X ❡ p ∈ M✳ ❊♥tã♦✿

  ✶✳ ω(p) 6= ∅❀ ✷✳ ω(p) é ❢❡❝❤❛❞♦❀ ✸✳ ω(p) é ✐♥✈❛r✐❛♥t❡ ♣❡❧♦ ✢✉①♦✱ ✐st♦ é✱ ω(p) é ❛ ✉♥✐ã♦ ❞❡ ór❜✐t❛s ❞❡ X❀ ✹✳ ω(p) é ❝♦♥❡①♦✳ n → ∞ n = X t (p) n

  ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ t ❡ p n ✳ ❈♦♠♦ M é ❝♦♠♣❛❝t❛ t❡♠♦s q✉❡ p ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ❝✉❥♦ ❧✐♠✐t❡ ♣❡rt❡♥❝❡ ❛ ω(p)✳ ▲♦❣♦✱ ω(p) 6= ∅✳

  ❱❛♠♦s ♣r♦✈❛r q✉❡ ω(p) é ❢❡❝❤❛❞♦✳ ❙✉♣♦♥❤❛✱ ♣♦r ❝♦♥tr❛❞✐çã♦✱ q✉❡ ❡①✐st❡ q ∈ ω (p) t❛❧ q✉❡ q /∈ ω(p)✳ ❈♦♠♦ q /∈ ω(p) ❡♥tã♦ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ V t (p); t ≥ T }

  ❞♦ ♣♦♥t♦ p ❞✐s❥✉♥t❛ ❞❡ {X ✱ ♣❛r❛ ❛❧❣✉♠ T > 0✳ ■st♦ ✐♠♣❧✐❝❛ q✉❡ V ∩ ω(p) = ∅

  ✱ ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s q ∈ ω(p)✳ P♦rt❛♥t♦✱ ω(p) é ❢❡❝❤❛❞♦✳

  ✷✷ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ ❙❡❥❛ q ∈ ω(p) ❡♥tã♦ ♣❡❧❛ ♦❜s❡r✈❛çã♦ s❡❣✉❡ q✉❡ O X (q) ∈ ω(p) ✳ ❆ss✐♠✱

  ✳ ❊✱ t❛♠❜é♠ X t n

  ✱ q✉❛♥❞♦ t n → ∞ ❝♦♠ n → ∞✱ ❡♥tã♦ ❡①✐st❡ t n < −T ❡ X t n

  ❙❛❜❡♥❞♦ q✉❡ X t n (x) → p

  ❉❛❞♦ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ p ❡ T ∈ R✱ ❝♦♠ T > 0✳ Pr❡❝✐s❛♠♦s ♠♦str❛r q✉❡ ❡①✐st❡ |t| > T t❛❧ q✉❡✿ X t (U ) ∩ U 6= ∅.

  X t n (x) → p ✳

  X t (V ) ∩ V 6= ∅. P♦rt❛♥t♦✱ y ∈ Ω(X)✳ ❊✱ ω(x) ⊂ Ω(X)✱ ∀x ∈ M✳ Pr♦✈❛r❡♠♦s q✉❡ α(x) ⊂ Ω(X)✳ ❙❡❥❛ p ∈ α(x) ❡♥tã♦ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ t n → −∞ ✱ n → ∞✱ t❛❧ q✉❡

  ❊♥tã♦✱ s❡ ❝♦♥s✐❞❡r❛♠♦s t = t n 1 > T t❡♠♦s q✉❡

  (x)) ∈ V ✳

  (X t n0 (x)) ∈ X t n1 (V ) ✱ ♣♦✐s X t n0

  (X t n0 (x)) ∈ V ❡ X t n1

  ✳ ❆ss✐♠✱ X t n1

  > t n > T t❛❧ q✉❡ X t n1 (X t n0 (x)) ∈ V

  (X t n0 (x)) → y ❛ss✐♠ ❡①✐st❡ ✉♠ t n 1

  ✱ q✉❛♥❞♦ t n → ∞ ❝♦♠ n → ∞✱ s❡❣✉❡ q✉❡ ❡①✐st❡ t n > T t❛❧ q✉❡ X t n0 (x) ∈ V

  ω(p) é ✐♥✈❛r✐❛♥t❡ ♣❡❧♦ ✢✉①♦✳

  ❈♦♠♦ X t n (x) → y

  ❉❛❞♦ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ V ❞❡ y ❡ T ∈ R✱ ❝♦♠ T > 0✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ ❡①✐st❡ |t| > T t❛❧ q✉❡✿ X t (V ) ∩ V 6= ∅.

  X t n (x) → y ✳

  → ∞ ✱ n → ∞✱ t❛❧ q✉❡

  ❡♥tã♦ ω(x) ⊂ Ω(X) ❡ α(x) ⊂ Ω(X)✳ ❱❛♠♦s ♣r♦✈❛r q✉❡ ω(x) ⊂ Ω(X)✳ ❙❡❥❛ y ∈ ω(X) ❡♥tã♦ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ t n

  (M ) ❡ x ∈ M

  ❯♠ r❡s✉❧t❛❞♦ ❛♥á❧♦❣♦ s❡ ✈❡r✐✜❝❛ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ α(p)✳ ❖❜s❡r✈❛çã♦ ✶✳✷✳✷✺✳ ❙❡❥❛♠ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛✱ X ∈ X r

  ❡✱ ♣♦rt❛♥t♦ ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡ X t n q✉❡ ❝♦♥✈❡r❣❡ ❛ ✉♠ ♣♦♥t♦ q ∈ K✳ ❊♥tã♦✱ q ∈ ω(p) ⊂ A ∪ B✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ P♦rt❛♥t♦ ω(p) é ❝♦♥❡①♦✳

  ∈ K ✱ s❡♥❞♦ K ❢❡❝❤❛❞♦ ❡ M ❝♦♠♣❛❝t♦ ❡♥tã♦ K é ❝♦♠♣❛❝t♦

  → ∞ ❝♦♠ X t n

  ❡ B t❛✐s q✉❡ ω(p) ⊂ A ∪ B ✱ A ∩ ω(p) 6= ∅✱ B ∩ ω(p) 6= ∅ ❡ A ∩ B = ∅✳ ❈♦♠♦ ❛ ór❜✐t❛ ❞❡ p s❡ ❛❝✉♠✉❧❛ ❡♠ ♣♦♥t♦s ❞❡ A ❡ B✱ ❞❛❞♦ T > 0 t❡♠♦s q✉❡ ❡①✐st❡ t > T t❛❧ q✉❡ X t (p) ∈ K = M − (A ∪ B) ✳ ▲♦❣♦✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ t n

  ❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡ ω(p) ♥ã♦ s❡❥❛ ❝♦♥❡①♦✳ P♦❞❡♠♦s ❡♥tã♦ ❡s❝♦❧❤❡r ❛❜❡rt♦s A

  (x) ∈ U ✳

  ✷✸ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ t (X t (x)) → p n < t n < −T t (X t (x)) ∈ n 1 ❊✱ X n0 ❛ss✐♠ ❡①✐st❡ ✉♠ t t❛❧ q✉❡ X n1 n0

  U ✳ t (X t (x)) ∈ U t (X t (x)) ∈ X t (U ) t (x)) ∈ U

  P♦rt❛♥t♦✱ X ❡ X ✱ ♣♦✐s X ✳ n1 n0 n1 n0 n1 n0 n < −T 1 ❈♦♥s✐❞❡r❡ t = t ❡♥tã♦ X t (U ) ∩ U 6= ∅.

  ▲♦❣♦✱ p ∈ Ω(X) ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ α(x) ⊂ Ω(X)✱ ♣❛r❛ q✉❛❧q✉❡r x ∈ M✳ ❉❡✈✐❞♦ ❛ ❡ss❛ ♦❜s❡r✈❛çã♦✱ ♦ ❝♦♥❥✉♥t♦ Ω(X) é ♥ã♦ ✈❛③✐♦ ❡♠ M✱ q✉❛♥❞♦ M ❢♦r

  ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛✱ ✉♠❛ ✈❡③ q✉❡ ♣❛r❛ ❝❛❞❛ x ∈ M ♦ ❝♦♥❥✉♥t♦ ω(x) é ♥ã♦ ✈❛③✐♦✳ ❖ ♠❡s♠♦ ♦❝♦rr❡ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ α(x)✳

  

✶✳✷✳✶ ❋✉♥çã♦ ❞❡ Pr✐♠❡✐r♦ ❘❡t♦r♥♦ ❞❡ P♦✐♥❝❛ré ❡

❍✐♣❡r❜♦❧✐❝✐❞❛❞❡ t

  (x) ◆❛ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦ ϕ ♣❛r❛ ✐♥❞✐❝❛r ❛ ❝✉r✈❛

  ✐♥t❡❣r❛❧ ❞❡ x ❛✈❛❧✐❛❞❛ ♥♦ t❡♠♣♦ t✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ t❡♠♦s q✉❡ t ϕ (x) = ϕ(t, x)

  ✳ n X Pr♦♣♦s✐çã♦ ✶✳✷✳✷✻✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠♣❛❝t❛ ❡ f ∈ r (M )

  ✳ t t t (x) = f (x) f (x) = f (ϕ (x)) x

  ✶✳ ❙❡❥❛ ϕ ✉♠❛ s♦❧✉çã♦ ❞❡ x ❡♥tã♦ Dϕ ♣❛r❛ q✉❛❧q✉❡r t✳ T ✷✳ ❙❡ γ é ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ ❞❡ ♣❡rí♦❞♦ T ❡ p ∈ γ ❡♥tã♦ ❛ ❞❡r✐✈❛❞❛ Dϕ p t❡♠ 1 ❝♦♠♦ ❛✉t♦✈❛❧♦r ❛ss♦❝✐❛❞♦ ❛♦ ❛✉t♦✈❡t♦r f(p)✳ ✸✳ ❙❡ p ❡ q sã♦ ❞♦✐s ♣♦♥t♦s ♥❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ γ ❞❡ ♣❡rí♦❞♦ T ❡♥tã♦ ❛s T T p q T p

  ❞❡r✐✈❛❞❛s Dϕ ❡ Dϕ sã♦ ❧✐♥❡❛r♠❡♥t❡ ❝♦♥❥✉❣❛❞❛s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ Dϕ T ❡ Dϕ q tê♠ ♦s ♠❡s♠♦s ❛✉t♦✈❛❧♦r❡s✳ t

  (x) = f (x) ❉❡♠♦♥str❛çã♦✳ ✶✳ ❈♦♠♦ ϕ é ✉♠❛ s♦❧✉çã♦ ❞❡ x s❡❣✉❡ q✉❡✿ t s t s t d d f (ϕ (x)) = ϕ (x)| s = ϕ ◦ ϕ (x)| s = Dϕ f (x). =t =0 x T T T ds ds

  (p) = p f (p) = f (ϕ (p)) = f (p) ✷✳ ❚❡♠♦s q✉❡ ϕ ❡♥tã♦ Dϕ p ✳ ❆ss✐♠✱ 1 é ❛✉t♦✈❛❧♦r T p f (p) ❞❡ Dϕ ❛ss♦❝✐❛❞♦ ❛♦ ❛✉t♦✈❡t♦r f(p)✳ r

  (p) ✸✳ ❙✉♣♦♥❤❛ q✉❡ q = ϕ ✳ ❊♥tã♦✱ T T r r r T q = ϕ (q) = ϕ ◦ ϕ (p) = ϕ (p) = ϕ ◦ ϕ (p) ⇒

  ✷✹ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ T r r T ⇒ ϕ ◦ ϕ (p) = ϕ ◦ ϕ (p).

  ▲♦❣♦✱ T r r T

  Dϕ Dϕ = Dϕ Dϕ q p p p T r T r −1 T T ⇒ Dϕ = Dϕ Dϕ (Dϕ ) . q p p p ❊♥tã♦✱ Dϕ p ❡ Dϕ q sã♦ ❧✐♥❡❛r♠❡♥t❡ ❝♦♥❥✉❣❛❞❛s ❡✱ ❛ss✐♠ tê♠ ♦s ♠❡s♠♦s

  ❛✉t♦✈❛❧♦r❡s✳ ❉❡✜♥✐çã♦ ✶✳✷✳✷✼✳ ❙❡❥❛ Σ ✉♠❛ ✈❛r✐❡❞❛❞❡ ♠❡r❣✉❧❤❛❞❛ ❡♠ M ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠ r

  (M ) ❡ X ∈ X ✱ r > 0✱ Σ é ✉♠❛ tr❛♥s✈❡rs❛❧ ♦✉ s❡çã♦ tr❛♥s✈❡rs❛❧ ❧♦❝❛❧ ❛♦ p Σ ❝❛♠♣♦ X ❡♠ p ∈ Σ ⊂ M s❡ ♦s ✈❡t♦r❡s ❞❡ ✉♠❛ ❜❛s❡ ❞❡ T ❡ ♦ ✈❡t♦r X(p) ❣❡r❛♠ T p M

  ✳ n r (M )

  ❉❡✜♥✐çã♦ ✶✳✷✳✷✽✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ❡ X ♦ ❡s♣❛ç♦ ❞♦s r n r ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❞❡ ❝❧❛ss❡ C ✱ r > 0✱ ❞❡ M ♠✉♥✐❞♦ ❞❡ ✉♠❛ t♦♣♦❧♦❣✐❛ C ✳ r

  (M ) ❉♦✐s ❝❛♠♣♦s X, Y ∈ X sã♦ t♦♣♦❧♦❣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s s❡ ❡①✐st❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ h : M → M q✉❡ ❧❡✈❛ ór❜✐t❛s ❞❡ X ❡♠ ór❜✐t❛s ❞❡ Y ♣r❡s❡r✈❛♥❞♦ ❛ ♦r✐❡♥t❛çã♦ ❞❛s tr❛❥❡tór✐❛s✱ ✐st♦ é✱ s❡ p ∈ M ❡ δ > 0✱ ❡①✐st❡ ǫ > 0 t❛❧ q✉❡ ♣❛r❛ 0 < t < δ t (p)) = Y (h(p)) t

  ✱ h(X ♣❛r❛ ❛❧❣✉♠ 0 < t < ǫ✳ ❉✐③❡♠♦s q✉❡ h é ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ t♦♣♦❧ó❣✐❝❛ ❡♥tr❡ X ❡ Y ✳

  ❉✐③❡♠♦s q✉❡ ♦s ❝❛♠♣♦s X ❡ Y sã♦ ❝♦♥❥✉❣❛❞♦s s❡ ❡①✐st❡ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ t (p)) = Y t (h(p)) t♦♣♦❧ó❣✐❝❛ h q✉❡ ♣r❡s❡r✈❛ ♦ ♣❛râ♠❡tr♦ t✱ ✐st♦ é✱ h(X ♣❛r❛ t♦❞♦ t ∈ R

  ❡ t♦❞♦ p ∈ M✳ r (M )

  ❉❡✜♥✐çã♦ ✶✳✷✳✷✾✳ ❙❡❥❛♠ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ X, Y ∈ X ✱ r > 0✱ ❡ p, q ∈ M✳ X ❡ Y sã♦ ❞✐t♦s t♦♣♦❧♦❣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s ❡♠ p ❡ q s❡ ❡①✐st❡♠ p q p q → W

  ✈✐③✐♥❤❛♥ç❛s V ❡ W ❞❡ p ❡ q r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ h : V q✉❡ ❧❡✈❛ ór❜✐t❛s ❞❡ X ❡♠ ór❜✐t❛s ❞❡ Y ♣r❡s❡r✈❛♥❞♦ ❛ ♦r✐❡♥t❛çã♦ ❞❛ ór❜✐t❛s ❡ h(p) = q

  ✳ r +1 ❖❜s❡r✈❛çã♦ ✶✳✷✳✸✵✳ ❙❡ f : M → N é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❝❧❛ss❡ C ✱ r > 0✱ m n r

  (M )

  X ♦♥❞❡ M ❡ N sã♦ ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❙❡❥❛ X ∈ X ✱ ❡♥tã♦ Y = f p X(p) r

  ❞❡✜♥✐❞♦ ♣♦r Y (q) = Df ✱ ❝♦♠ q = f(p) é ✉♠ ❝❛♠♣♦ ❞❡ ❝❧❛ss❡ C ❡♠ N✱ −1 X = Df ◦ X ◦ f ♣♦✐s f ✳ ❙❡ α : I → M é ✉♠❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ ❞❡ X✱ ❡♥tã♦ f ◦ α : I → N

  é ✉♠❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ ❡♠ Y ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ f ❧❡✈❛ tr❛❥❡tór✐❛s ❞❡

  X ❡♠ tr❛❥❡tór✐❛s ❞❡ Y ✳ m

  X ❙❡ x : U ⊂ M → V ⊂ R é ✉♠❛ ❝❛rt❛ ❡♠ M✱ Y = x ∗ é ✉♠ ❝❛♠♣♦ ❞❡ r

  ✈❡t♦r❡s ❞❡ ❝❧❛ss❡ C ❡♠ V ✳ ❉✐③❡♠♦s q✉❡ Y é ❛ ❡①♣r❡ssã♦ ❞❡ X ♥❛ ❝❛rt❛ ❧♦❝❛❧ x✳ n ❚❡♦r❡♠❛ ✶✳✷✳✸✶✳ ✭❋❧✉①♦ ❚✉❜✉❧❛r✮ ❙❡❥❛♠ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ r X ∈ X (M ) 1 ✱ r > 0 ❡ p ∈ M ✉♠ ♣♦♥t♦ r❡❣✉❧❛r ❞❡ X✱ ✐st♦ é✱ X(p) 6= 0✳ ❙❡❥❛♠ n n i

  C = {(x , ..., x ) ∈ R ; | x |< 1, 1 ≤ i ≤ n} C ❡ X ✉♠ ❝❛♠♣♦ ❞❡✜♥✐❞♦ ❡♠ C ♣♦r r

  X C (x) = (1, 0, ..., 0) p → C p ✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❝❧❛ss❡ C ✱ h : V ✱ ♦♥❞❡ V é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ p ❡♠ M✱ ❧❡✈❛♥❞♦ tr❛❥❡tór✐❛s ❞❡ X ❡♠ tr❛❥❡tór✐❛s C ❞❡ X ✳ n

  ⊂ R ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ x : U → U ✉♠❛ ❝❛rt❛ t❛❧ q✉❡ p ∈ U ❝♦♠ x(p) = 0✳ r

  X ❙❡❥❛ x ♦ ❝❛♠♣♦ ❞❡ ❝❧❛ss❡ C ✐♥❞✉③✐❞♦ ♣♦r X ❡♠ U ✳ ❈♦♠♦ X(p) 6= 0 t❡♠♦s

  ✷✺ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ −1 X(0) = Dx p X(x (0)) = Dx p X(p) 6= 0 p q✉❡ x ♣♦✐s Dx é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❙❡❥❛ n

  ϕ : [−τ, τ ] × V → U ∗ ∗ X ; hw, x X(0)i = 0 ♦ ✢✉①♦ ❧♦❝❛❧ ❞❡ x ❡ s❡❥❛ H = {w ∈ R ✳ n −1

  ❆✜r♠❛çã♦✿ H é ✐s♦♠♦r❢♦ ❛ R ✳ ∗ X(0) n 1 ❖ s✉❜❡s♣❛ç♦ H ❣❡r❛❞♦ ♣♦r x t❡♠ ❞✐♠❡♥sã♦ ✶ ❡♠ R ✳ ❙❡❥❛ H ♦ n 1 s✉❜❡s♣❛ç♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ H ❡♠ R ✱ ❧♦❣♦ ❞✐♠❡♥sã♦ ❞❡ H é ✐❣✉❛❧ ❛ n−1✳ ❊ ✉♠❛ n 1 ∗ X(0)

  ❜❛s❡ ❞❡ R é ❞❛❞❛ ♣❡❧♦ n − 1 ✈❡t♦r❡s ❞❛ ❜❛s❡ ❞❡ H ✉♥✐ã♦ ❝♦♠ ♦ ✈❡t♦r x ❞❛ ❜❛s❡ ❞❡ H ✳ P❡❧♦ ♣r♦❝❡ss♦ ❞❡ ♦rt♦❣♦♥❛❧✐③❛çã♦ ❞❡ ●r❛♠ ❙❝❤♠✐❞t ♣♦❞❡♠♦s t♦r♥❛r ❡ss❡s ✈❡t♦r❡s ✉♠ ❛ ✉♠ ♦rt♦❣♦♥❛✐s✱ ❛ss✐♠ H t❡rá ❝♦♠♦ ❜❛s❡ n − 1 ✈❡t♦r❡s n −1

  X(0) ♦rt♦❣♦♥❛✐s ❛ x ∗ ✳ P♦rt❛♥t♦✱ H é ✐s♦♠♦r❢♦ ❛ R ✳

  ∩ H , ..., e n } 1 ❙❡❥❛ ψ : [−τ, τ] × S → U ♦♥❞❡ S = V ✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ❜❛s❡ {e n

  } ⊂ {0} × H = (1, 0, ..., 0) , ..e n

  ❞❡ R × H✱ q✉❡ é ✐s♦♠♦r❢♦ ❛ R ✱ ♦♥❞❡ e 1 ❡ {e 2 ✱ s❡❣✉❡ q✉❡✿ ∂ϕ

  Dψ e = (t, 0)| t = x X(ϕ(0, 0)) = x X(0) (0,0) 1 =0 ∗ ∗ ✭♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ✢✉①♦ ❧♦❝❛❧✮ ∂t

  Dψ e j = e j , 2 ≤ j ≤ n, (0,0) m ♣♦✐s ψ(0, y) = y ∀y ∈ S.

  : R × H → R ▲♦❣♦ Dψ (0,0) é ✉♠ ✐s♦♠♦r✜s♠♦✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❋✉♥çã♦

  ■♥✈❡rs❛✱ ψ é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ (0, 0) ❡♠ [−τ, τ] × S s♦❜r❡ n ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ 0 ❡♠ R ✳ P♦rt❛♥t♦✱ s❡ ǫ > 0 é s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ C ǫ = {(t, x) ∈ R × S; | t |< ǫ, | x |< ǫ} ψ : C ǫ → U ǫ r ❡ e é ❛ r❡str✐çã♦ ❞❡ ψ ❛ C

  ψ ❡♥tã♦ e é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ C s♦❜r❡ s✉❛ ✐♠❛❣❡♠ ❡♠ U ✳

  ψ C ǫ ❆❧é♠ ❞✐ss♦✱ e ❧❡✈❛ ór❜✐t❛s ❞♦ ❝❛♠♣♦ ♣❛r❛❧❡❧♦ X ǫ ❡♠ C ❡♠ ór❜✐t❛s ❞❡ x ∗ ✳ ❈♦♥s✐❞❡r❡ ♦ ❞✐❢❡♦♠♦r✜s♠♦ C ❞❛❞♦ ♣♦r f(y) = ǫy ❡ ❞❡✜♥❛ −1 −1 X f : C → C ǫ r e h = x ψf : C → M −1 é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ C s♦❜r❡ s✉❛ ✐♠❛❣❡♠✳ ❊♥tã♦ r e h : x ψ(C ǫ ) → C

  é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ C ✳ r (M ) ∈ γ

  ❙❡❥❛ γ ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ ❞❡ ✉♠ ❝❛♠♣♦ X ∈ X ✳ P♦r ✉♠ ♣♦♥t♦ x ❝♦♥s✐❞❡r❡ ✉♠❛ tr❛♥s✈❡rs❛❧ ♦✉ s❡çã♦ tr❛♥s✈❡rs❛❧ Σ ❛♦ ❝❛♠♣♦ X✳

x

P(x) x Σ

  ❋✐❣✉r❛ ✶✳✽✳ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ P♦✐♥❝❛ré

  ✷✻ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ ❆ 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✳

  ❉❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦s ❝❛♠♣♦s X ❡ −X s❡❣✉❡ q✉❡ P é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❞❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x ❡♠ Σ✳ r

  (M ) ❉❡✜♥✐çã♦ ✶✳✷✳✸✷✳ ❯♠ ✢✉①♦ t✉❜✉❧❛r ❞❡ X ∈ X é ✉♠ ♣❛r (F, f) ♦♥❞❡ F é r n n −1

  = I × I = ✉♠ ❛❜❡rt♦ ❞❡ M ❡ f é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ C ❞❡ F s♦❜r❡ ♦ ❝✉❜♦ I n i −1 {(x, y) ∈ R × R ; | x |< 1 ||< 1, 1 ≤ i ≤ n − 1}

  ❡ || y ✱ ♦♥❞❡ n é ❛ ❞✐♠❡♥sã♦ ❞❛ ✈❛r✐❡❞❛❞❡ M✳

  X X(x, y) = ❙❡ f ❞❡♥♦t❛ ♦ ❝❛♠♣♦ ✐♥❞✉③✐❞♦ ♣♦r f✱ ✐st♦ é✱ f −1 −1

  Df X(f (x, y)) f (x,y) ∗

  X ❡♥tã♦ f é ♣❛r❛❧❡❧♦ ❛♦ ❝❛♠♣♦ ❝♦♥st❛♥t❡ (x, y) 7→ (1, 0)✳

  ❖ ❛❜❡rt♦ F ❞❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ é ❝❤❛♠❛❞♦ ✉♠❛ ❝❛✐①❛ ❞❡ ✢✉①♦ ♣❛r❛ ♦ ❝❛♠♣♦

  X ✳ ◆♦ ❚❡♦r❡♠❛ t❡♠♦s q✉❡ s❡ p ∈ M é ✉♠ ♣♦♥t♦ r❡❣✉❧❛r ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r

  ✉♠❛ ❝❛✐①❛ ❞❡ ✢✉①♦ q✉❡ ❝♦♥té♠ p✳ ❚❡♦r❡♠❛ ✶✳✷✳✸✸✳ ✭❋❧✉①♦ ❚✉❜✉❧❛r ▲♦♥❣♦✮ ❙❡❥❛ γ ⊂ M ✉♠ ❛r❝♦ ❞❡ ✉♠❛ tr❛❥❡tór✐❛ ❞❡ X ❝♦♠♣❛❝t♦ ❡ ♥ã♦ ❢❡❝❤❛❞♦✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ✢✉①♦ t✉❜✉❧❛r (F, f) ❞❡ X t❛❧ q✉❡ γ ⊂ F ✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ α : [−ǫ, a + ǫ] → M ✉♠❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ ❞❡ X t❛❧ q✉❡ ′ ′ α([0, a]) = γ )

  ❡ α(t) 6= α(t s❡ t 6= t ✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦♠♣❛❝t♦ eγ = α([−ǫ, a + ǫ])✳ ❈♦♠♦ ♦s ♣♦♥t♦s ❞❡ eγ sã♦ r❡❣✉❧❛r❡s✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❋❧✉①♦ ❚✉❜✉❧❛r ❡①✐st❡ ✉♠❛ ❝♦❜❡rt✉r❛ ❞❡ eγ ♣♦r ❝❛✐①❛s ❞❡ ✢✉①♦✳ ❈♦♠♦ eγ é ❝♦♠♣❛❝t♦✱ s❡❣✉❡ ❡ss❛ ❝♦❜❡rt✉r❛ ❛❞♠✐t❡ ✉♠ ♥ú♠❡r♦ ❞❡ ▲❡❜❡s❣✉❡✱ ❡♥tã♦ ❝♦♥s✐❞❡r❡ δ > 0 ♦ ♥ú♠❡r♦ ❞❡ 1 , ..., F k } ▲❡❜❡s❣✉❡ ❞❡st❛ ❝♦❜❡rt✉r❛✳ ❙❡❥❛ {F ✉♠❛ ❝♦❜❡rt✉r❛ ✜♥✐t❛ ♣♦r ❝❛✐①❛s ❞❡ ✢✉①♦ ❞❡ ❞✐â♠❡tr♦ ♠❡♥♦r q✉❡ δ/2✳ ❉❡✈✐❞♦ ❛ ❡s❝♦❧❤❛ ❞♦ ❞✐â♠❡tr♦ ❞❛s ❝❛✐①❛s ❞❡ i j i j ∩ F = ∅ ∪ F ✢✉①♦✱ t❡♠♦s q✉❡ s❡ F ❡♥tã♦ F ❡stá ❝♦♥t✐❞♦ ♥✉♠❛ ♠❡s♠❛ ❝❛✐①❛ i ❞❡ ✢✉①♦ ❞❡ X✳ ❯s❛♥❞♦ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ ♣♦❞❡♠♦s✱ ❞✐♠✐♥✉✐r ♦s F ✱ s❡ ♥❡❝❡ssár✐♦✱ i i i −1 +1 ❛❣r✉♣á✲❧♦s ❞❡ ♠♦❞♦ q✉❡ ❝❛❞❛ F ✐♥t❡rs❡t❡ ❛♣❡♥❛s F ❡ F ✳ F 1 F 2 F 3

  α(a) α(0) α(a + ε) α(-ε)

  ❋✐❣✉r❛ ✶✳✾

  ✷✼ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ 1 < t < ... < t m = a + ǫ i = α(t i 2 ❙❡❥❛♠ −ǫ = t t❛✐s q✉❡ p ) ∈ F ∩ eγ ❡ ❞❡♥♦t❡♠♦s n −1 n −1

  = {(0, y) ∈ I × I ; | y j |< d, 1 ≤ j ≤ n − 1} i , f i ) ♣♦r I d i = f (I ✳ ❙❡❥❛♠ (F ♦s ✢✉①♦s n −1 t✉❜✉❧❛r❡s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛s ❝❛✐①❛s F ✱ 1 ≤ i ≤ k✳ ❚❡♠♦s q✉❡ Σ 1 1 d é ✉♠❛ 1 1 ∈ Σ 1 s❡çã♦ tr❛♥s✈❡rs❛❧ ❛ x✱ ✉♠❛ ✈❡③ q✉❡ f é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧ ❡ p ✳ ❙❡❥❛

  ∈ Σ Σ i = X t −t i ❡♥tã♦ Σ é ✉♠❛ tr❛♥s✈❡rs❛❧ ❛ X t❛❧ q✉❡ p ✳ ❈♦♥s✐❞❡r❛♥❞♦ 1 (Σ) i i i d > 0 i ⊂ F i s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ t❡♠♦s q✉❡ Σ ✳

  Σ 3 Σ 2 Σ p 1 1

p p

2 3

  ❋✐❣✉r❛ ✶✳✶✵ t (p ) P❛r❛ ❝❛❞❛ p ∈ eγ t♦♠❡ t ∈ [0, a + 2ǫ] t❛❧ q✉❡ p = X 1 1 ❡ ❝♦♥s✐❞❡r❡ ❛ s❡çã♦

  Σ p = X t (Σ ) 1 p ∩ Σ q = ∅ ✳ P❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❋❧✉①♦ ❚✉❜✉❧❛r t❡♠♦s q✉❡ Σ s❡ p 6= q ❡

  [ Σ p q✉❡ F = é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ γ✳ p γ ∈e F i

  i Σ p i γ~

  ❋✐❣✉r❛ ✶✳✶✶ r p ◆❛ ✈✐③✐♥❤❛♥ç❛ ❡stá ❞❡✜♥✐❞❛ ✉♠❛ ✜❜r❛çã♦ C ❝✉❥❛ ✜❜r❛ s♦❜r❡ ♦ ♣♦♥t♦ p é Σ ✳ p

  ■st♦ é✱ ❛ ♣r♦❥❡çã♦ π 1 r : F → eγ q✉❡ ❛ ❝❛❞❛ z ∈ F ❛ss♦❝✐❛ ♦ ♣♦♥t♦ p t❛❧ q✉❡ z ∈ Σ t (x ) ∈ Σ 1 é ✉♠❛ ❛♣❧✐❝❛çã♦ C ✳ ❚❡♠♦s q✉❡✱ s❡ z ∈ F ❡♥tã♦ z = X ❝♦♠ x ✱ ❛ss✐♠ π (z) = X t (p 1 1

  ) ∈ eγ✳ r 2 : F → Σ 1 ❈♦♥s✐❞❡r❡ ❛ ♣r♦❥❡çã♦ C ✱ π q✉❡ ❛ ❝❛❞❛ ♣♦♥t♦ z ∈ F ❛ss♦❝✐❛ 1 p t (p 1 )

  ❛ ✐♥t❡rs❡çã♦ ❞❛ ór❜✐t❛ ❞❡ z ❝♦♠ Σ ✱ ✐st♦ é✱ s❡ z ∈ Σ ❡ p = X ❡♥tã♦ n −1 π (z) = X (z) : Σ → I 2 −t 1 2 1

  ✳ ❙❡❥❛♠ g : eγ → [−1, 1] ❡ g ❞♦✐s ❞✐❢❡♦♠♦r✜s♠♦s✳ n −1 1 ◦ π ◦ π 1 (z), g 2 2 (z)) ❉❡✜♥✐♠♦s✱ ❡♥tã♦ f : F → I × I ❞❛❞❛ ♣♦r f(z) = (g ✳ ▲♦❣♦✱ (F, f )

  é ✉♠ ✢✉①♦ t✉❜✉❧❛r q✉❡ ❝♦♥té♠ γ✳ ❖❜s❡r✈❛çã♦ ✶✳✷✳✸✹✳ ❖ ❞✐❢❡♦♠♦r✜s♠♦ f ❡♥❝♦♥tr❛❞♦ ❛❝✐♠❛ ❧❡✈❛ ór❜✐t❛s ❞♦ ❝❛♠♣♦ n n −1

  → R ❝♦♥st❛♥t❡ C : I × I ❞❛❞♦ ♣♦r C(x, y) = (1, 0)✳ ●❡r❛❧♠❡♥t❡✱ f ♥ã♦

  X ♣r❡s❡r✈❛ ♦ ♣❛râ♠❡tr♦ t✱ ✐st♦ é✱ f ♥ã♦ é ✐❣✉❛❧ ❛♦ ❝❛♠♣♦ ❝♦♥st❛♥t❡ C✳ ◆♦ n −1 f : e F → (−b, b) × I ❡♥t❛♥t♦✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ˜ ✱ ♦♥❞❡ b > 0 f ∗

  X ✱ t❛❧ q✉❡ ˜ s❡❥❛ ♦ ❝❛♠♣♦ ❝♦♥st❛♥t❡✳

  [ X t (p) ⊂ F

  ❈♦♠ ❡❢❡✐t♦✱ s❡❥❛ p ∈ γ ❡ b > 0 t❛❧ q✉❡ γ ⊂ ✳ ❙❡❥❛ t ∈(−b,b) Σ p ⊂ F

  ✉♠❛ s❡çã♦ tr❛♥s✈❡rs❛❧ ❛ X ♣❡❧♦ ♣♦♥t♦ p✱ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛ ♣❛r❛ [

  F = X t (p) F (z) ∈ Σ p −t q✉❡ e ❡st❡❥❛ ❝♦♥t✐❞♦ ❡♠ F ✳ ❙❡ z ∈ e ❡ X ✱ ❞❡✜♥✐♠♦s t ∈(−b,b) n r −1 ˜ f (z) = (t, hX (z)) p → I −t

  ✱ ♦♥❞❡ h : Σ é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ C ✳

  ✷✽ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ r f

  ❆ss✐♠✱ ˜ é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ C ✳ ❚❡♠♦s q✉❡✱ −1 n −1 h : I → Σ p −1 α 7→ h (α)

  ❡♥tã♦✱ −1 −1 n [ ˜ f : (−b, b) × I → e F = −1 t ∈(−b,b) X (p) t (t, α) 7→ X t (h (α)). t (h (α)) −1

  ❙❡❥❛✱ α : (−ǫ, ǫ) → M ✉♠❛ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧ t❛❧ q✉❡ α(0) = X ′ −1 −1 −1 (0) = X(X t (h (α))) f ◦ α(0) = ˜ f (X t (h (α)) = (t, h(h (α))) =

  ❡ α ✳ ▲♦❣♦✱ ˜ n n −1 −1 ⊂ R −1

  (t, α) ∈ (−b, b) × I f ˜ (X(X t (h (α)) ❡♥tã♦✱ D ˜ f é ❞❛❞♦ ♣❡❧♦ ✈❡t♦r (t,α)

  (1, 0) f

  X ✱ ❛ss✐♠ ˜ é ♦ ❝❛♠♣♦ C✳ r

  (M ) Pr♦♣♦s✐çã♦ ✶✳✷✳✸✺✳ ❙❡❥❛♠ γ ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ ❞❡ ✉♠ ❝❛♠♣♦ X ∈ X ✱ n ♦♥❞❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ❡ Σ é ✉♠❛ tr❛♥s✈❡rs❛❧ ❛ X ♣♦r ✉♠ ♣♦♥t♦ p ∈ γ : U ⊂ Σ → Σ Σ Σ

  ✳ ❙❡ P é ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ P♦✐♥❝❛ré ❡♥tã♦ P é ✉♠ r ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❝❧❛ss❡ C ❞❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ V ❞❡ p s♦❜r❡ ✉♠ ❛❜❡rt♦ ❞❡ Σ✳ 1 , f ) , f ) 1 2 2

  ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ (F ✉♠ ✢✉①♦ t✉❜✉❧❛r ❝♦♥t❡♥❞♦ p ❡ (F ✉♠ ✢✉①♦ 1 ∪ F 2 1 2 t✉❜✉❧❛r ❧♦♥❣♦ t❛❧ q✉❡ γ ⊂ F ❝♦♠♦ ♥❛ ✜❣✉r❛✳ ❙❡❥❛♠ Σ ❡ Σ ❛s ❝♦♠♣♦♥❡♥t❡s −1 n −1 2 1 = f ({−1} × I ) ❞♦ ❜♦r❞♦ ❞❡ F q✉❡ sã♦ tr❛♥s✈❡rs❛✐s ❛ X✱ ✐st♦ é✱ Σ −1 n −1 2 ❡

  Σ = f ({1} × I ) 2 2 ✳ : V ⊂ Σ → Σ : Σ → Σ : Σ → Σ

  ❉❡♥♦t❛r❡♠♦s ♣♦r π 1 1 ✱ π 2 1 2 ❡ π 3 2 ❛s ♣r♦❥❡çõ❡s ❛♦ ❧♦♥❣♦ ❞❛s tr❛❥❡tór✐❛s ❞❡ X✱ ♦♥❞❡ V é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ♣❡q✉❡♥❛ ❞❡ p✳ F 1

  

Σ

Σ 2 F 2 (q) Σ 1

p

q π 1 (q) π π 2 1 (q) π 3 π 2 π

1

γ

  ❋✐❣✉r❛ ✶✳✶✷ Σ = π ◦ π ◦ π 3 2 1 ❆ss✐♠ P ✳ ❯s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞♦ ❋❧✉①♦ ❚✉❜✉❧❛r ▲♦♥❣♦ t❡♠♦s 1 2 3 Σ Σ r r q✉❡ π ✱ π ❡ π sã♦ ❞❡ ❝❧❛ss❡ C ✱ ❧♦❣♦ P é ❞❡ ❝❧❛ss❡ C ✳ ❈♦♠♦ P t❡♠ ✉♠❛ ✐♥✈❡rs❛ r

  ❞❡ ❝❧❛ss❡ C ✱ q✉❡ é ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ P♦✐♥❝❛ré ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ❝❛♠♣♦ −X✱

  ✷✾ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ Σ r s❡❣✉❡ q✉❡ P é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❝❧❛ss❡ C ❞❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ V ❞❡ p s♦❜r❡ ✉♠ ❛❜❡rt♦ ❞❡ Σ✳

  ❉❡ ❢♦r♠❛ ♠❛✐s ❣❡r❛❧✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ P♦✐♥❝❛ré ♣❛r❛ ❛s ❡①tr❡♠✐❞❛❞❡s ❞❡ ✉♠ ❛r❝♦ ❝♦♠♣❛❝t♦ ❡ ♥ã♦ ❢❡❝❤❛❞♦ ❞❡ ✉♠❛ tr❛❥❡tór✐❛ ❞❡ ✉♠ ❝❛♠♣♦ r X ∈ X (M )

  ✳ ❙❡❥❛♠ p, q ❛s ❡①tr❡♠✐❞❛❞❡s ❞❡ss❡ ❛r❝♦ ❝♦♠♣❛❝t♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ 1 2 ❋❧✉①♦ ❚✉❜✉❧❛r ▲♦♥❣♦✱ ♣♦❞❡♠♦s ❝♦♥str✉✐r tr❛♥s✈❡rs❛✐s Σ ❡ Σ ♣❛ss❛♥❞♦ ♣❡❧♦s τ (p) ♣♦♥t♦s p ❡ q✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❚❡♠♦s q✉❡ q = X ♣❛r❛ ❛❧❣✉♠ τ ∈ R✱ ❞❡✈✐❞♦ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦ ❝❛♠♣♦✱ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ V ❞❡ p s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛ ♦s ♣♦♥t♦s ❞❡st❛ ✈✐③✐♥❤❛♥ç❛ ✐♥t❡rs❡❝t❛♠ Σ 2 ♥✉♠ t❡♠♣♦ ♣ró①✐♠♦ ❛ τ ✭s❡ ♥❡❝❡ssár✐♦ ❝♦♥s✐❞❡r❡ ❛ ♣r✐♠❡✐r❛ ✐♥t❡rs❡çã♦✮✳ p Σ 1 Σ q 2 x P(x)

  ❋✐❣✉r❛ ✶✳✶✸ ❉❡✜♥✐♠♦s ❛ ❢✉♥çã♦ ♦✉ tr❛♥s❢♦r♠❛çã♦ ❞❡ P♦✐♥❝❛ré ♣❡❧❛ ❛♣❧✐❝❛çã♦ P :

  V ⊂ Σ → Σ 1 2 t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ ♣♦♥t♦ x ∈ V ❛ss♦❝✐❛ ♦ ♣♦♥t♦ P (x)✱ ♦♥❞❡ P (x) é ❛ ♣r✐♠❡✐r❛ ✐♥t❡rs❡çã♦ ❞❛ ór❜✐t❛ ❞❡ x ❝♦♠ Σ 2 ✳ ❈♦♠♦ ❢❡✐t♦ ❛♥t❡r✐♦r♠❡♥t❡✱ t❡♠♦s r q✉❡ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ P♦✐♥❝❛ré s❡rá ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❝❧❛ss❡ C ❞❡ ✉♠❛ 2

  ✈✐③✐♥❤❛♥ç❛ p s♦❜r❡ ✉♠ ❛❜❡rt♦ ❞❡ Σ ✳ n r (M )

  ❉❡✜♥✐çã♦ ✶✳✷✳✸✻✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠♣❛❝t❛✱ X ∈ X ✱ r ≥ 1 ❡ γ é ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ ❞❡ ♣❡rí♦❞♦ T ❝♦♠ p ∈ γ✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ T ) p 1 , ..., λ n −1

  t❡♠♦s q✉❡ ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ (DX sã♦ 1, λ ✳ ❊♥tã♦ ♦s n − 1 1 −1 , ..., λ n ❛✉t♦✈❛❧♦r❡s λ sã♦ ❝❤❛♠❛❞♦s ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❝❛r❛❝t❡ríst✐❝♦s ♦✉ ✭❛✉t♦✈❛❧♦r❡s✮ ❞❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛✳ ◆♦t❡ q✉❡ t❛♠❜é♠ ♣❡❧❛ Pr♦♣♦s✐çã♦ ♦s ❛✉t♦✈❛❧♦r❡s ♥ã♦ ❞❡♣❡♥❞❡♠ ❞❛ ❡s❝♦❧❤❛ ❞♦ ♣♦♥t♦ p ∈ γ✳ r

  (M ) ❙❡❥❛ γ ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ ❞❡ ♣❡rí♦❞♦ T ❞❡ ✉♠ ❝❛♠♣♦ X ∈ X ✱ r > 0 n

  ♥✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠♣❛❝t❛ M ✳ ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❞❡❝♦♠♣♦s✐çã♦ ❞♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ r❡str✐t♦ ❛ γ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ❡ ✐♥✈❛r✐❛♥t❡ ❡♠ M✿ u X s

  T p M = E ⊕ E ⊕ E , t ) p E = E (p) σ σ p p p ♣❛r❛ ❝❛❞❛ ♣♦♥t♦ p ∈ γ✱ ❝♦♠ D(X p ✱ ♣❛r❛ σ = u, s, X✳ ❆q✉✐ ⊕ ❞❡♥♦t❛ s♦♠❛ ❞✐r❡t❛✳ T ) p T : M → M , · · · , λ n 1 −1

  ❈♦♥s✐❞❡r❡ (DX ❛ ❞❡r✐✈❛❞❛ ❡♠ p ❞❡ X ❡✱ s❡❥❛♠ 1, λ s T ) p k 6= 1 ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ (DX ✱ t❛✐s q✉❡ λ ✱ ∀k = 1, ..., n − 1✳ ❖s s✉❜✜❜r❛❞♦s E p u p T ) p p s

  ❡ E ❡stã♦ ❛ss♦❝✐❛❞♦s ❛♦s ❛✉t♦✈❛❧♦r❡s ❞❡ (DX ❞❡ ♠♦❞♦ q✉❡ ♦ s✉❜✜❜r❛❞♦ E u i i |< 1 ❡stá ❛ss♦❝✐❛❞♦ ❛♦s ❛✉t♦✈❛❧♦r❡s λ t❛✐s q✉❡ | λ ✱ ❡♥q✉❛♥t♦ E p ❡stá ❛ss♦❝✐❛❞♦ j j |> 1 p X

  ❛♦s ❛✉t♦❧❛✈♦r❡s λ t❛✐s q✉❡ | λ ✳ ❖ s✉❜✜❜r❛❞♦ E ❣❡r❛❞♦ ♣❡❧♦ ❝❛♠♣♦ ❡stá

  ✸✵ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ X p = 1 ❛ss♦❝✐❛❞♦ ❛♦ ❛✉t♦✈❛❧♦r 1✱ ❝♦♠♦ X(p) 6= 0✱ ♣❛r❛ t♦❞♦ p ∈ γ t❡♠♦s q✉❡ dimE X

  = hX(p)i ❡✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r E p ✱ ♦♥❞❡ hX(p)i ❞❡♥♦t❛ ♦ s✉❜❡s♣❛ç♦ ❣❡r❛❞♦ ♣❡❧♦ ✈❡t♦r X(p)✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s ✈❡r

  ❯♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ ❝♦♠ ❡st❛s ♣r♦♣r✐❡❞❛❞❡s é ❞✐t❛ s❡r ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ ❤✐♣❡r❜ó❧✐❝❛✳ ❉❡✜♥✐çã♦ ✶✳✷✳✸✼✳ ❯♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ γ ❤✐♣❡r❜ó❧✐❝❛ é ❞✐t❛ s❡r t✐♣♦ s❡❧❛ q✉❛♥❞♦ s s dimE 6= 0 6= 0 p ❡ dimE p ✱ p ∈ γ✳

  ❉❡✜♥✐çã♦ ✶✳✷✳✸✽✳ ❖s ❝♦♥❥✉♥t♦s✿ ss W (p) = {q ∈ M ; d(X t (p), X t (q)) → 0, q✉❛♥❞♦ t → ∞}

  ❡ uu W (p) = {q ∈ M ; d(X t (p), X 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✳

  ❉❛❞♦ ǫ > 0✱ ❛ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❧♦❝❛❧ ❞❡ t❛♠❛♥❤♦ ǫ ❞♦ ♣♦♥t♦ p ∈ M é ♦ ss (p)

  ❝♦♥❥✉♥t♦ ❞❡♥♦t❛❞♦ ♣♦r W ǫ ❞♦s ♣♦♥t♦s y ∈ M t❛✐s q✉❡ t →+∞ lim d(X t (p), X t (y)) = 0 ❡ d(X t (p), X t (y)) ≤ ǫ, ∀t ≥ 0.

  ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❡✜♥❡✲s❡ ❛ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧ ❧♦❝❛❧ ❞❡ t❛♠❛♥❤♦ ǫ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M✳

  ❆ ♥♦çã♦ ❞❡ ❤✐♣❡r❜ó❧✐❝♦✱ ❛ss✐♠ ❝♦♠♦ ❛s ✈❛r✐❡❞❛❞❡s ❡stá✈❡✐s ❡ ✐♥stá✈❡✐s ❧♦❝❛✐s ♣♦❞❡♠ s❡r ❣❡♥❡r❛❧✐③❛❞❛s ♣❛r❛ ✉♠ ❝♦♥❥✉♥t♦ Λ ⊂ M ❝♦♠♣❛❝t♦ ❡ ✐♥✈❛r✐❛♥t❡ ♣❡❧♦ ✢✉①♦✱ ♦ ♠❡s♠♦ ♦❝♦rr❡ ♣❛r❛ ♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r ❡ ❛s ❞❡✜♥✐çõ❡s r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❤✐♣❡r❜♦❧✐❝✐❞❛❞❡✱ ✈❡r P❛r❛ ♥♦ss♦ tr❛❜❛❧❤♦ ❝♦♥s✐❞❡r❛r❡♠♦s Λ = γ✳ ❚❡♦r❡♠❛ ✶✳✷✳✸✾✳ ✭❚❡♦r❡♠❛ ❞❛ ❱❛r✐❡❞❛❞❡ ❊stá✈❡❧ ♣❛r❛ ❋❧✉①♦s✮✳ ❙❡❥❛ Λ ⊂ M t

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

  ♦s q✉❛✐s sã♦ t❛♥❣❡♥t❡s ❛ E p ❡ E p ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r

  ❆ss✐♠ ❛s ✈❛r✐❡❞❛❞❡s ❡stá✈❡❧ ❡ ✐♥stá✈❡❧ ❢♦rt❡ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M ♣❛r❛ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❝♦♠♦ ss ss [

  W (p) = t ≥0 X (W (X t (p)) −t ǫ

  ✸✶ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ uu uu [ W (p) = t ≥0 X (W (X (p)). t −t ǫ

  ❉❡✜♥✐♠♦s ❛ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❡ ✐♥stá✈❡❧ ❞❡ ✉♠ ♣♦♥t♦ p ∈ M ♣❛r❛ ♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ❝♦♠♦ s❡♥❞♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ♦s ❝♦♥❥✉♥t♦s s ss [

  W (p) = W (X t (p)) t ∈R u uu [ W (p) = W (X t (p)). t ∈R t (p) (p) s u ❙❡ Λ é ✉♠ ❝♦♥❥✉♥t♦ ❤✐♣❡r❜ó❧✐❝♦ ♣❛r❛ ✉♠ ✢✉①♦ X ✱ ❡♥tã♦ W ❡ W s X u X

  ⊕ E ⊕ E sã♦ ✈❛r✐❡❞❛❞❡s t❛♥❣❡♥t❡s ❛ E ❡ E ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ ❞❡♣❡♥❞❡♠ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞❡ p✳ ▼❛✐♦r❡s ❞❡t❛❧❤❡s ✈❡r r

  (M ) ❈♦♥s✐❞❡r❡ M ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ X ∈ X ✱ r ≥ 1✱ γ ✉♠❛ ór❜✐t❛

  ♣❡r✐ó❞✐❝❛ ❤✐♣❡r❜ó❧✐❝❛ t✐♣♦ s❡❧❛ ❞❡ ♣❡rí♦❞♦ T > 0✳ ◆❡st❡ ❝❛s♦✱ ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ (DX ) , λ |< 1 |> 1 T p 1 2 1 2

  ✱ ❝♦♠ p ∈ γ✱ sã♦ 1, λ ✳ ❉❡✈❡♠♦s t❡r q✉❡ | λ ❡ | λ ✱ ❛ss✐♠ λ , λ T ) p 1 2 sã♦ ♥ú♠❡r♦s r❡❛✐s✳ ▲♦❣♦✱ (DX ♣♦ss✉✐ três ❛✉t♦✈❛❧♦r❡s r❡❛✐s ❡ ❞✐st✐♥t♦s 3

  , v , v } p M i 1 2 ❞♦✐s ❛ ❞♦✐s✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❜❛s❡ {v ❞❡ T ✭✐s♦♠♦r❢♦ ❛ R ✮✱ ♦♥❞❡ v ✱ 0 ≤ i ≤ 2

  , λ sã♦ ♦s ❛✉t♦✈❡t♦r❡s ❛ss♦❝✐❛❞♦s ❛♦s ❛✉t♦✈❛❧♦r❡s 1, λ X 1 2 ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ p = X(p)

  ❆ss✐♠✱ E é ♦ s✉❜❡s♣❛ç♦ ❣❡r❛❞♦ ♣❡❧♦ ❛✉t♦✈❛❧♦r v ✱ ❛ss♦❝✐❛❞♦ ❛♦ s 1 1 u ❛✉t♦✈❛❧♦r 1✱ E p é ♦ s✉❜❡s♣❛ç♦ ❣❡r❛❞♦ ♣♦r v ❛ss♦❝✐❛❞♦ ❛♦ ❛✉t♦✈❛❧♦r λ ✱ E p é ♦ 2 2 s✉❜❡s♣❛ç♦ ❣❡r❛❞♦ ♣♦r v ❛ss♦❝✐❛❞♦ ❛♦ ❛✉t♦✈❛❧♦r λ ✳ E s p E X(p) p u

p

  ❋✐❣✉r❛ ✶✳✶✹ T ) p ❉❡✈✐❞♦ ❛s ❤✐♣ót❡s❡s ♣♦❞❡♠♦s ❡s❝r❡✈❡r (DX ❝♦♠♦✿

   

  1  0 λ 

  (DX T ) p = 1 p ) T 6= 0 6= 0 (p) λ 2 1 2 ss ❈♦♠♦ (DX é ✉♠ ✐s♦♠♦r✜s♠♦✱ t❡♠✲s❡ q✉❡ λ ❡ λ ✳ ❙❡❥❛ W s

  ❛ ✈❛r✐❡❞❛❞❡ ❢♦rt❡ ❞♦ ♣♦♥t♦ p ∈ γ✱ q✉❡ é t❛♥❣❡♥t❡ ❛♦ s✉❜❡s♣❛ç♦ E p ✭✈❡r ❚❡♦r❡♠❛ T (W (p)) = W (p) T (W (p)) ss ss ss ❚❡♠♦s q✉❡ X ✱ ♣♦✐s s❡ y ∈ X ❡♥tã♦ ss y = X (x) (p) (y) T −T

  ❝♦♠ x ∈ W ❡♥tã♦ x = X ✱ ❧♦❣♦✿ d(X t (x), X t (p)) → 0, t → ∞ ⇒

  ✸✷ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ ⇒ d(X t (X (y), X t (X (p)) → 0, t → ∞ ⇒ −T −T

  ⇒ d(X t (y), X t (p)) → 0, t − T → ∞ ⇒ −T −T ss ⇒ y ∈ W T (W (p)) ⊂ W (p) (p) ss ss ss (p).

  P♦rt❛♥t♦✱ X ✳ ❊✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ s❡ x ∈ W ❡♥tã♦ d(X t (x), X t (p)) → 0, t → ∞ ✳ ❙❡❥❛ s = t − T ❛ss✐♠ s → ∞ ❡✱ d(X (x), X (p)) → 0, s → ∞ ⇒ s s

  ⇒ d(X t (x), X t (p)) → 0, t → ∞ ⇒ −T −T ⇒ d(X t (X (x)), X t (X (p))) → 0 ⇒ −T −T

  ⇒ d(X t (X −T (x)), X t (p)) → 0 ⇒ ss ⇒ X −T (x) ∈ W (p) ⇒ ss T (W (p)) = W (p) T (p) ss ss ⇒ x ∈ X T (W (p)). ss

  ▲♦❣♦✱ X ✳ ❆❣♦r❛✱ ❢❛③❡♥❞♦ ❛ r❡str✐çã♦ ❞❡ X ❛ W t❡♠♦s q✉❡✿ ss ss

  X T : W (p) → W (p) ❡✱ s s s

  | → E (DX T ) p E : E p p p t❛❧ q✉❡ s s ss (DX T ) p | E v = λ v, ∀v ∈ E . p 1 p s

  (p) p ❈♦♠♦ W t❡♠ ❞✐♠❡♥sã♦ ✉♠ ❡ é t❛♥❣❡♥t❡ ❛♦ s✉❜❡s♣❛ç♦ E ❣❡r❛❞♦ ♣❡❧♦ ss 1 (p)

  ❛✉t♦✈❡t♦r v t❡♠♦s q✉❡ W é ❤♦♠❡♦♠♦r❢♦ ❛ ✉♠ ✐♥t❡r✈❛❧♦ ❞❛ r❡t❛✳ ❆ss✐♠✱ T ) p | E s ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r (DX ♣♦s✐t✐✈❛ ♦✉ ♥❡❣❛t✐✈❛ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ✈❛❧♦r λ p ss 1 ✳ 1 (p) > 0 T | W

  ❈❛s♦✱ λ t❡r❡♠♦s q✉❡ f = X é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡✳ W ss (p) p=0 _

  γ

  • ❋✐❣✉r❛ ✶✳✶✺✳ ❱❛r✐❡❞❛❞❡ ❊stá✈❡❧ ❋♦rt❡

  ✸✸ ✶✳✷✳ ◆❖➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ❉■◆➶▼■❈❖❙ ss (p)

  ❈♦♥s✐❞❡r❡ q✉❡ p = 0 ♥♦ ✐♥t❡r✈❛❧♦ r❡❛❧ ❛♦ q✉❛❧ W é ❤♦♠❡♦♠♦r❢♦✱ ❝♦♠♦ ✐❧✉str❛❞♦ ♥❛ ❋✐❣✉r❛ ♦♥❞❡ + ✐♥❞✐❝❛ ❛ ♣❛rt❡ ♣♦s✐t✐✈❛ ❡ − ❛ ♣❛rt❡ ♥❡❣❛t✐✈❛ ❞♦ ✐♥t❡r✈❛❧♦ r❡❛❧✳ ❈♦♠♦ f(p) = f(0) = 0 ❡✱ f é ❝r❡s❝❡♥t❡ t❡♠♦s q✉❡ x ≤ 0 ⇒ f (x) ≤ 0

  ❡✱ t❛♠❜é♠ f(y) ≥ 0 ✱ ∀y ≥ 0✳ ❆ ❋✐❣✉r❛ ❞❡s❝r❡✈❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ór❜✐t❛ ❞❡ ❞♦✐s ♣♦♥t♦s x, y ♥❛ ss

  (p) ✈❛r✐❡❞❛❞❡ t❛✐s q✉❡ x < 0 < y✳ ◆♦t❡ q✉❡✱ ♥❡st❡ ❝❛s♦✱ ♦ s❛t✉r❛çã♦ ❞❡ W ♣❡❧♦ s

  (γ) ✢✉①♦ ♥♦s ❞á ✉♠❛ ✈❛r✐❡❞❛❞❡ ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠ ❛♥❡❧ ✭♦✉ ❝✐❧✐♥❞r♦✮✱ ❛ss✐♠ W s❡rá ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠ ❛♥❡❧✳ _ (p) x W ss

  • +

    y

    γ

  ❋✐❣✉r❛ ✶✳✶✻✳ ❱❛r✐❡❞❛❞❡ ❊stá✈❡❧ ◆♦ ❝❛s♦✱ q✉❡ f é ❞❡❝r❡s❝❡♥t❡ t❡♠♦s q✉❡ s❡ x ≤ 0 ❡♥tã♦ f(x) ≥ f(0) = 0 ss

  (p) ❡✱ y ≥ 0 ⇒ f(y) ≤ 0✳ ●❡♦♠❡tr✐❝❛♠❡♥t❡✱ ♦ s❛t✉r❛❞♦ ♣❡❧♦ ✢✉①♦ ❞❡ W s❡rá s

  (γ) ❤♦♠❡♦♠♦r❢♦ ❛ ✉♠❛ ❢❛✐①❛ ❞❡ ▼ö❡❜✐✉s✱ ♥❡st❡ ❝❛s♦ W s❡rá ❤♦♠❡♦♠♦r❢♦ ❛ ✉♠❛ ❢❛✐①❛ ❞❡ ▼ö❡❜✐✉s✳ W ss (p)

  _ x

f (x)

2

3

f (x)

f(x)

γ

  • ❋✐❣✉r❛ ✶✳✶✼

  ❈❛♣ít✉❧♦ ✷ ◆♦çõ❡s ❞❡ ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛ ❡ ❆s♣❡❝t♦s ❚♦♣♦❧ó❣✐❝♦s

  ◆❡st❡ ❝❛♣ít✉❧♦ ❛❜♦r❞❛r❡♠♦s ❝♦♥❝❡✐t♦s ✐♥tr♦❞✉tór✐♦s s♦❜r❡ t♦♣♦❧♦❣✐❛ ❛❧❣é❜r✐❝❛ ❡ ❛❧❣✉♥s ❛s♣❡❝t♦s t♦♣♦❧ó❣✐❝♦s✱ q✉❡ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ s❡rã♦ ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞♦ tr❛❜❛❧❤♦✳

  ✷✳✶ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛

  ❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡ss❛ s❡çã♦ s❡rá ❞❡✜♥✐r ❡ ❛♣r❡s❡♥t❛r r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ♦ ♣r✐♠❡✐r♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s s♦❜r❡ ♦ ❛ss✉♥t♦ ✐♥❞✐❝❛♠♦s ❛s r❡❢❡rê♥❝✐❛s

  ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s✱ f, g : M → N ❞✉❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s ❡ I = [0, 1]✳ ❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❉✐③❡♠♦s q✉❡ f ❡ g sã♦ ❤♦♠♦tó♣✐❝❛s q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛

  H : M × I → N t❛❧ q✉❡ H(x, 0) = f(x) ❡ H(x, 1) = g(x) ∀x ∈ M✳ ❆ ❛♣❧✐❝❛çã♦ H é ❝❤❛♠❛❞❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ❡ g✳ ◆♦t❛çã♦✿ H : f ≃ g ♦✉ f ≃ g✳

  ❆ ❤♦♠♦t♦♣✐❛ ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ✉♠❛ ❞❡❢♦r♠❛çã♦ ❞❛ ❛♣❧✐❝❛çã♦ f ❛té ❛ ❛♣❧✐❝❛çã♦ g ♥✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ I = [0, 1]✱ ♦♥❞❡ ♥♦ t❡♠♣♦ ③❡r♦ t❡♠♦s ❛ ❛♣❧✐❝❛çã♦ f ❡ ♥♦ t❡♠♣♦ ✶ t❡♠♦s ❛ ❛♣❧✐❝❛çã♦ g✳

  ❱❛❧❡ r❡ss❛❧t❛r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ ❝♦♥tr❛❞♦♠í♥✐♦ ♥❛ ❤♦♠♦t♦♣✐❛✱ ♣♦✐s é ♥❡❧❡ q✉❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❞❡❢♦r♠❛çã♦ ♦❝♦rr❡✳ ❉❡✜♥✐çã♦ ✷✳✶✳✷✳ ❈♦♥s✐❞❡r❡ ❞♦✐s ❤♦♠❡♦♠♦r✜s♠♦ f : M → N ❡ g : M → N✳ ❉✐③❡♠♦s q✉❡ f ❡ g sã♦ ✐s♦tó♣✐❝❛s q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛

  H : M × I → N t❛❧ q✉❡ H(x, 0) = f(x) ❡ H(x, 1) = g(x) ∀x ∈ M ❡✱ ♣❛r❛ ❝❛❞❛ t ∈ I t❡♠✲s❡ q✉❡ H t : M → N

  é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✳

  ✸✺ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ ◆♦t❡ q✉❡ t♦❞❛ ✐s♦t♦♣✐❛ é t❛♠❜é♠ ✉♠❛ ❤♦♠♦t♦♣✐❛✳

  Pr♦♣♦s✐çã♦ ✷✳✶✳✸✳ ❆ r❡❧❛çã♦ ❞❡ ❤♦♠♦t♦♣✐❛✱ f ≃ g é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s ❞❡ M ❡♠ N✳ ❉❡♠♦♥str❛çã♦✳ ✶✳ ❘❡✢❡①✐✈❛✿

  ❙❡❥❛ f : M → N ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛✱ ❡♥tã♦ ❝♦♥s✐❞❡r❡✱ F : M × I → N ❞❛❞❛ ♣♦r✿ F (x, t) = f (x).

  ❆ss✐♠✱ F é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ❡ f✳ P♦rt❛♥t♦✱ ≃ é r❡✢❡①✐✈❛✳ ✷✳ ❙✐♠étr✐❝❛✿ ❙❡❥❛♠ f, g : M → N ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s✱ t❛✐s q✉❡ f ≃ g✱ ❧♦❣♦ ❡①✐st❡ ✉♠❛

  ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛✿ F : M × I → N t❛❧ q✉❡ F (x, 0) = f(x) ❡ F (x, 1) = g(x)✳

  ❉❡✜♥❛✱ K : M × I → N ❞❛❞❛ ♣♦r✿ K(x, t) = F (x, 1 − t).

  ❊♥tã♦✱ K(x, 0) = F (x, 1) = g(x) K(x, 1) = F (x, 0) = f (x)

  ❡✱ K é ❝♦♥tí♥✉❛✱ ♣♦✐s F é ❝♦♥tí♥✉❛✳ ▲♦❣♦✱ K é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ g ❡ f✱ ♦✉ s❡❥❛✱ g ≃ f✳ P♦rt❛♥t♦✱ ≃ é s✐♠étr✐❝❛✳ ✸✳❚r❛♥s✐t✐✈❛✿ ❙❡❥❛♠ f, g, h : M → N ❝♦♥tí♥✉❛s t❛✐s q✉❡ F : f ≃ g ❡ G : g ≃ h✳ ❊♥tã♦✿

  F (x, 0) = f (x), F (x, 1) = g(x) G(x, 0) = g(x), G(x, 1) = h(x). ❉❡✜♥❛

  L : M × I → N t❛❧ q✉❡ 1 F (x, 2t), , s❡ 0 ≤ t ≤ 2 L(x, t) = 1 ≤ t ≤ 1. G(x, 2t − 1), 1 s❡ 2

  ) = F (x, 1) = g(x) = H(x, 0) ❈♦♠♦ L(x, ✱ s❡❣✉❡ q✉❡ L é ❝♦♥tí♥✉❛✳ ❆ss✐♠✱ L 2

  é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ❡ h✳ ▲♦❣♦✱ ≃ é tr❛♥s✐t✐✈❛✳ P♦rt❛♥t♦✱ ≃ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳

  ✸✻ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ ′ ′ : M → N : N → S

  Pr♦♣♦s✐çã♦ ✷✳✶✳✹✳ ❙❡❥❛♠ f, f ❡ g, g ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s✳ ′ ′ ′ ′ ◦ f

  ❙❡ f ≃ f ❡ g ≃ g ❡♥tã♦ g ◦ f ≃ g ✳ ■st♦ é✱ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ❛♣❧✐❝❛çõ❡s ❤♦♠♦tó♣✐❝❛s ♣r❡s❡r✈❛ ❤♦♠♦t♦♣✐❛s✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠✿ F : M × I → N

  ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥t❡ f ❡ f ✱ ❡♥tã♦✿ F (x, 0) = f (x) (x)

  ❡ F (x, 1) = f ❡ G : N × I → S ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ g ❡ g ✱ ❛ss✐♠✿ G(x, 0) = g(y) (y).

  ❡ G(y, 1) = g ❉❡✜♥❛✱ L : M × I → S ❞❛❞❛ ♣♦r✿

  L(x, t) = G(F (x, t), t) L

  é ❝♦♥tí♥✉❛ ✭❝♦♠♣♦st❛ ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s✮ ❡✿ L(x, 0) = G(F (x, 0), 0) = G(f (x), 0) = g(f (x)) = g ◦ f (x) ′ ′ ′ ′ ′ L(x, 1) = G(F (x, 1), 1) = G(f (x), 1) = g (f (x)) = g ◦ f (x). ′ ′ ′ ′

  ◦ g ◦ f P♦rt❛♥t♦✱ L é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ◦ g ❡ f ✱ ♦✉ s❡❥❛✱ g ◦ f ≃ g ✳

  ❉❡✜♥✐çã♦ ✷✳✶✳✺✳ ❆s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ s❡❣✉♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❤♦♠♦t♦♣✐❛ sã♦ ❞❡♥♦♠✐♥❛❞❛s ❝❧❛ss❡s ❞❡ ❤♦♠♦t♦♣✐❛✳ n ❊①❡♠♣❧♦ ✷✳✶✳✻✳ ❯♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❝♦♥tí♥✉♦ t❛♥❣❡♥t❡ ❡♠ S é ✉♠❛ ❛♣❧✐❝❛çã♦ n n n +1

  → R ❝♦♥tí♥✉❛ v : S t❛❧ q✉❡ < x, v(x) >= 0 ∀x ∈ S ✳ ❙❡ ❡①✐st❡ ✉♠ ❝❛♠♣♦ n ❝♦♥tí♥✉♦ ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s ❡ ♥ã♦✲♥✉❧♦s ❡♠ S ❡♥tã♦ ❛ ❛♣❧✐❝❛çã♦ ❛♥tí♣♦❞❛ n n α : S → S

  é ❤♦♠♦tó♣✐❝❛ ❛ ✐❞❡♥t✐❞❛❞❡✳ n n +1 n → R

  ❉❡ ❢❛t♦✱ v : S ❝♦♥tí♥✉♦✱ t❛♥❣❡♥t❡ ❡ ♥ã♦✲♥✉❧♦ ❡♠ S ✱ ❞❡✜♥✐♠♦s ❛ n n → S

  ❛♣❧✐❝❛çã♦ f : S ❞❛❞❛ ♣♦r✿ x + v(x) f (x) = . k x + v(x) k n n

  ❊♥tã♦✱ f é ❝♦♥tí♥✉❛ ❡ f(x) 6= x✱ ∀x ∈ S ✭❛ss✐♠✱ f(x) 6= −α(x) ∀x ∈ S ✮✳ n ▲♦❣♦✱ (1 − t)f(x) + tα(x) 6= 0✱ ∀t ∈ I = [0, 1] ❡ ∀x ∈ S ✱ t❡♠♦s q✉❡✿ n n

  F : S × I → S (1 − t)f (x) + tα(x) F (x, t) = . k (1 − t)f (x) + tα(x) k

  ❙❡❣✉❡ q✉❡✱ f (x) F (x, 0) = = f (x) k f (x) k

  ✸✼ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ α(x) F (x, 1) = = α(x). k α(x) k

  ❆ss✐♠✱ F é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ❡ α✱ ❡♥tã♦ f ≃ α✳ P♦r ♦✉tr♦ ❧❛❞♦✱ n n × I → S

  ❞❡✜♥✐♥❞♦ H : S t❛❧ q✉❡ x + tv(x) H(x, t) = . k x + tv(x) k

  ▲♦❣♦✱ x n H(x, 0) = = id S k x k x + v(x) H(x, 1) = = f (x). n n k x + v(x) k

  H S S ≃ f é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ id ❡ f✳ ❊♥tã♦✱ f ≃ α ❡ id ✱ ❝♦♠♦ ≃ é ✉♠❛ n S ≃ α r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡ id ✳

  ❉❡✜♥✐çã♦ ✷✳✶✳✼✳ ❯♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ f : M → N ❝❤❛♠❛✲s❡ ✉♠❛ N ❡q✉✐✈❛❧ê♥❝✐❛ ❤♦♠♦tó♣✐❝❛ q✉❛♥❞♦ ❡①✐st❡ g : N → M ❝♦♥tí♥✉❛ t❛❧ q✉❡ g◦f ≃ id M ❡ f ◦ g ≃ id ✳ ❉✐③❡♠♦s q✉❡ g é ✉♠ ✐♥✈❡rs♦ ❤♦♠♦tó♣✐❝♦ ❞❡ f ❡ q✉❡ M ❡ N t❡♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❤♦♠♦t♦♣✐❛✳ ◆♦t❛çã♦✿ f : M ≡ N ♦✉ M ≡ N✳ 1

  × R ❊①❡♠♣❧♦ ✷✳✶✳✽✳ ❖ ❝✐❧✐♥❞r♦✱ S ✱ t❡♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❤♦♠♦t♦♣✐❛ ❞♦ ❝ír❝✉❧♦✱ 1 S

  ✳

  1

  {0} S x 1 1 ❋✐❣✉r❛ ✷✳✶ 1 1 1

  ×R ≡ S ×{0} ≡ S ×R → S ×{0} ❱❛♠♦s ♠♦str❛r q✉❡ S ✳ ❈♦♥s✐❞❡r❡✱ f : S

  ❞❡✜♥✐❞❛ ♣♦r 1 f (x, t) = (x, 0). 1 × {0} → S × R

  ❊✱ ❛ ✐♥❝❧✉sã♦ i : S ✳ ❉❡✜♥❛✿ 1 1 H : (S × R) × I → S × R t❛❧ q✉❡ H((x, t), s) = (x, ts).

  ✸✽ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ H

  é ❝♦♥tí♥✉❛ ❡✱ H((x, t), 0) = (x, 0) = (g ◦ f )(x, t) 1 H((x, t), 1) = id S (x, t). ×R S 1 ×R

  ❊♥tã♦✱ H é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ g ◦ f ❡ id ✳ ❆❧é♠ ❞✐ss♦✱ f ◦ i(x, 0) = 1 1 1 × R ≡ S × {0}.

  (x, 0) = id S (x, 0). ×{0} ▲♦❣♦✱ S 1 ❊①❡♠♣❧♦ ✷✳✶✳✾✳ ❆ ❢❛✐①❛ ❞❡ ▼ö❡❜✐✉s t❡♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ S ✳

  ❱❡r ❉❡✜♥✐çã♦ ✷✳✶✳✶✵✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ M é ❞✐t❛ ❝♦♥trát✐❧ q✉❛♥❞♦ t❡♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ ✉♠ ♣♦♥t♦✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✶✶✳ M é ❝♦♥trát✐❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡ id : M → M

  é ❤♦♠♦tó♣✐❝❛ ❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥st❛♥t❡ ❞❡ M✳ ❉❡♠♦♥str❛çã♦✳ (⇒) ❈♦♠♦ M é ❝♦♥trát✐❧ t❡♠♦s q✉❡ f : M → {x} é ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❤♦♠♦tó♣✐❝❛ ✭ x ∈ M✮✱ ❛ss✐♠ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ g : {x} → M M t❛❧ q✉❡ g é ♦ ✐♥✈❡rs♦ ❤♦♠♦tó♣✐❝♦ ❞❡ f✱ ❡♥tã♦ g ◦ f ≃ id ✳

  ▼❛s✱ g ◦ f : M → {x} ❧♦❣♦ g ◦ f é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥st❛♥t❡ ❞❡ M✳ ❙❡♥❞♦✱ g ◦ f ≃ id M ✱ s❡❣✉❡ q✉❡ ❛ id : M → M é ❤♦♠♦tó♣✐❝❛ ❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥st❛♥t❡

  ❡♠ M✳ (⇐) M

  ❙❡❥❛ f : M → {x}✱ x ∈ M✱ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥st❛♥t❡ t❛❧ q✉❡ f ≃ id ✳ M ≃ id M ◦id M M ≃ id M ❊♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ s❡❣✉❡ q✉❡ f◦id ✱ ♦✉ s❡❥❛✱ f◦id ✳ M M M M M ◦ f ≃ id ◦ id ◦ f ≃ id ❊✱ t❛♠❜é♠ q✉❡ id ✱ ❧♦❣♦ id ✳ M M M M ≃ id ◦ f ≃ id

  ❈♦♠♦ f ◦ id ❡ id ✱ t❡♠♦s q✉❡ f é ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❤♦♠♦tó♣✐❝❛ ❛ss✐♠ M ❡ {x} t❡♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❤♦♠♦t♦♣✐❛✳ ▲♦❣♦✱ M é ❝♦♥trát✐❧✳ n ❊①❡♠♣❧♦ ✷✳✶✳✶✷✳ R é ❝♦♥trát✐❧✳ n n n

  → R ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ R ✱ id : R ✱ ❡ ❛♣❧✐❝❛çã♦ ❝♦♥st❛♥t❡ n n n n

  → R g : R t❛❧ q✉❡ g(x) = p✱ ∀x ∈ R ✳ ❈♦♠♦ R é ❝♦♥✈❡①♦ s❡❣✉❡ q✉❡ ♦ s❡❣♠❡♥t♦ n

  (1 − t)x + tp ✱ t ∈ [0, 1] ❡stá ✐♥t❡✐r❛♠❡♥t❡ ❝♦♥t✐❞♦ ❡♠ R ✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❞❡✜♥✐r✿ n n

  × I → R H : R

  ❞❛❞❛ ♣♦r H(x, t) = (1 − t)x + tp.

  ❊♥tã♦✱ H(x, 0) = x = id(x) H(x, 1) = p = g(x). n n

  → R ❈♦♠♦ H é ❝♦♥tí♥✉❛ ❡♥tã♦ H é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ ❛ id : R ❡ n n n

  → R ❛♣❧✐❝❛çã♦ ❝♦♥st❛♥t❡ g : R ✱ ❛ss✐♠ ♣❡❧❛ Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡ R é ❝♦♥trát✐❧✳

  ✸✾ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ Pr♦♣♦s✐çã♦ ✷✳✶✳✶✸✳ ❙❡ M ♦✉ N é ❝♦♥trát✐❧ ❡♥tã♦ t♦❞❛ ❛♣❧✐❝❛çã♦ f : M → N é ❤♦♠♦tó♣✐❝❛ ❛ ✉♠❛ ❝♦♥st❛♥t❡✳ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ M s❡❥❛ ❝♦♥trát✐❧✱ ❡♥tã♦ ♣❡❧❛ Pr♦♣♦s✐çã♦ ❡①✐st❡ ✉♠❛ ❤♦♠♦t♦♣✐❛✿ M H : M × I → M ❡♥tr❡ ❛ id ❡ ❛♣❧✐❝❛çã♦ ❝♦♥st❛♥t❡ g : M → M✱ t❛❧ q✉❡ g(x) = p✱ ∀x ∈ M✳ ❚❡♠♦s q✉❡✱

  H(x, 0) = id M (x) = x, ∀x ∈ M H(x, 1) = g(x) = p, ∀x ∈ M. ❙❡❥❛ f : M → N ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ q✉❛❧q✉❡r✳ ❈♦♥s✐❞❡r❡✿

  F : M × I → N ❞❛❞❛ ♣♦r F (x, t) = (f ◦ H)(x, t) = f (H(x, t)).

  ❆ss✐♠✱ F (x, 0) = f (H(x, 0)) = f (x) F (x, 1) = f (H(x, 1)) = f (p).

  ▲♦❣♦✱ F é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ❡ ❛ ❝♦♥st❛♥t❡ f(p)✳ ❆❣♦r❛✱ s❡ ❝♦♥s✐❞❡r❛♠♦s N ❝♦♥trát✐❧ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❤♦♠♦t♦♣✐❛✿ K : N ×I →

  N N ❡♥tr❡ ❛ id ❡ ❛ ❛♣❧✐❝❛çã♦ ❝♦♥st❛♥t❡ c : N → N✱ t❛❧ q✉❡ c(x) = q ∀x ∈ N✳

  ❉❡✜♥❛✿ L : M × I → N t❛❧ q✉❡

  L(x, t) = K(f ((x), t). ❊♥tã♦✱

  L(x, 0) = K(f (x), 0) = id N (f (x)) = f (x) L(x, 1) = K(f (x), 1) = c(f (x)) = q. ❙❡❣✉❡ q✉❡✱ L é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ❡ ✉♠❛ ❝♦♥st❛♥t❡✳ P♦rt❛♥t♦✱ s❡ M ♦✉

  N é ❝♦♥trát✐❧ t❡♠♦s q✉❡ q✉❛❧q✉❡r ❛♣❧✐❝❛çã♦ f : M → N ❝♦♥tí♥✉❛ é ❤♦♠♦tó♣✐❝❛ ❛

  ✉♠❛ ❝♦♥st❛♥t❡✳ ❉✐r❡♠♦s q✉❡ (M, A) é ✉♠ ♣❛r ❞❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s q✉❛♥❞♦ A ⊂ M ❢♦r

  ✉♠ s✉❜❡s♣❛ç♦ ❞❡ M✳❉❛❞♦s ♦s ♣❛r❡s (M, A) ❡ (N, B) ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ f : (M, A) → (N, B) t❛❧ q✉❡ f(A) ⊂ B✳

  ❉❡✜♥✐çã♦ ✷✳✶✳✶✹✳ ❉❛❞❛s ❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s f, g : (M, A) → (N, B) ✉♠❛ ❤♦♠♦t♦♣✐❛ ❞❡ ♣❛r❡s ❡♥tr❡ f ❡ g é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛✿

  H : (M × I, A × I) → (N, B) t❛❧ q✉❡ H(x, 0) = f(x)✱ H(x, 1) = g(x) ∀x ∈ M✳

  ✹✵ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ ❉❡✜♥✐çã♦ ✷✳✶✳✶✺✳ ❉❛❞❛s f, g : M → N ❝♦♥tí♥✉❛s✱ ❞✐③❡♠♦s q✉❡ f é ❤♦♠♦tó♣✐❝❛ ❛ g r❡❧❛t✐✈❛♠❡♥t❡ ❛ ✉♠ s✉❜❡s♣❛ç♦ A ⊂ M q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❤♦♠♦t♦♣✐❛ H : f ≃ g t❛❧ q✉❡✿

  H(x, t) = f (x) = g(x), ∀x ∈ A. ❉❡♥♦t❛♠♦s ♣♦r✿ f ≃ g (rel A)✳

  P♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ ♣❛r❛ ❡①✐st✐r ✉♠❛ ❤♦♠♦t♦♣✐❛ r❡❧❛t✐✈❛ ❛ A ⊂ M ❡♥tr❡

  f, g : M → N é ♥❡❝❡ssár✐♦ q✉❡ f(x) = g(x), ∀x ∈ A✳ n n

  −{0} → R −{0} ❊①❡♠♣❧♦ ✷✳✶✳✶✻✳ ❆ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡ id : R é ❤♦♠♦tó♣✐❝❛ n n x

  − {0} → R − {0} à ❛♣❧✐❝❛çã♦ r❛❞✐❛❧ r : R ❞❛❞❛ ♣♦r r(x) = r❡❧❛t✐✈❛♠❡♥t❡ ❛♦ n −1 kxk s✉❜❡s♣❛ç♦ S ✳ x n

  ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡ ♦ s❡❣♠❡♥t♦ ❞❛❞♦ ♣♦r (1−t)x+t ✱ ❝♦♠ x ∈ R ❡ t ∈ [0, 1]✱ n n kxk ❡stá ✐♥t❡✐r❛♠❡♥t❡ ❝♦♥t✐❞♦ ❡♠ R ✭ ✉♠❛ ✈❡③ q✉❡ R é ❝♦♥✈❡①♦✮✳ x n ❆✜r♠❛çã♦✿ (1 − t)x + t ♥ã♦ ❝♦♥té♠ ❛ ♦r✐❣❡♠ ❞❡ R ✱ ✐st♦ é✱ ❡stá ❝♦♥t✐❞♦ n kxk

  − {0} ❡♠ R ✳ x n

  ❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ (1 − t)x + t ❝♦♥té♠ ❛ ♦r✐❣❡♠ ❞❡ R ✱ ❡♥tã♦✿ kxk x (1 − t)x + t = 0 ♣❛r❛ ❛❧❣✉♠ t ∈ [0, 1]. k x k

  ❊♥tã♦✱

  1

  1 x((1 − t) + t ) = 0 ⇒ ((1 − t) + t ) = 0. k x k k x k

  ❙❡ k x k= 1 ❡♥tã♦ 1 = 0 ✉♠ ❛❜s✉r❞♦✳ ▲♦❣♦ k x k6= 1✳ ❆ss✐♠✱ k x k t = . k x k −1

  ❈❛s♦ ✶✳ ❙❡ k x k< 1 t❡rí❛♠♦s q✉❡ (k x k −1) < 0 ❛ss✐♠ t < 0✱ ✉♠ ❛❜s✉r❞♦ ♣♦✐s t ∈ [0, 1] ✳

  ❈❛s♦ ✷✳ ❙❡ k x k> 1 ❡♥tã♦✿ 0 <k x k −1 <k x k k x k −1

  < 1 k x k k x k > 1. k x k −1

  ▲♦❣♦✱ t > 1 ✭✉♠ ❛❜s✉r❞♦ ♣♦✐s t ∈ [0, 1]✮✳ x n − {0}

  ❚❡♠♦s q✉❡✱ (1 − t)x + t ❡stá ❝♦♥t✐❞♦ ❡♠ R ✳ kxk

  ✹✶ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ ❆ss✐♠ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛✿ n n

  H : R − {0} × I → R − {0} t❛❧ q✉❡ x H(x, t) = (1 − t)x + t . k x k

  ❊♥tã♦✱ H(x, 0) = x = id(x) x

  H(x, 1) = = r(x). k x k n n n

  − {0} → R − {0} − {0} → R n ▲♦❣♦✱ H é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ id : R ❡ r : R − {0}

  ✳ n −1 ❆❣♦r❛✱ s❡ x ∈ S t❡♠♦s q✉❡ id(x) = r(x) ❡✱ n −1 H(x, t) = x = id(x) = r(x), ∀x ∈ S . n −1

  ) P♦rt❛♥t♦✱ id ≃ r (rel S ✳

  ❉❡✜♥✐çã♦ ✷✳✶✳✶✼✳ ❙❡❥❛ A ⊂ M✱ A é ✉♠ r❡tr❛t♦ ❞❡ M s❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ s♦❜r❡❥❡t✐✈❛ r : M → A t❛❧ q✉❡ ❝♦♠ ❛ ✐♥❝❧✉sã♦ i : A → M ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛✿ r //

  M A OO >> i id A A A ❖✉ s❡❥❛✱ r ◦ i = id ✳ ❆ ❢✉♥çã♦ r é ❞✐t❛ r❡tr❛çã♦ ❡♥tr❡ M ❡ A✳

  ❉❡✜♥✐çã♦ ✷✳✶✳✶✽✳ ❙❡❥❛ A ⊂ M✳ A é ❞✐t♦ r❡tr❛t♦ ♣♦r ❞❡❢♦r♠❛çã♦ ❞❡ M s❡ ❡①✐st❡ ✉♠❛ r❡tr❛çã♦ r : M → A t❛❧ q✉❡ i ◦ r ≃ id M . i : A → M

  ❞❡♥♦t❛ ✐♥❝❧✉sã♦ ❞❡ A ❡♠ M✳ 1 1 × {0} ⊂ S × R

  ❊①❡♠♣❧♦ ✷✳✶✳✶✾✳ S é ✉♠ r❡tr❛t♦ ♣♦r ❞❡❢♦r♠❛çã♦ ❞♦ ❝✐❧✐♥❞r♦✱ 1 S × R ✳

  ❇❛st❛ ❝♦♥s✐❞❡r❛r r = f ♥♦ ❡①❡♠♣❧♦ ❖❜s❡r✈❛çã♦ ✷✳✶✳✷✵✳ ❙❡ A ⊂ M é ✉♠ r❡tr❛t♦ ♣♦r ❞❡❢♦r♠❛çã♦ ❞❡ M✱ ❡♥tã♦ M ≡ A

  ✳ M A ❉❡ ❢❛t♦✱ t❡♠♦s q✉❡ i ◦ r ≃ id ❡ r ◦ i = id ✱ ♦♥❞❡ r : M → A ❞❡♥♦t❛ ❛ ❢✉♥çã♦ r❡tr❛çã♦ ❡ i ❞❡♥♦t❛ ❛ ✐♥❝❧✉sã♦ ❞❡ A ❡♠ M✳ ▲♦❣♦✱ M ≡ A✳

  ✹✷ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆

  ✷✳✶✳✶ ●r✉♣♦ ❋✉♥❞❛♠❡♥t❛❧

  ❉❡✜♥✐çã♦ ✷✳✶✳✷✶✳ ❉✐r❡♠♦s q✉❡ ❞♦✐s ❝❛♠✐♥❤♦s α, β : I → M sã♦ ❝❛♠✐♥❤♦s ❤♦♠♦tó♣✐❝♦s ❝♦♠ ❡①tr❡♠♦s ✜①♦s q✉❛♥❞♦ t✐✈❡r♠♦s α ≃ β (rel ∂I)✳ ❆ss✐♠ ✉♠❛ ❤♦♠♦t♦♣✐❛ H : α ≃ β é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛✿

  H : I × I → M t❛❧ q✉❡ H(s, 0) = α(s), H(s, 1) = β(s)

  H(0, t) = α(0) = β(0) H(1, t) = α(1) = β(1). P❛r❛ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ t, s ∈ I✳ ❉❡♥♦t❛r❡♠♦s ♣♦r [α] ❛ ❝❧❛ss❡ ❞❡ ❤♦♠♦t♦♣✐❛

  ❞♦ ❝❛♠✐♥❤♦ α✱ ✐st♦ é✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s q✉❡ ♣♦ss✉❡♠ ❛s ♠❡s♠❛s ❡①tr❡♠✐❞❛❞❡s ❞❡ α ❡ q✉❡ sã♦ ❤♦♠♦tó♣✐❝♦s ❛ α ❝♦♠ ❡①tr❡♠♦s ✜①♦s ❞✉r❛♥t❡ ❛ ❤♦♠♦t♦♣✐❛✳

  ❖❜s❡r✈❡ q✉❡ ♣❛r❛ ❞♦✐s ❝❛♠✐♥❤♦s α ❡ β s❡❥❛♠ ❝❛♠✐♥❤♦s ❤♦♠♦tó♣✐❝♦s é ♥❡❝❡ssár✐♦ q✉❡ t❡♥❤❛♠ ❛ ♠❡s♠❛ ❡①tr❡♠✐❞❛❞❡✱ ♦✉ s❡❥❛✱ α(0) = β(0) ❡ α(1) = β(1)✳

  ❯♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ é q✉❛♥❞♦ ♦s ❝❛♠✐♥❤♦s α ❡ β sã♦ ❢❡❝❤❛❞♦s ✭✐st♦ é✱ α(0) = β(0) = α(1) = β(1) = x ✮✱ ❡♥tã♦ α ≃ β s❡ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛✿

  H : I × I → M t❛❧ q✉❡ H(s, 0) = α(s), H(s, 1) = β(s) H(0, t) = H(1, t) = x . ❉❡✜♥✐çã♦ ✷✳✶✳✷✷✳ ❙❡❥❛♠ α, β : I → M ❝❛♠✐♥❤♦s q✉❡ ♦ ✜♠ ❞❡ α ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ❡①tr❡♠✐❞❛❞❡ ❞❡ β✱ ♦✉ s❡❥❛✱ α(1) = β(0)✳ ❉❡✜♥✐♠♦s ♦ ♣r♦❞✉t♦ αβ ♦✉ ❥✉st❛♣♦s✐çã♦ ❝♦♠♦ s❡♥❞♦ ♦ ❝❛♠✐♥❤♦ q✉❡ ❝♦♥s✐st❡ ❡♠ ♣❡r❝♦rr❡r α ❡ ❞❡♣♦✐s β✳ ❆ss✐♠ αβ : I → M é ♦ ♥♦✈♦ ❝❛♠✐♥❤♦ ❞❛❞♦ ♣♦r✿ 1

  α(2s), , s❡ 0 ≤ s ≤ 2 αβ(s) = 1 β(2s − 1), ≤ s ≤ 1. s❡ 2

  ❈♦♠♦ α(1) = β(0) ❛s r❡❣r❛s ❛❝✐♠❛ ❞❡✜♥❡♠ ❜❡♠ ✉♠❛ ❛♣❧✐❝❛çã♦ αβ : I → M 1 1 , t❛❧ q✉❡ αβ| [0, ] ❡ αβ| [ 1] sã♦ ❝♦♥tí♥✉❛s✳ ❊♥tã♦✱ αβ é ❝♦♥tí♥✉❛ ❡ ❛ss✐♠ é ✉♠ 2 2 ❝❛♠✐♥❤♦ q✉❡ ❝♦♠❡ç❛ ❡♠ α(0) ❡ t❡r♠✐♥❛ ❡♠ β(1)✳ −1

  : I → M ❉❡✜♥✐çã♦ ✷✳✶✳✷✸✳ ❖ ❝❛♠✐♥❤♦ ✐♥✈❡rs♦ ❞❡ α : I → M é ♦ ❝❛♠✐♥❤♦ α ❞❛❞♦ ♣♦r✿ −1 α (s) = a(1 − s), 0 ≤ s ≤ 1. −1

  = α◦j ❙❡❥❛ j : I → I ❛ ❢✉♥çã♦ ❞❛❞❛ ♣♦r j(s) = 1−s✳ ❊♥tã♦ α ✳ ❉❡♥♦t❛r❡♠♦s x x (s) = x

  ♣♦r ε ♦ ❝❛♠✐♥❤♦ ❝♦♥st❛♥t❡ ❡♠ x ∈ M✱ ✐st♦ é✱ ε ✱ ∀s ∈ I✳ ❙✉❛ ❝❧❛ss❡ ❞❡ x ] ❤♦♠♦t♦♣✐❛ s❡rá ❞❡♥♦t❛❞❛ ♣♦r [ε ✳

  ✹✸ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ ❙❡❥❛♠ α, β : I → M ❝❛♠✐♥❤♦s t❛✐s q✉❡ α(1) = β(0)✱ ❡♥tã♦✿ ′ ′ ′ ′ ′ −1 −1

  β ≃ (α ) Pr♦♣♦s✐çã♦ ✷✳✶✳✷✹✳ ❙❡ α ≃ α ❡ β ≃ β ❡♥tã♦ αβ ≃ α ❡ α ✳ ′ ′ ❉❡♠♦♥str❛çã♦✳ ❙❡ H : α ≃ α ❡ K : β ≃ β sã♦ ❤♦♠♦tó♣✐❝♦s✳ ❉❡✜♥❛

  L : I × I → M ❞❛❞❛ ♣♦r✿ 1 H(2s, t), , t ∈ I s❡ 0 ≤ s ≤ 2 L(s, t) = 1 K(2s − 1, t), ≤ s ≤ 1 t ∈ I. s❡ 2

  ❈♦♠♦ H(1, t) = K(0, t) = α(1) = β(0)✱ ∀t ∈ I✱ t❡♠♦s q✉❡ L ❡stá ❜❡♠ 1 1 , ❞❡✜♥✐❞❛✳ ❆❧é♠ ❞✐ss♦✱ L ([0, ]×I) ❡ L ([ 0]×I) sã♦ ❝♦♥tí♥✉❛s ❡♥tã♦ L é ❝♦♥tí♥✉❛ ❡♠ 2 2 I × I

  ✳ ❚❡♠♦s q✉❡✿

  1 L(s, 0) = H(2s, 0) = α(s), s ∈ [0, ]

  2

  1 L(s, 0) = K(2s − 1, 0) = β(s), s ∈ [0, ].

  2 ❆ss✐♠✱ L(s, 0) = αβ(s)✳ ′ ′

  β (s) ❆♥❛❧♦❣❛♠❡♥t❡✱ t❡♠✲s❡ q✉❡ L(s, 1) = α ✳ ❊✱ ′ ′ ′ ′ L(0, t) = H(0, t) = α (0) = α(0) = β(0) = αβ(0) = β (0) = α β (0).

  ❚❡♠♦s q✉❡✱ ′ ′ ′ ′ ′ ′ L(0, t) = αβ(0) = α β (0) L(1, t) = H(1, t) = α (1) = α(1) = β(1) = αβ(1) = β (1) = α β (1).

  ❊♥tã♦✱ ′ ′ L(0, t) = αβ(1) = α β (1). ′ ′ β

  ❈♦♠ ✐ss♦ ❝♦♥❝❧✉í♠♦s q✉❡ L é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ αβ ❡ α ✳ P♦rt❛♥t♦✱ ′ ′ αβ ≃ α β

  ✳ ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡✿

  G : I × I → M ❞❡✜♥❛❞❛ ♣♦r G(s, t) = H(1 − s, t).

  ❆ss✐♠✱ −1 G(s, 0) = H(1 − s, 0) = α −1

  G(s, 1) = H(1 − s, 1) = (α ) −1 ′ ′ G(0, t) = H(1, t) = α(1) = α (0) = α (1) = (α )−1(0) −1 ′ ′ G(1, t) = H(0, t) = α(0) = α (1) = α (0) = (α )−1(1).

  ✹✹ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ −1 −1 )

  ❈♦♠♦ G é ❝♦♥tí♥✉❛ t❡♠♦s q✉❡ G é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ α ❡ (α ✳ −1 −1 ≃ (α )

  P♦rt❛♥t♦✱ α ✳ ❈♦♥s✐❞❡r❡ ❛s ❝❧❛ss❡s [α] ❡ [β] ❞♦s ❝❛♠✐♥❤♦s ❡♠ M q✉❡ t❡♠ ♦r✐❣❡♠ ♥✉♠ ♣♦♥t♦ x ∈ M

  ❡ t❡r♠✐♥❛♠ ♥✉♠ ♣♦♥t♦ y ∈ M ❡ q✉❡ t❡♠ ♦r✐❣❡♠ ❡♠ y ❡ t❡r♠✐♥❛♠ ♥✉♠ ♣♦♥t♦ z ∈ M✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡✜♥✐çã♦ ✷✳✶✳✷✺✳ ❉❡✜♥✐♠♦s ❛ ❝❧❛ss❡ ❞♦ ♣r♦❞✉t♦ [αβ] ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s ❤♦♠♦tó♣✐❝♦s ❛♦ ❝❛♠✐♥❤♦ αβ✱ ♦♥❞❡ α ∈ [α] ❡ β ∈ [β]✳ P♦r ❞❡✜♥✐çã♦✿ [α][β] = [αβ]✳ −1 −1 ❉❡✜♥✐♠♦s t❛♠❜é♠✱ ❛ ❝❧❛ss❡ ✐♥✈❡rs❛ ❞❡ [α] ❝♦♠♦ s❡♥❞♦ ❛ ❝❧❛ss❡ ❞♦ ❝❛♠✐♥❤♦ α ]

  ✱ ♦♥❞❡ α ∈ [α]✱ ❞❡♥♦t❛❞❛ ♣♦r [α ❆ Pr♦♣♦s✐çã♦ ♠♦str❛ q✉❡ [αβ] ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞♦s ❝❛♠✐♥❤♦s −1

  α ∈ [α] ] ❡ β ∈ [β] ❡✱ t❛♠❜é♠ q✉❡ [α ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ α ∈ [α]✳ −1

  ] P♦rt❛♥t♦✱ [αβ] ❡ [α ❡stã♦ ❜❡♠ ❞❡✜♥✐❞♦s✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✷✻✳ ❙❡❥❛♠ α, β, γ : I → M ❝❛♠✐♥❤♦s t❛✐s q✉❡ ✉♠ ❞❡❧❡s t❡r♠✐♥❛♠ ♦♥❞❡ ♦ s❡❣✉✐♥t❡ ❝♦♠❡ç❛✳ ❈♦♥s✐❞❡r❡ s✉❛s ❝❧❛ss❡s ❞❡ ❤♦♠♦t♦♣✐❛ [α]✱ [α]✱ [γ] x y

  ❡ α(0) = x α(1) = y✱ ♦♥❞❡ x, y ∈ M✳ ❙❡❥❛♠ ε ❡ ε ♦s ❝❛♠✐♥❤♦s ❝♦♥st❛♥t❡s x ] y ] ❞❡ x ❡ ❞❡ y✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❡ [ε ✱ [ε ❛s ❝❧❛ss❡s ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ss❡s ❝❛♠✐♥❤♦s ❝♦♥st❛♥t❡s✳ ❊♥tã♦✿ −1

  ] = [ε x ] ✶✳ [α][α ✳ −1

  ][α] = [ε y ] ✷✳ [α ✳ x ][α] = [α] = [α][ε y ] ✸✳ [ε ✳ ✹✳ ([α][β])[γ] = [α]([β][γ])✳

  ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ ❝♦♥s✐❞❡r❛♠♦s ♥❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✱ ♦s ❝❛♠✐♥❤♦s α, β, γ : I → M

  ❢❡❝❤❛❞♦s t❛✐s q✉❡ α(0) = α(1) = β(0) = β(1) = γ(0) = γ(1) = x✱ ❝♦♠ x ∈ M ✱ t❡rí❛♠♦s q✉❡✿ −1

  [α][α ] = [ε x ] −1 [α ][α] = [ε x ] [ε x ][α] = [α] = [α][ε x ].

  ❆ss✐♠ ❝♦♠ ❡st❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❛ ♣r♦♣r✐❡❞❛❞❡ 4 ❞❛ Pr♦♣♦s✐çã♦ ❛s ❝❧❛ss❡s ❞❡ ❤♦♠♦t♦♣✐❛s ❞♦s ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s ♥✉♠ ♣♦♥t♦ x ∈ M ❝♦♥st✐t✉✐ ✉♠ ❣r✉♣♦ ❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡ ♣r♦❞✉t♦ ❞❡ ❝❛♠✐♥❤♦s✳ ❈♦♠ ✐ss♦ ♣♦❞❡♠♦s ❞❛r ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳

  ) ∈ M ❉❡✜♥✐çã♦ ✷✳✶✳✷✼✳ ❈♦♥s✐❞❡r❡ ♣❛r❡s ❞♦ t✐♣♦ (M, x ✱ x s❡rá ❝❤❛♠❛❞♦

  ) ♣♦♥t♦ ❜ás✐❝♦ ❞❡ M✳ ❖s ❝❛♠✐♥❤♦s α : (I, ∂I) → (M, x s❡rã♦ ❝❤❛♠❛❞♦s ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s ❝♦♠ ❜❛s❡ ♥♦ ♣♦♥t♦ x ✳ ❖ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦s ♣❡❧❛s ❝❧❛ss❡s ❞❡ ❤♦♠♦t♦♣✐❛s ❞❡ ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s ❝♦♠ ❜❛s❡ ♥♦ ♣♦♥t♦ x é ❝❤❛♠❛❞♦ ❣r✉♣♦ 1 (M, x )

  ❢✉♥❞❛♠❡♥t❛❧ ❞❡ M ❝♦♠ ❜❛s❡ ♥♦ ♣♦♥t♦ x ✱ ❞❡♥♦t❛❞♦ ♣♦r π ✳

  ✹✺ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ , x ∈ M 1 1

  ❚❡♦r❡♠❛ ✷✳✶✳✷✽✳ ❉❛❞♦s x ✱ s❡ x ❡ x ♣❡rt❡♥❝❡♠ ❛ ♠❡s♠❛ ❝♦♠♣♦♥❡♥t❡ 1 (M, x ) (M, x ) 1 1 ❝♦♥❡①❛ ♣♦r ❝❛♠✐♥❤♦s ❞❡ M ❡♥tã♦ π ❡ π sã♦ ✐s♦♠♦r❢♦s✳ 1

  ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ x ❡ x ♣❡rt❡♥❝❡♠ ❛ ♠❡s♠❛ ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①❛ ♣♦r ❝❛♠✐♥❤♦s ❞❡ M✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❝❛♠✐♥❤♦ γ : I → M q✉❡ ❧✐❣❛ x ❛ x −1 1 ✱ ♦✉ 1

  αγ s❡❥❛✱ γ(0) = x ❡ γ(1) = x ✳ ❙❡ α é ✉♠ ❝❛♠✐♥❤♦ ❢❡❝❤❛❞♦ ❞❡ x ❡♥tã♦ γ é ✉♠ 1 ❝❛♠✐♥❤♦ ❢❡❝❤❛❞♦ ❞❡ x ✳ −1 −1

  αγ ≃ γ α γ ❊♥tã♦✱ s❡ α ≃ α ✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡ γ ✳ ′ ′ −1 −1

  ] αγ] ≃ [γ α γ] ▲♦❣♦✱ s❡ [α] = [α s❡❣✉❡ q✉❡ [γ ✱ ♣♦rt❛♥t♦ ❞❛❞♦ ✉♠ ❡❧❡♠❡♥t♦ −1 −1 [α] ∈ π (M, x ) αγ] αγ] ∈ 1

  ❛ ❛♣❧✐❝❛çã♦ [α] 7−→ [γ ✐♥❞✉③ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ [γ π 1 (M, x 1 )

  ✳ ❉❡♥♦t❛r❡♠♦s ❡ss❛ ❛♣❧✐❝❛çã♦ ♣♦r✿ P γ : π (M, x ) → π (M, x ) 1 1 1 −1 ❞❛❞❛ ♣♦r 1 (M, x ) P ([α]) = [γ αγ]. γ

  ❉❛❞♦s [α], [β] ∈ π t❡♠♦s q✉❡ −1 P ([α][β]) = [γ ][α][β][γ] γ −1 −1

  = [γ ][α][γ][γ ][β][γ] −1 −1 = ([γ ][α][γ])([γ ][β][γ]) γ = P γ ([α])P γ ([β]).

  ❆ss✐♠✱ P é ✉♠ ❤♦♠♦♠♦r✜s♠♦✳ ❈♦♥s✐❞❡r❡ ❛ ❛♣❧✐❝❛çã♦✿ −1 P : π (M, x ) → π (M, x ) γ 1 −1 −1 1 1 t❛❧ q✉❡ −1 P ([β]) = [γβγ ]. γ γ

  ❚❡♠♦s q✉❡ P γ é ❛ ✐♥✈❡rs❛ ❞❡ P ✳ ❉❡ ❢❛t♦✱ ♣♦✐s −1 −1 −1 P ◦ P γ ([α]) = P ([γ αγ]) γ γ −1 −1

  = [γγ αγγ ] γ ◦ P ([β]) = [β] ◦ P γ = id π γ γ −1 −1 = [α]. 1 (M,x ) ❆♥❛❧♦❣❛♠❡♥t❡✱ t❡♠✲s❡ q✉❡ P ✳ ❊♥tã♦✱ P −1 γ π ◦ P = id 1 (M,x 1 )

  ❡ P γ ✳ γ P♦rt❛♥t♦✱ P é ✉♠ ✐s♦♠♦r✜s♠♦✳

  ❈♦r♦❧ár✐♦ ✷✳✶✳✷✾✳ ❙❡ M é ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s ❡♥tã♦ ♣❛r❛ q✉❛✐sq✉❡r ♣♦♥t♦s , x ∈ M (M, x ) (M, x ) 1 1 1 1

  ❜ás✐❝♦s x t❡♠✲s❡ q✉❡ π ❡ π sã♦ ✐s♦♠♦r❢♦s✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ M é ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s✱ ❡♥tã♦ ♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①❛ ♣♦r ❝❛♠✐♥❤♦s ✭♦ ♣ró♣r✐♦ M✮✱ ❛ ❝♦♥❝❧✉sã♦ s❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ❚❡♦r❡♠❛

  ❖ ❝♦r♦❧ár✐♦ ❛❝✐♠❛ ♠♦str❛ q✉❡ q✉❛♥❞♦ M é ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s ♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ✐♥❞❡♣❡♥❞❡ ❞♦ ♣♦♥t♦ ❜❛s❡ ❡s❝♦❧❤✐❞♦✱ ❛ss✐♠ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ♦ 1 (M )

  ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ♣♦r π s❡♠ ♥❡♥❤✉♠❛ r❡❢❡rê♥❝✐❛ ❡①♣❧í❝✐t❛ ❛♦ ♣♦♥t♦ ❜ás✐❝♦✳

  ✹✻ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ 1 (M, x) = {[ε x ]} x ] ❉❡✜♥✐çã♦ ✷✳✶✳✸✵✳ ❙❡ π ✱ ♣❛r❛ t♦❞♦ x ∈ M✱ ♦♥❞❡ [ε ❞❡♥♦t❛ ❛ ❝❧❛ss❡ ❞❡ ❤♦♠♦t♦♣✐❛ ❞♦ ❝❛♠✐♥❤♦ ❝♦♥st❛♥t❡ ♥♦ ♣♦♥t♦ x✱ ❞✐③❡♠♦s q✉❡ M é s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ t♦❞♦ ❝❛♠✐♥❤♦ ❢❡❝❤❛❞♦ α : I → M x

  ❝♦♠ ❜❛s❡ ♥✉♠ ♣♦♥t♦ x ∈ M t❡♠✲s❡ q✉❡ α ≃ ε ✳ ❊①❡♠♣❧♦ ✷✳✶✳✸✶✳ 1. ❙❡ M é ❝♦♥trát✐❧ ❡♥tã♦ M é s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦✳ n 2.

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

  ❉❡✜♥✐çã♦ ✷✳✶✳✸✷✳ ❯♠❛ ❛♣❧✐❝❛çã♦ f : M → N ❝♦♥tí♥✉❛ ✐♥❞✉③ ✉♠ = f (x )

  ❤♦♠♦♠♦r✜s♠♦✭y ✮✿ f : π (M, x ) → π (N, y ) # 1 1 ❞❡✜♥✐❞♦ ♣♦r✿ f # ([α]) = [f ◦ α] f #

  é ❝❤❛♠❛❞♦ ❤♦♠♦♠♦r✜s♠♦ ✐♥❞✉③✐❞♦ ♣❡❧❛ ❛♣❧✐❝❛çã♦ f✳ ❈♦♠♦ α ≃ α s❡❣✉❡ q✉❡ ❡①✐st❡ ✉♠❛ ❤♦♠♦t♦♣✐❛✿

  F : I × I → M t❛❧ q✉❡ F (s, 0) = α(s), F (s, 1) = α (s) F (0, t) = F (1, t) = x .

  ❈♦♥s✐❞❡r❡✿ H : I × I → N

  ❞❛❞❛ ♣♦r✿ H(s, t) = f ◦ F (s, t).

  H é ❝♦♥tí♥✉❛✱ ♣♦✐s é ❛ ❝♦♠♣♦st❛ ❞❡ ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s✳ ❊✱ t❛♠❜é♠✿

  H(s, 0) = f ◦ F (s, 0) = f ◦ α(s) H(s, 0) = f ◦ F (s, 0) = f ◦ α (s)

  H(0, t) = f ◦ F (0, t) = f (x ) = y H(1, t) = f ◦ F (1, t) = f (x ) = y . ❊♥tã♦✱ f ◦ α ≃ f ◦ α ✱ ❛ss✐♠ f # ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳ ❆❧é♠ ❞✐ss♦✱ f ◦ (αβ) = (f ◦ α)(f ◦ β)✳ ❉❡ ❢❛t♦✱ αβ : I → M t❛❧ q✉❡✿ 1

  α(2s), , s❡ 0 ≤ s ≤ 2 αβ(s) = 1 β(2s − 1), ≤ s ≤ 1. s❡ 2

  ✹✼ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ ▲♦❣♦✱ f ◦ (αβ) : I → N é ❞❛❞❛ ♣♦r✿ 1 f ◦ α(2s), , s❡ 0 ≤ s ≤ 2 f ◦ (αβ)(s) = 1 f ◦ β(2s − 1), ≤ s ≤ 1. s❡ 2

  ❆ss✐♠✱ f ◦ (αβ)(s) = (f ◦ α(s))(f ◦ β(s))✱ ∀s ∈ I✳ P♦rt❛♥t♦✱ f ◦ (αβ) = (f ◦ α)(f ◦ β) ([αβ]) = f ([α])f ([β]) # # # #

  ✳ ❊♥tã♦ f ✱ ❝♦♥❝❧✉í♠♦s q✉❡ f é ✉♠ ❤♦♠♦♠♦r✜s♠♦✳ # : π (M, x ) → π (M, x ) 1 1

  ❙❡ id : M → M é ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡ ❡♥tã♦ id é ♦ ❤♦♠♦♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡✳ ❖❜s❡r✈❛çã♦ ✷✳✶✳✸✸✳ ❚❡♠♦s q✉❡ s❡ h : M → N é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❡♥tã♦ h : π (M, x ) → π (N, y ) ) = y # 1 1

  ✱ h(x ✱ é ✉♠ ✐s♦♠♦r✜s♠♦✳ # : π 1 (M, x ) → π 1 (N, y ) ❉❡ ❢❛t♦✱❝♦♠♦ h : M → N é ❝♦♥tí♥✉❛ ❥á t❡♠♦s q✉❡ h ✱ h(x ) = y ([α]) = [h ◦ α] #

  ✱ ❞❛❞♦ ♣♦r h é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ✐♥❞✉③✐❞♦ ♣♦r h✳ −1 −1 : N → M −1 −1 ❙❡♥❞♦ h ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❡①✐st❡ h ❡ h é ❝♦♥tí♥✉❛✱ ❧♦❣♦ −1 h : π (N, y ) → π (M, x ) ([β]) = [h ◦ β] # ❞❛❞♦ ♣♦r h # é ✉♠ ❤♦♠♦♠♦r✜s♠♦ 1 −1 1

  ✐♥❞✉③✐❞♦ ♣♦r h ✳ ❊♥tã♦✱ −1 −1 h ◦ h ([α]) = h ([h ◦ α]) # # # −1

  = [h ◦ h ◦ α] = [α].

  ❊✱ −1 −1 h ◦ h ([β]) = h ([h ◦ β]) # # # −1 = [h ◦ h ◦ β] = [β]. # : π (M, x ) → π (M, y ) 1 1 −1 ▲♦❣♦✱ h é ✐♥✈❡rtí✈❡❧ ❡ s✉❛ ✐♥✈❡rs❛ é ❞❛❞❛ ♣♦r h : π 1 (N, y ) → π 1 (M, x ) # : π 1 (M, x ) → π 1 (M, y ) # ✳ P♦rt❛♥t♦✱ h é ✉♠

  ✐s♦♠♦r✜s♠♦✳ ❈♦♠ ✐ss♦✱ ♦❜s❡r✈❛✲s❡ q✉❡ ✈❛r✐❡❞❛❞❡s ❤♦♠❡♦♠♦r❢❛s ♣♦ss✉❡♠ ❣r✉♣♦s ❣r✉♣♦s ❢✉♥❞❛♠❡♥t❛✐s ✐s♦♠♦r❢♦s✳ ❉❡✜♥✐çã♦ ✷✳✶✳✸✹✳ ❉✐③❡♠♦s q✉❡ ❞♦✐s ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s α, βI → M sã♦ ❧✐✈r❡♠❡♥t❡ ❤♦♠♦tó♣✐❝♦s q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ H : I ×I → M t❛❧ q✉❡

  H(s, 0) = α(s), H(s, 1) = β(s) H(0, t) = H(1, t).

  P❛r❛ q✉❛✐sq✉❡r s, t ∈ I✳ ❆ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s✐❣♥✐✜❝❛ q✉❡ ♣❛r❛ t♦❞♦ t ∈ I ♦ t t : I → M (s) = H(s, t) ❝❛♠✐♥❤♦ H ❞❛❞♦ ♣♦r H é ✉♠ ❝❛♠✐♥❤♦ ❢❡❝❤❛❞♦✳ ◆♦t❡ q✉❡ ❛ r❡❧❛çã♦ ❞❡ ❝❛♠✐♥❤♦s ❧✐✈r❡♠❡♥t❡ ❤♦♠♦tó♣✐❝♦s é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳

  ✹✽ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ ❊①❡♠♣❧♦ ✷✳✶✳✸✺✳ ❖s ❝❛♠✐♥❤♦s α ❡ γ ♥♦ ❝✐❧✐♥❞r♦ sã♦ ❧✐✈r❡♠❡♥t❡ ❤♦♠♦tó♣✐❝♦s✱ ✈❡r ❋✐❣✉r❛

  γ α

  ❋✐❣✉r❛ ✷✳✷✳ ▲✐✈r❡♠❡♥t❡ ❍♦♠♦tó♣✐❝♦s ❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r r❡❧❛❝✐♦♥❛ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❝❛♠✐♥❤♦s ❧✐✈r❡♠❡♥t❡

  ❤♦♠♦tó♣✐❝♦s ❝♦♠ ❝❛♠✐♥❤♦s ❤♦♠♦tó♣✐❝♦s✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✸✻✳ ❙❡❥❛♠ α, β : I → M ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s✱ ❝♦♠ ❜❛s❡ ♥♦s ♣♦♥t♦s x ❡ y r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖s ❝❛♠✐♥❤♦s α ❡ β sã♦ ❧✐✈r❡♠❡♥t❡ ❤♦♠♦tó♣✐❝♦s −1 s❡ ❡ s♦♠❡♥t❡ s❡ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ γ : I → M✱ ❧✐❣❛♥❞♦ x ❛ y t❛❧ q✉❡ α ≃ γβγ ✳ ❖❜s❡r✈❛çã♦ ✷✳✶✳✸✼✳ ❙❡❥❛ M ❝♦♥❡①❛ ♣♦r ❝❛♠✐♥❤♦s ❡ α, β : I → M ❞♦✐s ❝❛♠✐♥❤♦s ❢❡❝❤❛❞♦s✱ ❝♦♠ ❜❛s❡ ♥♦s ♣♦♥t♦s x ❡ y✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ sã♦ ❧✐✈r❡♠❡♥t❡ ❤♦♠♦tó♣✐❝♦s ❡✱ ♦ ❝❛♠✐♥❤♦ β é ❤♦♠♦tó♣✐❝♦ ❛ ✉♠❛ ❝♦♥st❛♥t❡✳ ❊♥tã♦ α t❛♠❜é♠ é ❤♦♠♦tó♣✐❝♦ ❛ ✉♠❛ ❝♦♥st❛♥t❡✳ x y

  ❱❛♠♦s ❞❡♥♦t❛r ♦s ❝❛♠✐♥❤♦s ❝♦♥st❛♥t❡s ❡♠ x ❡ y ♣♦r ε ❡ ε ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ −1 P❡❧❛ Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡ α ≃ γβγ ✱ ♦♥❞❡ γ é ♦ ❝❛♠✐♥❤♦ q✉❡ ❧✐❣❛ ♦s ♣♦♥t♦s x ❡ y✱ ✐st♦ é✱ γ(0) = x ❡ γ(1) = y✳ ❆ss✐♠✱ −1 −1 −1 −1 β ≃ ε y ⇒ γβ ≃ cε y ⇒ γβγ ≃ γε y γ . y γ x x ≃ ε ≃ α

  ❈♦♠♦ γε ❡ γβγ t❡♠♦s q✉❡ α ≃ ε ✳ P♦rt❛♥t♦✱ α é ❤♦♠♦tó♣✐❝♦ ❛ ✉♠❛ ❝♦♥st❛♥t❡✳ ❈♦r♦❧ár✐♦ ✷✳✶✳✸✽✳ ❙❡❥❛♠ f, g : M → N ❝♦♥tí♥✉❛s ❡ ❤♦♠♦tó♣✐❝❛s✳ ❊♥tã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ✐♥❞✉③✐❞♦s✿ f : π (M, x) → π (N, y ) # 1 1 g # : π 1 (M, x) → π 1 (N, y 1 ).

  ◦g y = f (x) 1 = g(x) # = P γ # γ : π 1 (N, y 1 ) → ❡ y ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ♣♦r f ✱ ♦♥❞❡ P

  π (N, y ) 1 é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡✜♥✐❞♦ ♥❛ ❢♦r♠❛ ❞♦ ❚❡♦r❡♠❛

  ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ H : M × I → N ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ❡ g✱ ❡♥tã♦ H(x, 0) = f (x)

  ❡ H(x, 1) = g(x)✳ ❖ ❝❛♠✐♥❤♦ γ : I → M ❞❛❞♦ ♣♦r γ(t) = H(x, t)

  ✹✾ ✷✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ❉❊ ❚❖P❖▲❖●■❆ ❆▲●➱❇❘■❈❆ 1 é ✉♠ ❝❛♠✐♥❤♦ ❧✐❣❛♥❞♦ γ(0) = H(x, 0) = f(x) = y ❛ γ(1) = H(1, x) = g(x) = y ✳ P❛r❛ t♦❞♦ ❝❛♠✐♥❤♦ ❢❡❝❤❛❞♦ α : I → M ❝♦♠ ❜❛s❡ ❡♠ x ❝♦♥s✐❞❡r❡ ❛ ❛♣❧✐❝❛çã♦✿

  F : I × I → N (s, t) 7→ H(α(s), t).

  ❚❡♠✲s❡ q✉❡✱ F (s, 0) = H(α(s), 0) = f (a(s)) = f ◦ α(s)

  F (s, 1) = H(α(s), 1) = g(a(s)) = g ◦ α(s) F (0, t) = H(α(0), t) = H(x, t) = H(α(1), t) = F (1, t). ❆ss✐♠✱ f ◦ α ❡ g ◦ α sã♦ ❧✐✈r❡♠❡♥t❡ ❤♦♠♦tó♣✐❝♦s✳ P❡❧❛ Pr♦♣♦s✐çã♦ s❡❣✉❡ −1 q✉❡ f ◦ α ≃ γ(g ◦ α)γ ✳ ❊♥tã♦✱ f ([α]) = [f ◦ α] # −1

  = [γ(g ◦ α)γ ] −1 = γg ([α])γ # = P (g (α)). γ # # # = P γ ◦ g

  P♦rt❛♥t♦✱ f ✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ é ❝♦♠✉t❛t✐✈♦✳ f # // π 1 (M, x) π g # P γ 1 (N, y ) 88

  π (N, y ) 1 1 Pr♦♣♦s✐çã♦ ✷✳✶✳✸✾✳ ❙❡❥❛♠ M ❡ N ❝♦♥❡①♦s ♣♦r ❝❛♠✐♥❤♦s ❡ f : M → N é ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❤♦♠♦tó♣✐❝❛✱ ♦✉ s❡❥❛✱ M ❡ N t❡♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❤♦♠♦t♦♣✐❛✳ ❊♥tã♦✱ f : π (M, x ) → π (N, y ) # 1 1

  = f (x ) ❝♦♠ y é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ f : M → N ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❤♦♠♦tó♣✐❝❛ ❡ g : N → M M N = f (x ) ∈ M s❡✉ ✐♥✈❡rs♦ ❤♦♠♦tó♣✐❝♦✱ ❡♥tã♦ f ◦g = id ❡ g◦f = id ✳ ❙❡❥❛ x ✱ y ✱ x = g(y ) = f (x ) 1 1 1

  ❡ y ✳ ❈♦♥s✐❞❡r❡ ♦s ❤♦♠♦♠♦r✜s♠♦s ✐♥❞✉③✐❞♦s✿ f # : π 1 1 (M, x ) → π 1 (N, y ) f : π (M, x ) → π (N, y ) # 1 1 1 1 g : π (N, y ) → π (M, x ). # 1 1 1 M ◦f = P γ : π (M, x ) →

  ❈♦♠♦ g◦f ≃ id ✱ ♣❡❧❛ ❈♦r♦❧ár✐♦ t❡♠♦s q✉❡ g # # 1

  ✺✵ ✷✳✷✳ ❆❙P❊❈❚❖❙ ❚❖P❖▲Ó●■❈❖❙

  π (M, x ) γ 1 1 ✱ ♦♥❞❡ P é ♦ ✐s♦♠♦r✜s♠♦ ❞❡✜♥✐❞♦ ❞❛ ❢♦r♠❛ ❞♦ ❚❡♦r❡♠❛ ❝♦♠ γ 1

  ✉♠ ❝❛♠✐♥❤♦ ❧✐❣❛♥❞♦ x ❛ x ✳ N ◦ g = P δ : 1 # ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❛ ❤♦♠♦t♦♣✐❛ f ◦ g ≃ id ❝♦♥❝❧✉í♠♦s q✉❡ f #

  π (N, y ) → π (N, y ) 1 1 1 é ♦ ✐s♦♠♦r✜s♠♦ ❞❡✜♥✐❞♦ ❞❛ ❢♦r♠❛ ❞♦ ❚❡♦r❡♠❛ ❝♦♠ δ 1

  ✉♠ ❝❛♠✐♥❤♦ ❧✐❣❛♥❞♦ y ❛ y ✳ ❆ss✐♠✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦✳ f # // π 1 (M, x ) π P γ xx g # // 1 (N, y ) P δ π (M, x ) π (N, y ) 1 1 f # 1 1 1 1 # # # # ◦ f

  ◦ g ❙❡♥❞♦ g ✉♠ ✐s♦♠♦r✜s♠♦ s❡❣✉❡ q✉❡ g é s♦❜r❡❥❡t✐✈❛✳ ❊✱ f # é # # #

  ✐s♦♠♦r✜s♠♦ ❧♦❣♦ g é s♦❜r❡❥❡t✐✈❛✳ ❊♥tã♦ g é ✉♠ ✐s♦♠♦r✜s♠♦ ❡✱ ♣♦rt❛♥t♦ f ❡ 1 f (M, x ) ≈ π (N, y ) # sã♦ t❛♠❜é♠ ✐s♦♠♦r✜s♠♦s✳ ❈♦♥❝❧✉í♠♦s q✉❡ π 1 1 ✳

  ❈♦r♦❧ár✐♦ ✷✳✶✳✹✵✳ ❙❡ A ⊂ M é ✉♠ r❡tr❛t♦ ♣♦r ❞❡❢♦r♠❛çã♦ ❞❡ M✱ ❡♥tã♦ ❛ r❡tr❛çã♦ r : M → A✱ x ∈ A✱ ✐♥❞✉③ ✉♠ ✐s♦♠♦r✜s♠♦✿ π 1 (M, x) ≈ π 1 (A, x). ❉❡♠♦♥str❛çã♦✳ P❡❧❛ ♦❜s❡r✈❛çã♦ t❡♠ s❡ q✉❡ M ≡ A✱ ❡♥tã♦ ♣❡❧❛ Pr♦♣♦s✐çã♦ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✹✶✳ ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s ❝♦♥❡①❛s ♣♦r ❝❛♠✐♥❤♦s ❡♥tã♦ π 1 (M × N ) 1 (M ) × π 1 (N )

  ❡ π sã♦ ✐s♦♠♦r❢♦s✳ 1 ❊①❡♠♣❧♦ ✷✳✶✳✹✷✳ ❙❡❥❛ T ♦ t♦r♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ t❡♠♦s q✉❡ S ❡ T sã♦ ✈❛r✐❡❞❛❞❡s 1 1 (S ) = Z

  ❝♦♥❡①❛s ♣♦r ❝❛♠✐♥❤♦s✱ ❡ π ✭ ✈❡r 1 1 1 × S 1 (T ) = π (S ) × 1

  ❈♦♠♦ T = S ✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ s❡❣✉❡ q✉❡ π 1 π (S ) = Z × Z 1

  ✳

  ✷✳✷ ❆s♣❡❝t♦s t♦♣♦❧ó❣✐❝♦s 3

  ❊♠ t♦❞♦ tr❛❜❛❧❤♦ ❝❤❛♠❛r❡♠♦s ❞❡ ❜♦❧❛✱ às ✈❡③❡s ❞❡♥♦t❛❞❛ ♣♦r B ✱ ✉♠❛ 3 ✈❛r✐❡❞❛❞❡ ❤♦♠❡♦♠♦r❢❛ à ❜♦❧❛ ✉♥✐tár✐❛ tr✐❞✐♠❡♥s✐♦♥❛❧ ❞♦ R ✱ ❡ ✉♠ t♦r♦ só❧✐❞♦ 2 1 2

  × S s❡rá ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ tr✐❞✐♠❡♥s✐♦♥❛❧ ❤♦♠❡♦♠♦r❢❛ à D ✱ D ❞❡♥♦t❛ 2 ♦ ❞✐s❝♦ ✉♥✐tár✐♦ ❞❡ ❞✐♠❡♥sã♦ ✷✳ ❆ ❡s❢❡r❛ S s❡rá ✉♠❛ ✈❛r✐❡❞❛❞❡ ❤♦♠❡♦♠♦r❢❛ 3

  ❛ ❡s❢❡r❛ ✉♥✐tár✐❛ ❞❡ ❞✐♠❡♥sã♦ ✷ ❡♠ R ✳ ❆s ✈❛r✐❡❞❛❞❡s ❝♦♥s✐❞❡r❛❞❛s ❛q✉✐ s❡rã♦ ✸✲✈❛r✐❡❞❛❞❡s ❡ ❝♦♥❡①❛s✱ ❡①❝❡t♦ q✉❛♥❞♦ ❢♦r ♠❡♥❝✐♦♥❛❞♦ ♦ ❝♦♥trár✐♦✳ ❊st❛ s❡çã♦ t❡♠ ❝♦♠♦ ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ❉❡✜♥✐çã♦ ✷✳✷✳✶✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ é ❢❡❝❤❛❞❛ s❡ ❢♦r ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛ ❡ s❡♠ ❜♦r❞♦✳ ❊①❡♠♣❧♦ ✷✳✷✳✷✳ ❖ t♦r♦ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛✳ ❉❡✜♥✐çã♦ ✷✳✷✳✸✳ ❯♠❛ s✉♣❡r❢í❝✐❡ S ♠❡r❣✉❧❤❛❞❛ ❡♠ M s❡♣❛r❛ M s❡ M − S ♥ã♦ é ❝♦♥❡①❛✳

  ✺✶ ✷✳✷✳ ❆❙P❊❈❚❖❙ ❚❖P❖▲Ó●■❈❖❙

  ❊①❡♠♣❧♦ ✷✳✷✳✹✳ ❯♠ t♦r♦ T ♠❡r❣✉❧❤❛❞♦ ❡♠ M ❜♦r❞❛♥❞♦ ✉♠ t♦r♦ só❧✐❞♦ ✭ST ⊂ M

  ✮✱ s❡♣❛r❛ ❛ ✈❛r✐❡❞❛❞❡ M✱ ♣r♦❞✉③✐♥❞♦ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s A, B ♣❛r❛ M − T

  ✱ ♦♥❞❡ A = int(ST ) ❡ B = M − ST 2 ⊂ M

  ❉❡✜♥✐çã♦ ✷✳✷✳✺✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ M é ❞✐t❛ ✐rr❡❞✉tí✈❡❧ s❡ t♦❞❛ ❡s❢❡r❛ S é 3 ⊂ M

  ❜♦r❞♦ ❞❡ ✉♠❛ ❜♦❧❛ B ✳ ❙❡ M ♥ã♦ é ✐rr❡❞✉tí✈❡❧ ❞✐③❡♠♦s q✉❡ M é r❡❞✉tí✈❡❧✳ 3 ❊①❡♠♣❧♦ ✷✳✷✳✻✳ ❖ t♦r♦ só❧✐❞♦ ❡ S sã♦ ✈❛r✐❡❞❛❞❡s ✐rr❡❞✉tí✈❡✐s✳

  ❱❡r ❉❡✜♥✐çã♦ ✷✳✷✳✼✳ ❯♠❛ s✉♣❡r❢í❝✐❡ S ⊂ M é ❞✐t❛ tr❛♥s✈❡rs❛❧ ❛♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X x S

  ❡♠ M s❡ X(x) /∈ T ✱ ♣❛r❛ t♦❞♦ x ∈ S✳ m m = {x ∈ R ; x ≥ 0} m

  ❉❡✜♥✐çã♦ ✷✳✷✳✽✳ ❈♦♥s✐❞❡r❡ ♦ s❡♠✐✲❡s♣❛ç♦ s✉♣❡r✐♦r H ✳ k ❯♠❛ m−✈❛r✐❡❞❛❞❡ ❝♦♠ ❜♦r❞♦ ❞❡ ❝❧❛ss❡ C ✱ k ≥ 0✱ M✱ é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ i : U i → V i ⊂ H } m ❞❡ ❍❛✉s❞♦r✛✱ ❝♦♠ ❜❛s❡ ❡♥✉♠❡rá✈❡❧ ♠✉♥✐❞♦ ❞❡ ✉♠ ❛t❧❛s {Φ k ❝✉❥❛s ♠✉❞❛♥ç❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s sã♦ ❞❡ ❝❧❛ss❡ C ✳ i (x) ∈ m m ❖ ❜♦r❞♦ ❞❡ M✱ ❞❡♥♦t❛❞♦ ♣♦r ∂M✱ sã♦ ♦s ♣♦♥t♦s x ∈ M t❛✐s q✉❡ Φ ∂H = {x ∈ R ; x m = 0}

  ✳ ❖ ❜♦r❞♦ ❞❡ M t❡♠ ❞✐♠❡♥sã♦ m − 1 ❡ intM é ✉♠❛ ✈❛r✐❡❞❛❞❡ s❡♠ ❜♦r❞♦✱ ♠❛✐s ❞❡t❛❧❤❡s ✈❡r Pr♦♣♦s✐çã♦ ✷✳✷✳✾✳ ❙❡ M é ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❝♦♠ ❜♦r❞♦✱ ❡♥tã♦ ∂M é ✉♠❛ s✉♣❡r❢í❝✐❡✳ Pr♦♣♦s✐çã♦ ✷✳✷✳✶✵✳ ❙❡ M é ✉♠❛ m−✈❛r✐❡❞❛❞❡ ♦r✐❡♥tá✈❡❧ ❝♦♠ ❜♦r❞♦ ❡♥tã♦ ∂M t❛♠❜é♠ é ♦r✐❡♥tá✈❡❧✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r 1 2

  ❉❡✜♥✐çã♦ ✷✳✷✳✶✶✳ ✭✧❈♦❧❛❣❡♠ ❞❡ ✈❛r✐❡❞❛❞❡s✧✮ ❙❡❥❛♠ M ❡ M ✸✲✈❛r✐❡❞❛❞❡s 1 2 ❝♦♠ ❜♦r❞♦s✱ A ⊂ ∂M ❡ B ⊂ ∂M ❛❜❡rt♦s ❡ ❢❡❝❤❛❞♦s ❡ φ : A → B ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✱ ❡♥tã♦ ❡s❝r❡✈❡♠♦s✿

  [ M M = M/τ. 1 φ 2 M 1 M 2 A B M 1 M Φ 2 ❋✐❣✉r❛ ✷✳✸ 1 M S 2 A = φ

  ❖♥❞❡ M = M ✭✉♥✐ã♦ ❞✐s❥✉♥t❛✮ ❡ τ : A ∪ B → A ∪ B t❛❧ q✉❡ τ| ❡ −1 τ | B = φ

  ✳

  ✺✷ ✷✳✷✳ ❆❙P❊❈❚❖❙ ❚❖P❖▲Ó●■❈❖❙

  ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ M ❡ N sã♦ ✸✲✈❛r✐❡❞❛❞❡s ❝♦♠ ❜♦r❞♦ ❡ ∂M = ∂N ❡ S id : ∂M → ∂N N

  ❛ ✐❞❡♥t✐❞❛❞❡ ❡♥tã♦ M id é ✉♠❛ ✈❛r✐❡❞❛❞❡ s❡♠ ❜♦r❞♦✳ 1 2 × D

  ❊①❡♠♣❧♦ ✷✳✷✳✶✷✳ ❈♦♥s✐❞❡r❡ ❞♦✐s t♦r♦s só❧✐❞♦s S ✱ ♣♦❞❡♠♦s ❝♦❧á✲❧♦s ♣❡❧♦ 1 2 S 1 2 1 2 1 2 ×D S ×D ×D ) → ∂(S ×D )

  ❜♦r❞♦ ❡ ♣r♦❞✉③✐r ✉♠❛ ✈❛r✐❡❞❛❞❡ S ✱ id : ∂(S 1 2 id × S

  ❤♦♠❡♦♠♦r❢❛ ❛ S ✳ ◆❡st❡ ❝❛s♦✱ ❡st❛♠♦s ✐❞❡♥t✐✜❝❛♥❞♦ ♣❛r❛❧❡❧♦s ❝♦♠ ♣❛r❛❧❡❧♦s 1 2 1 1 × D × S

  ) = S ❡ ♠❡r✐❞✐❛♥♦s ❝♦♠ ♠❡r✐❞✐❛♥♦s ❡♠ ∂(S ✳ 1 2 S 1 2 1 2 1 2

  × D S × D × D ) → ∂(S × D ) ❊✱ S f ✱ f : ∂(S t❛❧ q✉❡ f(x, y) = (y, x) 3

  ♣r♦❞✉③ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❤♦♠❡♦♠♦r❢❛ ❛ S ✳ ❖ ❞✐❢❡♦♠♦r✜s♠♦ f ✐♥❞✐❝❛ q✉❡ ❛ ❝♦❧❛❣❡♠ 1 2 × D ) =

  é ❢❡✐t❛ ❛tr❛✈és ❞❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ♣❛r❛❧❡❧♦s ❝♦♠ ♠❡r✐❞✐❛♥♦s ❡♠ ∂(S 1 1 S × S ✳ ❱❡r 3

  ❊①❡♠♣❧♦ ✷✳✷✳✶✸✳ P♦❞❡♠♦s ♦❜t❡r S ♣❡❧❛ ❝♦❧❛❣❡♠ ❞❡ ❞✉❛s ❜♦❧❛s ♣❡❧♦ ❜♦r❞♦✳ ❱❡r ❉❡✜♥✐çã♦ ✷✳✷✳✶✹✳ ❙❡❥❛ S ✉♠❛ s✉♣❡r❢í❝✐❡ ❢❡❝❤❛❞❛ ♠❡r❣✉❧❤❛❞❛ ❡♠ M✳ ❉✐③❡♠♦s q✉❡ S é ✷✲❧❛❞♦s s❡ ❡①✐st❡ ✉♠ ♠❡r❣✉❧❤♦ h : S × [−1, 1] → M t❛❧ q✉❡ h(x, 0) = x✳

  ■♥t✉✐t✐✈❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❡♥①❡r❣❛r ❛ ♣r♦♣r✐❡❞❛❞❡ ✷✲❧❛❞♦s ❝♦♠♦ s❡♥❞♦ ✉♠❛ ✧❢♦❧❣❛✧q✉❡ ❛ s✉♣❡r❢í❝✐❡ S t❡♠ ♥❛ ✈❛r✐❡❞❛❞❡ M ♣❛r❛ s❡ ❞❡s❧♦❝❛r✱ q✉❡r ❞✐③❡r q✉❡ S

  ❛❞♠✐t❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❝♦♥❡①❛ ❡♠ M✳ ❖❜s❡r✈❛♠♦s✱ t❛♠❜é♠ q✉❡ ❡♠ ❝❛❞❛ t ∈ [−1, 1] ✜①❛❞♦ ❛ ✐♠❛❣❡♠ ❞♦ ♠❡r❣✉❧❤♦ h(S × {t}) é ❤♦♠❡♦♠♦r❢❛ ❛ s✉♣❡r❢í❝✐❡

  S ✳ ●❡♦♠❡tr✐❝❛♠❡♥t❡✱ é ❝♦♠♦ s❡ ❡♠ ❝❛❞❛ t ∈ [−1, 1] ❡♥①❡r❣❛r♠♦s ❛ s✉♣❡r❢í❝✐❡ S

  ❞❡♣♦✐s ❞❡ ✉♠ ❞❡s❧♦❝❛♠❡♥t♦ ♥❛ ✈❛r✐❡❞❛❞❡✳ ❆ ✐♠❛❣❡♠ ❞❡ S × [−1, 1] ♣❡❧♦ ♠❡r❣✉❧❤♦ h é ❝❤❛♠❛❞♦ ✉♠ ❜✐❝♦❧❛r ❞❡ S✳ ▼❛✐s

  ❣❡r❛❧✱ ❡♠ ✈❡③ ❞♦ ✐♥t❡r✈❛❧♦ [−1, 1] ♣♦❞❡♠♦s t❡r ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ I = [a − ǫ, a + ǫ]✱

  a, ǫ ∈ R ❡ ǫ > 0✱ t❛❧ q✉❡ h(x, a) = x✱ ✉♠❛ ✈❡③ q✉❡ t♦❞♦s ✐♥t❡r✈❛❧♦s ❢❡❝❤❛❞♦s sã♦

  ❞✐❢❡♦♠♦r❢♦s✳ ❖❜s❡r✈❛çã♦ ✷✳✷✳✶✺✳ ❙❡❥❛ T ✉♠ t♦r♦ ✷✲❧❛❞♦s ♥✉♠❛ 3−✈❛r✐❡❞❛❞❡ M ❡ h : T × [−1, 1] → M

  = M − h(T × (−1, 1)) ♦ ♠❡r❣✉❧❤♦ ❡♥tã♦ ❛ 3−✈❛r✐❡❞❛❞❡ M

  é ❝❤❛♠❛❞❛ ❞❡ ♦ r❡s✉❧t❛❞♦ ❞♦ ❝♦rt❡ ❞❡ M ❛♦ ❧♦♥❣♦ ❞❡ T ✱ ❡ M − h(T × [−1, 1]) é ❤♦♠❡♦♠♦r❢❛ ❛ M − T ✱ ❞❡♥♦t❛r❡♠♦s q✉❡ M − h(T × [−1, 1]) ∼ M − T ✳ ▼❛✐s ❞❡t❛❧❤❡s ✈❡r M

  • - M M +

  caso conexo caso desconexo

  ❋✐❣✉r❛ ✷✳✹✳ ❋✐❣✉r❛ ❜❛s❡❛❞❛ ❡♠ ❆♦ r❡❛❧✐③❛r♠♦s ❡st❡ ❝♦rt❡ ♣♦❞❡♠♦s ♦❜t❡r M ❝♦♥❡①❛ ♦✉ ♥ã♦✳ ❈❛s♦ s❡❥❛

  ❞❡s❝♦♥❡①❛ ♦❜t❡♠♦s ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s✳ ❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ♦❜t❡♠♦s ❝♦♠♦

  ✺✸ ✷✳✷✳ ❆❙P❊❈❚❖❙ ❚❖P❖▲Ó●■❈❖❙

  ❜♦r❞♦ ❞❡ M ❞✉❛s ❝ó♣✐❛s ❞♦ t♦r♦ T ✭s❡ M é ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❝♦♠ ❜♦r❞♦ ❡♥tã♦ ♣r♦❞✉③✐♠♦s ♠❛✐s ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ❜♦r❞♦ ♣❛r❛ M ✮✳

  ❆♦ ❝♦❧❛r♠♦s ♦s ❜♦r❞♦s ♣r♦❞✉③✐❞♦s ✈✐❛ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ ′ ′ ♥♦✈❛ 3−✈❛r✐❡❞❛❞❡ M ✭❡♠ ❣❡r❛❧✱ M ♥ã♦ é ❤♦♠❡♦♠♦r❢❛ ❛ M✮✱ ❡st❡ ❝❛s♦ t❛♠❜é♠ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ❝✐r✉r❣✐❛ ❞❡ ❉❡❤♥✳ ❱❡r ❉❡✜♥✐çã♦ ✷✳✷✳✶✻✳ ❙❡❥❛ S ✉♠❛ s✉♣❡r❢í❝✐❡✱ ❞✐③❡♠♦s q✉❡ S ❡stá ❞❡✈✐❞❛♠❡♥t❡ ♠❡r❣✉❧❤❛❞❛ ❡♠ M s❡ S ❡stá ♠❡r❣✉❧❤❛❞❛ ❡♠ M ❡ ∂M ∩ S = ∂S✳ 2 2

  ❉❡✜♥✐çã♦ ✷✳✷✳✶✼✳ ❯♠❛ s✉♣❡r❢í❝✐❡ S ⊂ M ✷✲❧❛❞♦s s❡♠ ❝♦♠♣♦♥❡♥t❡ S ♦✉ D é ❞✐t❛ s❡r ✐♥❝♦♠♣r❡ssí✈❡❧ s❡ ♣❛r❛ t♦❞♦ ❞✐s❝♦ D ⊂ M ❝♦♠ D ∩ S = ∂D ❡①✐st❡ ′ ′

  ⊂ S = ∂D ♦✉tr♦ ❞✐s❝♦ D ❝♦♠ ∂D ✳ ❙❡ S ♥ã♦ é ✐♥❝♦♠♣r❡ssí✈❡❧ ❞✐③❡♠♦s q✉❡ é ❝♦♠♣r❡ssí✈❡❧✳ S S D' D D

  ❋✐❣✉r❛ ✷✳✺ ❖❜s❡r✈❛çã♦ ✷✳✷✳✶✽✳ ❯♠❛ s✉♣❡r❢í❝✐❡ S ⊂ intM tr❛♥s✈❡rs❛❧ ❛ ✉♠ ❝❛♠♣♦ ❞❡ 1

  ✈❡t♦r❡s X ❞❡ ❝❧❛ss❡ C ♥✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ M é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✷✲❧❛❞♦s✳ ❈♦♥s✐❞❡r❡ ♦ ✢✉①♦ ϕ : (−ǫ, ǫ) × M → M ♦ ✢✉①♦ ❛ss♦❝✐❛❞♦ ❛♦ ❝❛♠♣♦ X✱ t❡♠♦s (−ǫ,ǫ)×S : (−ǫ, ǫ) × S → M q✉❡ ϕ| é ✉♠ ♠❡r❣✉❧❤♦✱ ✉♠❛ ✈❡③ q✉❡ S é tr❛♥s✈❡rs❛❧ ❛♦

  ❝❛♠♣♦ ❡ ❛ tr❛♥s✈❡rs❛❧✐❞❛❞❡ é ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❛❜❡rt❛✱ ✐st♦ é✱ é ♣r❡s❡r✈❛❞❛ ♥✉♠❛ ♣❡q✉❡♥❛ ✈✐③✐♥❤❛♥ç❛ ✭❚❡♦r❡♠❛ ❞❛ ❚r❛♥s✈❡rs❛❧✐❞❛❞❡ ♦✉ ❚❡♦r❡♠❛ ❞❡ ❚♦❤♠ ✮✳ Pr♦♣♦s✐çã♦ ✷✳✷✳✶✾✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ✷✲❧❛❞♦s S ⊂ M é ✐♥❝♦♠♣r❡ssí✈❡❧ s❡ ♦ ❤♦♠♦♠♦r✜s♠♦ ✐♥❞✉③✐❞♦ ♣❡❧❛ ✐♥❝❧✉sã♦ ❢♦r ✐♥❥❡t✐✈♦✳ 1 1 3

  × S ❊①❡♠♣❧♦ ✷✳✷✳✷✵✳ ❚♦❞♦ t♦r♦ T = S ❡♠ S é ❝♦♠♣r❡ssí✈❡❧✳ 3 3 1 (T ) = Z × Z (S ) = {0} 1

  ❉❡ ❢❛t♦✱ ❝♦♠♦ π ✱ ❡ S é s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①❛ ❧♦❣♦ π ✱ # : Z × Z → {0} ❡♥tã♦ ♦ ❤♦♠♦♠♦r✜s♠♦ ✐♥❞✉③✐❞♦ ♣❡❧❛ ❛♣❧✐❝❛çã♦ ✐♥❝❧✉sã♦✿ i ♥ã♦ é 3

  ✐♥❥❡t✐✈♦✳ P❡❧❛ Pr♦♣♦s✐çã♦ s❡❣✉❡ q✉❡ T ❡♠ S é ❝♦♠♣r❡ssí✈❡❧✳ ❖s s❡❣✉✐♥t❡s t❡♦r❡♠❛s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ n −1

  ❚❡♦r❡♠❛ ✷✳✷✳✷✶✳ ✭❚❡♦r❡♠❛ ❞❛ ❙❡♣❛r❛çã♦ ❞❡ ❇r♦✉✇❡r✮✿ ❙❡ K é n n − K t♦♣♦❧♦❣✐❝❛♠❡♥t❡ ✉♠❛ ❡s❢❡r❛ ❞❡ ❞✐♠❡♥sã♦ n − 1 ❡♠ R ✱ ❡♥tã♦ R t❡♠

  ❡①❛t❛♠❡♥t❡ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❡ K é ♦ ❜♦r❞♦ ❞❡ ❝❛❞❛ ✉♠❛✳ ❖ t❡♦r❡♠❛ ❛❝✐♠❛ ♣❛r❛ ❝❛s♦ n = 2 é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ t❡♦r❡♠❛ ❞❛ ❝✉r✈❛ ❞❡

  ❏♦r❞❛♥✳

  ✺✹ ✷✳✷✳ ❆❙P❊❈❚❖❙ ❚❖P❖▲Ó●■❈❖❙ 1 2

  → S ❚❡♦r❡♠❛ ✷✳✷✳✷✷✳ ✭❚❡♦r❡♠❛ ❞❡ ❙❝❤ö♥✢✐❡s✮ ❉❛❞♦ ✉♠ ♠❡r❣✉❧❤♦ f : S ✱ 2 1 2

  − f (S ) ♦ ❢❡❝❤♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ S é ❤♦♠❡♦♠♦r❢❛ ❛♦ ❞✐s❝♦ D ✳ Pr♦♣♦s✐çã♦ ✷✳✷✳✷✸✳ ❙❡❥❛ T ✉♠ t♦r♦ ✷✲❧❛❞♦s ❡♠ ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ M ✐rr❡❞✉tí✈❡❧✱ ❡♥tã♦ T é ✐♥❝♦♠♣r❡ssí✈❡❧ ♦✉ T é ❜♦r❞♦ ❞❡ ✉♠ t♦r♦ só❧✐❞♦ ♦✉ ❡stá ❝♦♥t✐❞♦ ❡♠ ✉♠❛ ❜♦❧❛ ❝♦♥t✐❞❛ ❡♠ M✳ ❉❡♠♦♥str❛çã♦✳ ❙❡ T é ✐♥❝♦♠♣r❡ssí✈❡❧ ❡♥tã♦ t❡♠♦s ♦ r❡s✉❧t❛❞♦✳ ❆❣♦r❛✱ s❡ T é ❝♦♠♣r❡ssí✈❡❧✱ ❡①✐st❡ ✉♠ ❞✐s❝♦ D ⊂ M ❝♦♠ D ∩ T = ∂D✱ t❛❧ q✉❡ ∂D ♥ã♦ ❜♦r❞❛ ✉♠ ❞✐s❝♦ ❡♠ T ✳ ❊♥tã♦ ♥✉♠❛ ♣❡q✉❡♥❛ ✈✐③✐♥❤❛♥ç❛ ❡♠ T t♦❞♦s ❝ír❝✉❧♦s ❜♦r❞❛♠ ✉♠ ❞✐s❝♦✱ r❡❛❧✐③❛♥❞♦ ✉♠❛ ❝✐r✉r❣✐❛ ❡♠ T ❛♦ ❧♦♥❣♦ ❞❡ss❡ ❞✐s❝♦ D ♦❜t❡♠♦s ✉♠❛ 2 s✉♣❡r❢í❝✐❡ ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠❛ ❡s❢❡r❛✱ ❝♦♠♦ M é ✐rr❡❞✉tí✈❡❧ ❞❡✈❡♠♦s t❡r q✉❡ S

  ❜♦r❞❛ ✉♠❛ ❜♦❧❛ B ❡♠ M✳ ❚❡♠♦s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿ D ∩ B = ∅ ♦✉ D ⊂ B✳ D D

  1 D D 2 B D

  1 D D 2 1º caso D D B 2º caso

  ❋✐❣✉r❛ ✷✳✻ ◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ✐st♦ é✱ D ∩ B = ∅✱ ❞❡s❢❛③❡♠♦s ❛ ❝✐r✉r❣✐❛ ❡ t❡♠♦s q✉❡ T é

  ❜♦r❞♦ ❞❡ ✉♠ t♦r♦ só❧✐❞♦✳ ◆♦ s❡❣✉♥❞♦ ❝❛s♦✱ q✉❛♥❞♦ D ⊂ B ❡♥tã♦ T ❡stá ❝♦♥t✐❞♦ ❡♠ B✳

  ❈♦♠♦ T é ✷✲❧❛❞♦s ♣♦❞❡♠♦s s✉♣♦r q✉❡ T ❡stá ♥♦ ✐♥t❡r✐♦r ❞❛ ❜♦❧❛ B✳ ❚❡♦r❡♠❛ ✷✳✷✳✷✹✳ ✭❚❡♦r❡♠❛ ❞♦ ❚♦r♦ ❙ó❧✐❞♦✮✳ ❙❡❥❛ T ✉♠ t♦r♦ 3

  ♠❡r❣✉❧❤❛❞♦ ❡♠ S ✱ ❡♥tã♦ T ❞✐✈✐❞❡ ❡st❡ ❡s♣❛ç♦ ❡♠ ❞✉❛s ♣❛rt❡s s❡♥❞♦ ♣❡❧♦ ♠❡♥♦s 2 1 × S

  ✉♠❛ ❞❡❧❛s ❤♦♠❡♦♠♦r❢❛ ❛♦ t♦r♦ só❧✐❞♦ D ✳ Pr♦♣♦s✐çã♦ ✷✳✷✳✷✺✳ ❙❡❥❛ S ✉♠❛ ❝♦❧❡çã♦ ✜♥✐t❛ ❞❡ s✉♣❡r❢í❝✐❡s ❞✐s❥✉♥t❛s ✐♥❝♦♠♣r❡ssí✈❡✐s✳ M é ✐rr❡❞✉tí✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡ M − S é ✐rr❡❞✉tí✈❡❧✳

  ✺✺ ✷✳✷✳ ❆❙P❊❈❚❖❙ ❚❖P❖▲Ó●■❈❖❙

  ❉❡♠♦♥str❛çã♦✳ (⇒) ❙❡❥❛ M ✐rr❡❞✉tí✈❡❧✱ ❡♥tã♦ t♦❞❛ ❡s❢❡r❛ ❡♠ M − S ⊂ M ❜♦r❞❛ ✉♠❛ ❜♦❧❛ B ❡♠ M✳ ❚❡♠♦s ❞♦✐s ❝❛s♦s ♣❛r❛ ❛♥❛❧✐s❛r ✿

  1. B ⊂ M − S ✱ ❡♥tã♦ ♦ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳

  2. B ❝♦♥té♠ S✱

  ◆♦ ❝❛s♦ 1 s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳ ◆♦ ❝❛s♦ 2✱ t❡rí❛♠♦s S ⊂ B ⊂ M ❛ss✐♠ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ S é ❝♦♠♣r❡ssí✈❡❧ ❡♠ B ✭✉♠❛ ✈❡③ q✉❡ B é ❝♦♥trát✐❧✮ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ S s❡rá ❝♦♠♣r❡ssí✈❡❧ ❡♠ M✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦ ❝♦♠ ❛ ❤✐♣ót❡s❡✳

  P♦rt❛♥t♦✱ M − S é ✐rr❡❞✉tí✈❡❧✳ 2 (⇐)

  ⊂ M ❙✉♣♦♥❤❛ q✉❡ M − S s❡❥❛ ✐rr❡❞✉tí✈❡❧✳ ❙❡❥❛ S ✉♠❛ ❡s❢❡r❛✱ ♣♦❞❡♠♦s 2 2

  ❛ss✉♠✐r q✉❡ S é tr❛♥s✈❡rs❛❧ ❛ S✳ ❆ss✐♠✱ S ∩ S ❝♦♥s✐st❡ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦s ❞❡ ❝ír❝✉❧♦s✳ 2 2

  ❈♦♥s✐❞❡r❡ ✉♠ ❝ír❝✉❧♦ ❡♠ S ∩ S ❜♦r❞❛♥❞♦ ✉♠ ❞✐s❝♦ D ❡♠ S ❞❡ t❛❧ ❢♦r♠❛ i q✉❡ D ∩ S = ∂D✱ ♦✉ s❡❥❛✱ intD ⊂ M − S✳ ❙❡❥❛ S ❛ ❝♦♠♣♦♥❡♥t❡ ❞❡ S t❛❧ i = ∂D i q✉❡ D ∩ S ✱ ♣❡❧❛ ✐♥❝♦♠♣r❡ss✐❜✐❧✐❞❛❞❡ ❞❡ S t❡♠♦s q✉❡ ∂D ❜♦r❞❛ ✉♠ ❞✐s❝♦ D ⊂ S i

  ✳ S i

  é ✷✲❧❛❞♦s ❡♥tã♦ ❝❛❞❛ ❝♦♠♣♦♥❡♥t❡ ❞❡ S é ✷✲❧❛❞♦s ❡♠ M✱ ✐st♦ é✱ S ❛❞♠✐t❡ i i ✉♠ ❜✐❝♦❧❛r V ❡♠ M✱ ❛ss✐♠ ♣♦❞❡♠♦s ❞❡s❧♦❝❛r ∂D ♥❡ss❛ ✈✐③✐♥❤❛♥ç❛ V ❡♠ ❞✐r❡çã♦ ❛♦ ✐♥t❡r✐♦r ❞♦ ❞✐s❝♦ D ❞❡ ♠♦❞♦ q✉❡ t♦❞♦s ♦s ❝ír❝✉❧♦s ❜♦r❞❛♠ ✉♠ ❞✐s❝♦ D ❡♠ ∗ ∗

  M − S ∩ D = ∂D i ⊂ D i = ∂D ❝♦♠ ∂D ❡ s❡❥❛ D ♦ ❞✐s❝♦ t❛❧ q✉❡ ∂D ✳

  ∪D ⊂ M −S i ❈♦♠♦ ❛ ❡s❢❡r❛ D ❜♦r❞❛ ✉♠❛ ❜♦❧❛ ❡♠ M −S ✱ s❡❣✉❡ q✉❡ ❛ ❡s❢❡r❛ D ∪ D

  ❜♦r❞❛ ✉♠❛ ❜♦❧❛ B ❡♠ M✳ ❉❡✈❡♠♦s t❡r q✉❡ B ∩ S = D ✱ ❝❛s♦ ❝♦♥trár✐♦✱ i t❡rí❛♠♦s S ❝♦♥t✐❞❛ ♥✉♠❛ ❜♦❧❛✱ ✉♠ ❛❜s✉r❞♦ ❝♦♠ ♦ ❢❛t♦ ❞❡ ❝❛❞❛ ❝♦♠♣♦♥❡♥t❡ ❞❡ S s❡r ✐♥❝♦♠♣r❡ssí✈❡❧✳

  Si Si D D' D' D* D* D i B B* V i S S

  2 2 2 ❋✐❣✉r❛ ✷✳✼ P♦r ✉♠❛ ✐s♦t♦♣✐❛ ❡♠ S ♣♦❞❡♠♦s ❧❡✈❛r ♦ ❞✐s❝♦ D ♣❛r❛ ♦ ✐♥t❡r✐♦r ❞❡ B ❡♠ i

  ❞✐r❡çã♦ ❛♦ ❞✐s❝♦ D ❡✱ ✉♠ ♣♦✉❝♦ ♠❛✐s ❛❧é♠ ♥❛ ✈✐③✐♥❤❛♥ç❛ V ❞❡ ♠♦❞♦ q✉❡ ♦ ♥♦✈♦ i 2 ∩ S i

  ❞✐s❝♦ D ♥ã♦ ✐♥t❡rs❡t❛ S ✱ ❡❧✐♠✐♥❛♥❞♦ ∂D ❡ q✉❛❧q✉❡r ♦✉tr♦ ❝ír❝✉❧♦ ❞❡ S q✉❡ ❡st❡❥❛ ♥♦ ✐♥t❡r✐♦r ❞❡ D ✳

  ✺✻ ✷✳✷✳ ❆❙P❊❈❚❖❙ ❚❖P❖▲Ó●■❈❖❙

  Si D' B D B' V i

  Vi S 2 novo D ❋✐❣✉r❛ ✷✳✽ 2

  ∩ S 2 ❘❡♣❡t✐♥❞♦ ❡st❡ ♣r♦❝❡ss♦ ❡❧✐♠✐♥❛♠♦s t♦❞♦s ♦s ❝ír❝✉❧♦s ❞❡ S ❛ss✐♠ t❡r❡♠♦s 2 S ⊂ M − S ✳ ❊♥tã♦ S ❜♦r❞❛ ✉♠❛ ❜♦❧❛ ❡ M é ✐rr❡❞✉tí✈❡❧✳

  ❈❛♣ít✉❧♦ ✸ ❘❡s✉❧t❛❞♦ Pr✐♥❝✐♣❛❧

  ◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛ q✉❡ ♠♦t✐✈♦✉ ❡st❡ ❡st✉❞♦✳ P❛r❛ ✐st♦✱ ✉t✐❧✐③❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ❡ ❝♦♥❝❡✐t♦s ❛♣r❡s❡♥t❛❞♦s ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s✱ ❜❡♠ ❝♦♠♦ ♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ♥❛ ♣r✐♠❡✐r❛ s❡çã♦ ❞❡ss❡ ❝❛♣ít✉❧♦✳ ❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s sã♦

  ✸✳✶ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s 1

  ❚❡♦r❡♠❛ ✸✳✶✳✶✳ ❙❡❥❛ T ✉♠ t♦r♦ tr❛♥s✈❡rs❛❧ ❛ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X✱ C ✱ ❡♠ ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ ❡ ✐rr❡❞✉tí✈❡❧ M✳ ❙❡ ❛ ✉♥✐ã♦ ❞❡ ór❜✐t❛s ❞❡ X q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛♠ T é ❝♦♥❡①❛ ❡♠ M✱ ❡♥tã♦ T é ✐♥❝♦♠♣r❡ssí✈❡❧✳

  ❆ ♣r♦✈❛ ❞❡ss❡ t❡♦r❡♠❛ é ❝♦♥s❡q✉ê♥❝✐❛ ❞♦s s❡❣✉✐♥t❡s ❞♦✐s ❧❡♠❛s✳ ▲❡♠❛ ✸✳✶✳✷✳ ❙❡ M é ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ ❡ ✐rr❡❞✉tí✈❡❧ ❡ T é ✉♠ t♦r♦ ✷✲❧❛❞♦s ♠❡r❣✉❧❤❛❞♦ ❡♠ M✱ ❡♥tã♦ T é ✐♥❝♦♠♣r❡ssí✈❡❧ ♦✉ s❡♣❛r❛ M✳ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ T ♥ã♦ é ✐♥❝♦♠♣r❡ssí✈❡❧✳ ❈♦♠♦ M é ✐rr❡❞✉tí✈❡❧ ❡ T

  é ✷✲❧❛❞♦s✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ s❡❣✉❡ q✉❡ T ❜♦r❞❛ ✉♠ t♦r♦ só❧✐❞♦ ♦✉ ❡stá ❝♦♥t✐❞♦ ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠❛ ❜♦❧❛ B ❡♠ M✳

  ◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ T s❡♣❛r❛ M ❡ t❡♠♦s ♦ r❡s✉❧t❛❞♦✳ ◆♦ s❡❣✉♥❞♦ ❝❛s♦✱ ✈❛♠♦s ♣r♦✈❛r q✉❡ T t❛♠❜é♠ s❡♣❛r❛ M✳ ′ ′

  P❛r❛ ✐ss♦ ❝♦♥s✐❞❡r❡ ✉♠❛ ❜♦❧❛ B ✱ ❝♦❧❛♥❞♦ ❛s ❜♦❧❛s B ❡ B ♣❡❧♦ ❜♦r❞♦ ✈✐❛ ✉♠ 3 ❞✐❢❡♦♠♦r✜s♠♦ ♦❜t❡♠♦s ✉♠❛ S q✉❡ ❝♦♥té♠ T ✳ ❙❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ❞♦ ❚♦r♦ ❙ó❧✐❞♦ 3 3 1 2 \ T q✉❡ T s❡♣❛r❛ S ✳ ❉❡♥♦t❡ ♣♦r H ❡ H ❛s ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ❞❡ S ✳

  ∩ T = ∅ ′ ′ ′ ′ ❈♦♠♦ T ⊂ int(B) t❡♠♦s q✉❡ B ✳ ❆ss✐♠✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ B ⊂ int(H ) \ B = B \ H \ B ) = T ∪ ∂B = ∂B 1 1 2 1

  ❡ ❡♥tã♦ H ❧♦❣♦ ∂(H ♣♦✐s ∂B ✳ 1 \ B ❆❧é♠ ❞✐ss♦✱ ∂(M \ B) = ∂B✳ ❊♥tã♦✱ ♣♦❞❡♠♦s ❝♦❧❛r H ❡ M \ B ❛♦ ❧♦♥❣♦ ❞❡ ∂B

  ❞❡ ❢♦r♠❛ ♦r❞❡♥❛❞❛ ♣❛r❛ ♦❜t❡r ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ A ❝♦♥❡①❛ ❝♦♠ ❜♦r❞♦ ∂A = T ✳

  ✺✽ ✸✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ 2 2 = T 2 ❆♥❛❧♦❣❛♠❡♥t❡✱ H é ❝♦♥❡①❛ ❡ s❛t✐s❢❛③ ∂H ✳ ❈♦❧❛♥❞♦ A ❡ H ❛♦ ❧♦♥❣♦ ❞❡

  T ❞❡ ✉♠ ♠♦❞♦ ❛❞❡q✉❛❞♦ ♣r♦❞✉③✐♠♦s M✳ ❆ss✐♠ M \ T t❡♠ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s✱

  A ❡ H 2 ✳ ❊♥tã♦✱ T s❡♣❛r❛ M ♥❡st❡ ❝❛s♦ t❛♠❜é♠✱ ❝♦♠♦ q✉❡rí❛♠♦s✳ 1 ❙❡ S ⊂ M é ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ tr❛♥s✈❡rs❛❧ ❛ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X✱

  C S ✱ ♥✉♠❛ 3−✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ M✱ ❞❡♥♦t❛r❡♠♦s ♣♦r σ ❛ ✉♥✐ã♦ ❞❛s ór❜✐t❛s ❞❡

  X q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛♠ S✱ ✐st♦ é✱ σ = {x ∈ M : O (x) ∩ S = ∅}. S X Pr♦♣♦s✐çã♦ ✸✳✶✳✸✳ ❈♦♥s✐❞❡r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ S ⊂ M tr❛♥s✈❡rs❛❧ ❛ ✉♠ 1 S

  ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X✱ C ✱ ♥✉♠❛ 3−✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ M✱ ❡♥tã♦ σ é ✐♥✈❛r✐❛♥t❡ t S S ♣❡❧♦ ✢✉①♦ X ❡ σ é ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ ❡♠ M✱ ❛ss✐♠ σ é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦✳ S

  ❉❡♠♦♥str❛çã♦✳ ❆ ✐♥✈❛r✐â♥❝✐❛ ♣❡❧♦ ✢✉①♦ s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ σ ✱ ✉♠❛ ✈❡③ q✉❡ s❡ x ∈ σ S t (x) ∈ σ S ❡♥tã♦ ♣❛r❛ q✉❛❧q✉❡r t ∈ R t❡♠♦s q✉❡ X ✱ ♣♦✐s ❛ ór❜✐t❛ ❞♦ ♣♦♥t♦ x (x) t S

  ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ór❜✐t❛ ❞♦ ♣♦♥t♦ X ✳ ❆ss✐♠✱ σ é ✉♠ ❝♦♥❥✉♥t♦ ✐♥✈❛r✐❛♥t❡ ♣❡❧♦ ✢✉①♦✳ S

  ❆❣♦r❛ ✈❛♠♦s ♠♦str❛r q✉❡ σ é ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦✳ ❙✉♣♦♥❤❛✱ ♣♦r S S S ❝♦♥tr❛❞✐çã♦✱ q✉❡ ❡①✐st❡ p ∈ σ ❡ p /∈ σ ✳ ❈♦♠♦ p /∈ σ ❡①✐st❡ ✉♠ t❡♠♣♦ T T (p) ∈ S t❛❧ q✉❡ y = X ✳ S y

  V x p U n S ❋✐❣✉r❛ ✸✳✶ S 6= ∅

  ❊✱ p ∈ σ ❛ss✐♠ ♣❛r❛ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ U ❞❡ p t❡♠✲s❡ q✉❡ U ∩ σ ✳ ❈♦♥s✐❞❡r❡ V ✉♠❛ ✈✐③✐♥❤❛♥ç❛ t✉❜✉❧❛r ❞♦ s❡❣♠❡♥t♦ ❞❡ ór❜✐t❛ ❞❡ p ❛té ♦ ♣♦♥t♦ y s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛ ❡ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❝♦♥té♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ p✱ n S n ∈ σ ∈ U ⊂ V

  ❡♥tã♦ ❡①✐st❡ x t❛❧ q✉❡ x ✳ P❡❧♦ ❚❡♦r❡♠❛ ❋❧✉①♦ ❚✉❜✉❧❛r ▲♦♥❣♦ ❞❡✈❡♠♦s t❡r q✉❡ ❛s ór❜✐t❛s ❞♦s ♣♦♥t♦s ❡♠ V ✐♥t❡rs❡❝t❛♠ S ♥✉♠ t❡♠♣♦ ♣ró①✐♠♦ n ❛♦ t❡♠♣♦ T ✱ ❡♠ ♣❛rt✐❝✉❧❛r ❛ ór❜✐t❛ ❞❡ x ❞❡✈❡ ✐♥t❡rs❡❝t❛r S✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ n ∈ σ S ❢❛t♦ ❞❡ x ✳ S S S

  P♦rt❛♥t♦✱ p ∈ σ ❡✱ σ é ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦✳ ❙❡♥❞♦ M ✉♠ ❝♦♠♣❛❝t♦ ❡ σ ✉♠ ❢❡❝❤❛❞♦ t❡♠♦s q✉❡ S é ✉♠ ❝♦♠♣❛❝t♦✳ 1

  ▲❡♠❛ ✸✳✶✳✹✳ ❙❡❥❛ T ✉♠ t♦r♦ tr❛♥s✈❡rs❛❧ ❛ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X✱ C ✱ ❡♠ ✉♠❛ T ✸✲✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ M✳ ❙❡ σ é ❝♦♥❡①❛✱ ❡♥tã♦ T ♥ã♦ s❡♣❛r❛ M✳

  ✺✾ ✸✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛✱ ♣♦r ❝♦♥tr❛❞✐çã♦✱ q✉❡ T s❡♣❛r❛ M ❡ ❞❡♥♦t❡ ♣♦r A ❡ B ❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ❞❡ M \ T ✳ T 6= ∅

  ❈♦♥s✐❞❡r❡ q✉❡ σ ❡ q✉❡ ♦ ❝❛♠♣♦ X ❛♣♦♥t❛ ♣❛r❛ ❞❡♥tr♦ ❞❡ A ❡♠ T = ∂A ❡ ♣❛r❛ ❢♦r❛ ❞❡ B ❡♠ T = ∂B✳ ❈♦♠♦ M \ T é ❞❡s❝♦♥❡①♦✱ s❡❣✉❡ q✉❡ ✉♠❛ ór❜✐t❛ ♣♦s✐t✐✈❛ ❝♦♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡♠ T ♥ã♦ ♣♦❞❡ r❡t♦r♥❛r ❛ T ✳ ❖ ♠❡s♠♦ ♦❝♦rr❡ ❝♦♠ ❛s ór❜✐t❛s ♥❡❣❛t✐✈❛s ❝♦♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡♠ T ✳

  = {x ∈ ❊♥tã♦ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ♣❡q✉❡♥❛ ❝♦♠♣❛❝t❛ ❞❡ T ❡♠ M✱ ❞✐❣❛♠♦s V

  X t (T ); t ∈ [−ǫ, ǫ]} t (T ) ✭ ǫ > 0✮✱ t❡♠♦s q✉❡ ♦ ❝❛♠♣♦ ❝♦♥t✐♥✉❛ tr❛♥s✈❡rs❛❧ ❡♠ X

  ✭t ∈ [−ǫ, ǫ]✮ ❛♣♦♥t❛♥❞♦ ♣❛r❛ ♦ ✐♥t❡r✐♦r ❞❡ A✱ ❞❛❞♦ ✉♠ ♣♦♥t♦ y ∈ Ω(X) t❛❧ q✉❡ ❛ ór❜✐t❛ ❞❡ y ✐♥t❡rs❡❝t❛ T ♥✉♠ ♣♦♥t♦ x✱ ❡♥tã♦ x ∈ Ω(X)✳ ❈♦♥s✐❞❡r❡ ♦ s❡❣♠❡♥t♦

  = X (x) = X ǫ (x) −ǫ 1 ❞❡ ór❜✐t❛ ❝♦♠♣❛❝t♦ ❞❡ x ❛té x ✱ t♦♠❛♥❞♦ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ t✉❜✉❧❛r V s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛ ❞❡ss❡ s❡❣♠❡♥t♦ ❞❡ ór❜✐t❛✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❋❧✉①♦ ❚✉❜✉❧❛r ▲♦♥❣♦✱ s❡❣✉❡ q✉❡ ❛s ór❜✐t❛s ❞♦s ♣♦♥t♦s ♥❡ss❛ ✈✐③✐♥❤❛♥ç❛ t❛♠❜é♠ t❡♠ ♦ ♠❡s♠♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ór❜✐t❛ ❞❡ x ❡✱ ❛❧é♠ ❞✐ss♦ M \ T é ❞❡s❝♦♥❡①♦✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ ♣❡q✉❡♥❛ ✈✐③✐♥❤❛♥ç❛ W ⊂ V ❞❡ x t❛❧ q✉❡ ♣❛r❛ t♦❞♦ | t |> 2ǫ t (W ) ∩ W = ∅ t❡♠✲s❡ q✉❡ X ✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦ ♣♦✐s x ∈ Ω(X)✳ B A T x

  V W x x 1

  ❋✐❣✉r❛ ✸✳✷ T X (p) ∪ α X (p) ⊂ Ω(X) ❉✐ss♦ ❝♦♥❝❧✉í♠♦s q✉❡ Ω(X) ⊂ σ ✳ ❡✱ ❝♦♠♦ ω ✱ ✭✈❡r X (p) ∪ α X (p) ⊂ σ T

  ♦❜s❡r✈❛çã♦ t❡♠♦s q✉❡ ω ♣❛r❛ ❝❛❞❛ p ∈ M ✭✐st♦ é ✈❡r❞❛❞❡ ❡♠ ♣❛rt✐❝✉❧❛r q✉❛♥❞♦ p ∈ T ✮✳ X (p) ⊂ int(A) T ∩ int(A) 6= ∅

  ❋✐①❡ p ∈ T ✳ P♦r ✉♠ ❧❛❞♦✱ ω ✳ ❊♥tã♦ σ ✳ P♦r ♦✉tr♦ X (p) ⊂ int(B) T ∩ int(B) 6= ∅ T ❧❛❞♦✱ α ✳ ❊♥tã♦ σ ✳ ❙❡❣✉❡ q✉❡ σ ♥ã♦ é ❝♦♥❡①♦✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❚❡♠♦s ♦ r❡s✉❧t❛❞♦✳ T = ∅

  ❈❛s♦✱ σ t❡♠♦s q✉❡ t♦❞❛s ❛s ór❜✐t❛s ✐♥t❡rs❡❝t❛♠ T ✳ ❈♦♥s✐❞❡r❡ ❛ ór❜✐t❛ ❞❡ ✉♠ ♣♦♥t♦ p ∈ T ✱ ❡♥tã♦ ❛ ór❜✐t❛ ♥❡❣❛t✐✈❛ ❞❡ p ♥ã♦ r❡t♦r♥❛ ❛ T ❡✱ ♣❡❧❛ ❡s❝♦❧❤❛

  (p) ⊂ int(B) ❞♦ ❝❛♠♣♦✱ O X ✳ ❙❡❥❛ q ∈ α(p) ❡ α(p) ⊂ int(B)✱ ♣♦rq✉❡ ♦ ❝❛♠♣♦ ❛♣♦♥t❛ ♣❛r❛ ❢♦r❛ ❞❡ B ❡♠ ∂B = T ✳

  ❆❧é♠ ❞✐ss♦ ❛ ór❜✐t❛ ❞♦ ♣♦♥t♦ q ❞❡✈❡rá ✐♥t❡rs❡❝t❛r T ❡✱ ❝♦♠♦ ♦ ❝❛♠♣♦ ❛♣♦♥t❛ X (q) ∩ int(A) 6= ∅ ♣❛r❛ ❞❡♥tr♦ ❞❡ A ❡♥tã♦ O ✳ ❖ ❝♦♥❥✉♥t♦ α(p) é ✐♥✈❛r✐❛♥t❡ ♣❡❧♦ ✢✉①♦✱ ❡♥tã♦ α(p) ∩ int(A) 6= ∅✱ ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s α(p) ⊂ int(B)✳ ▲♦❣♦✱ T ♥ã♦ s❡♣❛r❛ M✱ ❝♦♠♦ q✉❡rí❛♠♦s✳

  P♦rt❛♥t♦✱ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s T ♥ã♦ s❡♣❛r❛ M✳

  ✻✵ ✸✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ s❡❣✉❡ q✉❡ T é T

  ✐♥❝♦♠♣r❡ssí✈❡❧ ♦✉ T s❡♣❛r❛ M✳ ❊ ♣♦r ❤✐♣ót❡s❡✱ σ é ❝♦♥❡①❛✱ ❡♥tã♦ ♣❡❧♦ ▲❡♠❛ t❡♠✲s❡ q✉❡ T ♥ã♦ s❡♣❛r❛ M✳ ❆ss✐♠✱ T é ✐♥❝♦♠♣r❡ssí✈❡❧✳ ❚❡♦r❡♠❛ ✸✳✶✳✺✳ ❙❡❥❛ M ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧✳ ❙✉♣♦♥❤❛ q✉❡ M 1

  ❛♣r❡s❡♥t❛ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X✱ C ✱ ❝♦♠ ✉♠ t♦r♦ tr❛♥s✈❡rs❛❧ T s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

  ✶✳ ❊①✐st❡ ✉♠❛ ú♥✐❝❛ ór❜✐t❛ O ❞❡ X q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛ T ✳ ✷✳ O é ❤✐♣❡r❜ó❧✐❝♦ ❡ ♥ã♦ é ❤♦♠♦tó♣✐❝♦ ❛ ✉♠ ♣♦♥t♦ ❡♠ M✳

  ❊♥tã♦✱ M é ✐rr❡❞✉tí✈❡❧✳ ❖❜s❡r✈❡ q✉❡✱ ♣❡❧♦ ❢❛t♦ ❞❛ ór❜✐t❛ O ♥ã♦ s❡r ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦✱ O ❞❡✈❡rá s❡r ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛✱ ♣♦✐s s✐♥❣✉❧❛r✐❞❛❞❡s sã♦ ❤♦♠♦tó♣✐❝❛s ❛ ✉♠ ♣♦♥t♦ ❡

  ór❜✐t❛s r❡❣✉❧❛r❡s✱ q✉❡ ♥ã♦ sã♦ ♣❡r✐ó❞✐❝❛s✱ sã♦ ✐♠❛❣❡♥s ❞❡ ❝✉r✈❛s ✐♥t❡❣r❛✐s ❝✉❥♦ ❞♦♠í♥✐♦ é ❝♦♥trát✐❧✱ ❝♦♠♦ ♥ã♦ s❡ tr❛t❛ ❞❡ ✉♠❛ ❤♦♠♦t♦♣✐❛ ♦♥❞❡ ♦s ❡①tr❡♠♦s sã♦ ✜①❛❞♦s ✭♣♦rq✉❡ ♥ã♦ é ✉♠ ❝❛♠✐♥❤♦ ❢❡❝❤❛❞♦✮✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ s❡❣✉❡ q✉❡ ❛ ór❜✐t❛ é ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦✳ ▲♦❣♦✱ ❛ ú♥✐❝❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ♣❛r❛ ❛ ór❜✐t❛ O é s❡r ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛✳

  P❛r❛ ♣r♦✈❛r ❡ss❡ t❡♦r❡♠❛ ❢❛r❡♠♦s ✉s♦ ❞♦s s❡❣✉✐♥t❡s ❧❡♠❛s✳ ▲❡♠❛ ✸✳✶✳✻✳ ❙❡❥❛ M ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛✳ ❙✉♣♦♥❤❛ q✉❡ M ❛♣r❡s❡♥t❡ ✉♠ t♦r♦ ♠❡r❣✉❧❤❛❞♦ q✉❡ ♥ã♦ s❡♣❛r❛ M t❛❧ q✉❡ ❛ ✈❛r✐❡❞❛❞❡ M ✱ ♦❜t✐❞❛ ♣♦r ✉♠ ❝♦rt❡ ❞❡ M ❛♦ ❧♦♥❣♦ ❞❡ T é ✐rr❡❞✉tí✈❡❧✳ ❊♥tã♦✱ M é ✐rr❡❞✉tí✈❡❧✳ ❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ♣r♦✈❛r q✉❡ T é ✐♥❝♦♠♣r❡ssí✈❡❧ ❡♠ M✳

  T é ✷✲❧❛❞♦s ❡♠ M✱ ♣♦r ❤✐♣ót❡s❡ t❡♠♦s q✉❡ M − T é ✐rr❡❞✉tí✈❡❧✳ ❙✉♣♦♥❤❛✱

  ♣♦r ❝♦♥tr❛❞✐çã♦✱ q✉❡ T é ❝♦♠♣r❡ssí✈❡❧ ❡♠ M ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❝✐r✉r❣✐❛ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ❞✐s❝♦ D ✭♦♥❞❡ D ∩ T = ∂D ❡ ∂D ♥ã♦ ❜♦r❞❛ ✉♠ ❞✐s❝♦ D ❡♠ T ✮ q✉❡ 2 2 tr❛♥s❢♦r♠❛ T ♥✉♠❛ ❡s❢❡r❛ S ✳ S ❜♦r❞❛ ✉♠❛ ❜♦❧❛ B ⊂ M✱ ❥á q✉❡ M − T é

  ✐rr❡❞✉tí✈❡❧✳ Pr♦❝❡❞❡♥❞♦ ❝♦♠♦ ♥❛ ♣r♦✈❛ ❞❛ Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡ T ❜♦r❞❛ ✉♠ t♦r♦ só❧✐❞♦ ♦✉ T ❡stá ❝♦♥t✐❞♦ ♥♦ ✐♥t❡r✐♦r ❞❡ B✳

  ❯s❛♥❞♦ ♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ ❞❛ ♣r♦✈❛ ❞♦ ▲❡♠❛ s❡❣✉❡ q✉❡ T s❡♣❛r❛ M✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦ ❝♦♠ ❛ ❤✐♣ót❡s❡ s♦❜r❡ T ✳

  P♦rt❛♥t♦✱ T é ✐♥❝♦♠♣r❡ssí✈❡❧ ❡♠ M✳ ❊✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ❝♦♥❝❧✉í♠♦s q✉❡ M é ✐rr❡❞✉tí✈❡❧✳ ▲❡♠❛ ✸✳✶✳✼✳ ❙❡❥❛ M ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ ❡ s❡❥❛ X ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s✱ 1 C

  ✱ ❝♦♠ ✉♠ t♦r♦ tr❛♥s✈❡rs❛❧ T ✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ór❜✐t❛ O ❞❡ X q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛ T ✳ ❙❡ O é ❤✐♣❡r❜ó❧✐❝♦✱ ❡♥tã♦ O é t✐♣♦ s❡❧❛✳ T = O ❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ♠♦str❛r q✉❡ ♦ ❝♦♥❥✉♥t♦ σ ♥ã♦ é ✉♠ s✉♠✐❞♦✉r♦ ♣❛r❛ ♦ ❝❛♠♣♦ X ❡ ♥❡♠ ✉♠ s✉♠✐❞♦✉r♦ ♣❛r❛ ♦ ❝❛♠♣♦ −X✳ ❉❡✜♥❛✱ ♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦✿ l T = {x ∈ T ; X t (x) ∩ T = ∅, ∀t > 0}

  ✻✶ ✸✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ T ■st♦ é✱ l é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❡ T ❝✉❥❛ ór❜✐t❛ ♣♦s✐t✐✈❛ ♥ã♦ r❡t♦r♥❛ ❛ T ✳ T

  ❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ σ é ✉♠ s✉♠✐❞♦✉r♦ ♣❛r❛ ♦ ❝❛♠♣♦ X✳ T ❈♦♠♦ σ é ✉♠ s✉♠✐❞♦✉r♦ ♣❛r❛ ♦ ❝❛♠♣♦ X ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ✐♥✈❛r✐❛♥t❡ t (U ) ⊂ U

  ♣♦s✐t✐✈❛✱ ✐st♦ é✱ X ✱ ∀t ≥ 0 ❡✱ \ T σ T = t ≥0 X t (U ). T ) >

  ❆❧é♠ ❞✐ss♦ σ é ❢❡❝❤❛❞♦ ✭✈❡r Pr♦♣♦s✐çã♦ ❡ T ❝♦♠♣❛❝t♦ ❡♥tã♦ d(T, σ ✳ ❊✱ s❡ 0 ≤ s ≤ l t❡♠♦s q✉❡ l = s + ǫ✱ ǫ ≥ 0 ❛ss✐♠✿

  X ǫ (U ) ⊂ U ⇒ X s (X ǫ (U )) ⊂ X s (U ) ⇒ l (U ) ⊂ X s (U ) ⇒ X l (U ) ⊂ X s (U ).

  ▲♦❣♦✱ X ✱ ♣❛r❛ t♦❞♦ 0 ≤ s ≤ l✳ n (U ) ∩ T = ∅ ❆ss✐♠ ❞❡✈❡♠♦s t❡r ✉♠ t❡♠♣♦ n ≥ 0 t❛❧ q✉❡ ♦ ❛❜❡rt♦ X ✳ ❈❛s♦ t (U ) ∩ T 6= ∅

  ❝♦♥trár✐♦✱ s❡ X ✱ ∀t ≥ 0 t❡♠✲s❡ q✉❡ X l (U ) ⊂ X s (U ), 0 ≤ s ≤ l ⇒

  ⇒ ∅ 6= X l (U ) ∩ T ⊂ X s (U ) ∩ T, 0 ≤ s ≤ l ⇒ \

  ⇒ t ≥0 X t (U ) ∩ T 6= ∅. ▼❛s✱

  \ σ T = t ≥0 X t (U ). T ∩ T 6= ∅

  ❆ss✐♠✱ σ ✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ P♦rt❛♥t♦ ❡①✐st❡ n ≥ 0 t❛❧ q✉❡ X n (U ) ∩ T

  ✳ P♦❞❡♠♦s ❡s❝r❡✈❡r✱ \ n (U ) σ T = t ≥n X t (U ).

  ❙❡❥❛ V = X ✱ ❞❛❞♦ t ≥ n ≥ 0 t❡♠♦s q✉❡ t = n + s✱ ♦♥❞❡ s = t − n ≥ 0✱ ❡♥tã♦

  X s (U ) ⊂ U ⇒ X n (X s (U )) ⊂ X n (U ) = V ⇒ ⇒ X ∀s ≥ 0. s (X n (U )) ⊂ V ⇒ X s (V ) ⊂ V, ❊✱

  \ σ = T s s ≥0 X (V ).

  P♦rt❛♥t♦ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r U ❛❜❡rt♦ ❡ t❛❧ q✉❡ U ∩ T = ∅✳

  ✻✷ ✸✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ T T t (x) / ∈ T ❙❡♥❞♦ σ ❢❡❝❤❛❞♦ ❡ U ❛❜❡rt♦ ❡♥tã♦ ❡①✐st❡ x ∈ U \ σ ✳ ❊✱ X ✱ ∀t > 0✱ t (U ) ⊂ U

  ♣♦✐s U ∩ T = ∅✱ X ✱ ∀t > 0 ❡ x ∈ U✳ T x > 0 P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ x /∈ σ ✱ s❡❣✉❡ q✉❡ ❡①✐st❡ ✉♠ ♣r✐♠❡✐r♦ t t❛❧ −t −t (x) ∈ T t (y) = X t (x) x ] q✉❡ y = X x ✳ ◆♦t❡ q✉❡ X x ✳ ❙❡ t ∈ (0, t t❡♠♦s q✉❡

  | t − t |< t ∈ T x x t (y) / x ❛ss✐♠ X ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ t ✳ t (y) = X t (x) ∈ U x > 0 x −t

  ❆❧é♠ ❞✐ss♦✱ X x ✱ ♣❛r❛ t♦❞♦ t − t ✱ ✐st♦ é✱ ∀t > t ❡♥tã♦ X (y) / ∈ T t x

  ✱ ∀t > t ✳ t (y) / ∈ T T T 6= ∅ ■st♦ ♣r♦✈❛ q✉❡ X ✱ ∀t > 0✱ ♣♦rt❛♥t♦ y ∈ l ✱ ♦✉ s❡❥❛✱ l ✳ T

  ❆✜r♠❛çã♦ ✶✿ l é ❢❡❝❤❛❞♦ ❡♠ T ✳ T ⊂ l T ∩ T T ∩ T ⊂ l T ❚❡♠♦s q✉❡ l ✳ ❆❣♦r❛✱ ✈❛♠♦s ♠♦str❛r q✉❡ l ✳ T n T n ∩ T ∈ l → p ❙❡❥❛ p ∈ l ❡♥tã♦ ❡①✐st❡ x ✱ n ∈ N✱ t❛❧ q✉❡ x ✳ ❙✉♣♦♥❤❛ q✉❡ p / ∈ l T

  ✱ ❡♥tã♦ ❛ ór❜✐t❛ ♣♦s✐t✐✈❛ ❞❡ p r❡t♦r♥❛ ❛ T ✱ ✐st♦ é✱ ❡①✐st❡ ✉♠ t❡♠♣♦ t > 0 t❛❧ t (p) ∈ T q✉❡ X ✳ T Xt(p) W x n' p V ❋✐❣✉r❛ ✸✳✸

  ❙❡❥❛ V ✈✐③✐♥❤❛♥ç❛ t✉❜✉❧❛r s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛ ❞♦ s❡❣♠❡♥t♦ ❞❡ ór❜✐t❛ ❞❡ p t (p) ❛té X ✱ ❝♦♥s✐❞❡r❡ W ✉♠ ❛❜❡rt♦ ❞❡ T ❝♦♥t✐❞♦ ❡♠ V t❛❧ q✉❡ p ∈ W t❡♠✲s❡ n ∈ W ⊂ V ∩ l T q✉❡ ❡①✐st❡ ✉♠ ♣♦♥t♦ x ✳ ❊♥tã♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❋❧✉①♦ ❚✉❜✉❧❛r n

  ▲♦♥❣♦✱ s❡❣✉❡ q✉❡ ❛ ór❜✐t❛ ♣♦s✐t✐✈❛ ❞❡ x ❞❡✈❡rá r❡t♦r♥❛r ❛ T ✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦ T T ∩ T ⊂ l T ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞♦ ❝♦♥❥✉♥t♦ l ✳ ❆ss✐♠✱ l ✳ T +

  P♦rt❛♥t♦✱ l é ❢❡❝❤❛❞♦ ❡♠ T ✳ (q) ∩ U 6= ∅ T

  ❆✜r♠❛çã♦ ✷✿ O

  • + X ✱ ∀q ∈ l ✳ t (q) ∩ U = ∅

  ❙✉♣♦♥❤❛ ♣♦r ❝♦♥tr❛❞✐çã♦ q✉❡ ❡①✐st❡ q ∈ l t❛❧ q✉❡ O X ❡♥tã♦ O X (q) ∩ U = ∅

  ✳ ❚❡♠♦s q✉❡ ω(q) 6= ∅✳ ❙❡❥❛ x ∈ ω(q) ❞❡✈❡♠♦s t❡r q✉❡ O X (x) ∩ U = ∅ X (q) ∩ U = ∅ s❡ O ✳ ❈♦♥s✐❞❡r❡ x ∈ ω(q) ❞❡ ♠♦❞♦ q✉❡ x /∈ T ❡ ❡①✐st❡ N > 0 N (x) ∈ T X (x) ⊂ ω(q)

  ❝♦♠ y = X ✭✐ss♦ é ♣♦ssí✈❡❧ ♣♦✐s O ✱ T é tr❛♥s✈❡rs❛❧ ❛♦ ❝❛♠♣♦ X ❡✱ ❛❧é♠ ❞✐ss♦ só ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ór❜✐t❛ q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛ T ✮✳

  ✻✸ ✸✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ T q x n y ❋✐❣✉r❛ ✸✳✹

  ❚♦♠❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ t✉❜✉❧❛r s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛ V ❞♦ s❡❣♠❡♥t♦ ❞❡ N (x) X (x) ∈ ω(q) n > 0 ór❜✐t❛ ❞❡ x ❛ y = X ✳ ❈♦♠♦ O ❡①✐st❡ ✉♠ t s✉✜❝✐❡♥t❡♠❡♥t❡ n = X t (q) ∈ V

  • + ❣r❛♥❞❡ ❝♦♠ q n ✳ P❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❋❧✉①♦ ❚✉❜✉❧❛r ▲♦♥❣♦✱ ❞❡✈❡♠♦s n

  ∈ V (q) t❡r q✉❡ q ✐♥t❡rs❡❝t❛ T ♥✉♠ t❡♠♣♦ ♣ró①✐♠♦ ❛ N✳ ❆ss✐♠ O t X ❞❡✈❡rá r❡t♦r♥❛r ❛ T ✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦ ♣♦✐s q ∈ l ✳ t (q) ∈ U

  ▲♦❣♦✱ ❡①✐st❡ ✉♠ t > 0 ❞❡ ♠♦❞♦ q✉❡ X ✳ ❊✱ ❞❡✈✐❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ U (q) ∈ U

  ♣❛r❛ t♦❞♦ t > t t❡♠✲s❡ q✉❡ X t ✳ T ❆✜r♠❛çã♦ ✸✿ l é ❛❜❡rt♦ ❡♠ T ✳ T B y X (y) t U T t (y) ∈ U ❋✐❣✉r❛ ✸✳✺

  ❙❡❥❛ y ∈ l ✱ ❡♥tã♦ ❡①✐st❡ t > 0 t❛❧ q✉❡ X ✱ ♣❡❧❛ ❛✜r♠❛çã♦ ❛♥t❡r✐♦r✱ t♦♠❡ t (y) ✉♠❛ ✈✐③✐♥❤❛♥ç❛ t✉❜✉❧❛r V ❞♦ s❡❣♠❡♥t♦ ❞❡ ór❜✐t❛ ❞❡ y ❛té X s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛✱ ❡①✐st❡ ✉♠ ❛❜❡rt♦ B ❞❡ T ❝♦♥t❡♥❞♦ y ❡♠ V ✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❋❧✉①♦ q > 0 ❚✉❜✉❧❛r ▲♦♥❣♦✱ ❞❡✈❡♠♦s t❡r q✉❡ ♣❛r❛ t♦❞♦ q ∈ B t❡♠♦s q✉❡ ❡①✐st❡ ✉♠ t t (q) ∈ U ❝♦♠ X q ✳ t (q) ∈ U q T = ∅ T

  ❆❧é♠ ❞✐ss♦✱ X ✱ ∀t > t ❡ U ∩ l ✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ q ∈ l ✱ ❛ss✐♠ B ⊂ l T

  ✳ T T ⊂ T P♦rt❛♥t♦✱ l é ❛❜❡rt♦ ❡♠ T ✳ ❙❡♥❞♦ T ❝♦♥❡①♦ ❡ l é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ T = T

  ✈❛③✐♦✱ ❛❜❡rt♦ ❡ ❢❡❝❤❛❞♦ ❡♠ T ✱ ❡♥tã♦ l ✳ T . ❆✜r♠❛çã♦ ✹✿ Ω(X) ⊂ σ T

  ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡ x ∈ Ω(X) ❡ x /∈ σ ✱ ❡♥tã♦ ❛ ór❜✐t❛ ❞❡ x ✐♥t❡rs❡❝t❛ T ♥✉♠ T T = T ♣♦♥t♦ y✳ ❈♦♠♦ Ω(X) é ✐♥✈❛r✐❛♥t❡ t❡♠♦s q✉❡ y ∈ Ω(X)✳ ▼❛s✱ l ❧♦❣♦ y ∈ l ✳ y > 0

  ❆ss✐♠✱ ❡①✐st❡ ✉♠ t ❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ W ❞❡ y ❝♦♠ W ∩ U = ∅ t❛❧ q✉❡ X t (W ) ⊂ U y t (W ) ⊂ U t (W ) ∩ W = ∅ y ✳ ▲♦❣♦✱ ♣❛r❛ t♦❞♦ t > t t❡♠✲s❡ q✉❡ X ❡✱ X ✳

  ✻✹ ✸✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ y t (W ) ∩ W 6= ∅ ❙❡ ♣❛r❛ t < −t t✐✈❡r♠♦s q✉❡ X ❡♥tã♦ ❡①✐st❡ p ∈ W t❛❧ q✉❡ X t (p) ∈ W y < −t (X t (p)) ∈ U −t

  ✱ ❝♦♠♦ 0 < t t❡♠✲s❡ q✉❡ p = X ♣♦rt❛♥t♦ W ∩ U 6= ∅ t (W ) ∩ W = ∅ y

  ✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❊♥tã♦✱ X ✱ ∀ | t |> t ✳ ■st♦ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦ ❞❡ y ∈ Ω(X) T

  ▲♦❣♦✱ Ω(X) ⊂ σ ✳ ❆❣♦r❛✱ s❡❥❛ x ∈ T ❡ q ∈ α(x)✱ ❝♦♠♦ α(x) ⊂ Ω(X) ✭♦❜s❡r✈❛çã♦ s❡❣✉❡ T −t n (x) ∈ U > 0 q✉❡ q ∈ Ω(X) ❡✱ ❛ss✐♠ q ∈ σ ✳ ❊♥tã♦✱ X n ✱ ♣❛r❛ ❛❧❣✉♠ t ❡✱ ′ ′ q = X (x) ∈ U ⇒ X t (q ) = x ∈ T. ′ ′ −t n n

  ∈ U t (q ) ∈ X t (U ) ⊂ U ❯♠ ❛❜s✉r❞♦✱ ♣♦✐s s❡ q ❡♥tã♦ X n n ❡ U ∩ T = ∅✳ T P♦rt❛♥t♦✱ σ ♥ã♦ é ✉♠ s✉♠✐❞♦✉r♦ ♣❛r❛ ♦ ❝❛♠♣♦ X✳ ❆♣❧✐❝❛♥❞♦ ✉♠ ❛r❣✉♠❡♥t♦ T

  ❛♥á❧♦❣♦ ❛♦ ❝❛♠♣♦ r❡✈❡rs♦ −X✱ t❡♠♦s q✉❡ σ ♥ã♦ é ✉♠ s✉♠✐❞♦✉r♦ ♣❛r❛ ♦ ❝❛♠♣♦ −X T

  ✳ ❆ss✐♠✱ σ é ❤✐♣❡r❜ó❧✐❝♦ t✐♣♦ s❡❧❛✳ ▲❡♠❛ ✸✳✶✳✽✳ ❙❡❥❛ M ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ♦r✐❡♥tá✈❡❧ ❝✉❥❛ ❢r♦♥t❡✐r❛ ∂M 1 2 1

  ❝♦♥s✐st❡ ❞❡ ❞♦✐s t♦r♦s T ✱ T ✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s Y ✱ C ✱ tr❛♥s✈❡rs❛❧ ❛ ∂M ✱ ❛♣r❡s❡♥t❛♥❞♦ ✉♠❛ ú♥✐❝❛ ór❜✐t❛ O q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛ ∂M ❡ s❛t✐s❢❛③❡♥❞♦✿ 1 2

  ✶✳ Y ❛♣♦♥t❛ ♣❛r❛ ❞❡♥tr♦ ❡♠ T ❡ ♣❛r❛ ❢♦r❛ ❡♠ T ✳ ✷✳ O é ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ ❤✐♣❡r❜ó❧✐❝❛ t✐♣♦ s❡❧❛✳ ✸✳ O ♥ã♦ é ❤♦♠♦tó♣✐❝♦ ❛ ✉♠ ♣♦♥t♦ ❡♠ M ✳

  ❊♥tã♦✱ M é ✐rr❡❞✉tí✈❡❧✳ ❆♥t❡s ❞❡ ✐♥✐❝✐❛r ❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❧❡♠❛✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ♣r♦♣♦s✐çã♦

  ❡ ❛❧❣✉♠❛s ♦❜s❡r✈❛çõ❡s q✉❡ s❡rã♦ út❡✐s ❞✉r❛♥t❡ ❛ s✉❛ ♣r♦✈❛✳ r (M )

  Pr♦♣♦s✐çã♦ ✸✳✶✳✾✳ ❙❡❥❛ M ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ❡ ♦r✐❡♥tá✈❡❧ X ∈ X t ❡ X ♦ ✢✉①♦ ❛ss♦❝✐❛❞♦ ❛ X ❝♦♠ ✉♠❛ s✉♣❡r❢í❝✐❡ S tr❛♥s✈❡rs❛❧✳ ❙❡ ♦ ❝♦♥❥✉♥t♦ ❞❛s ór❜✐t❛s ❞❡ X q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛♠ X ❝♦♥s✐st❡ ❡♠ ✉♠❛ ú♥✐❝❛ ór❜✐t❛ O ♣❡r✐ó❞✐❝❛ j i

  ⊂ S ∩ W (O) ❤✐♣❡r❜ó❧✐❝❛ t✐♣♦ s❡❧❛✱ ❡♥tã♦ ❡①✐st❡♠ ❝✉r✈❛s ❢❡❝❤❛❞❛s s✐♠♣❧❡s C ✭i = u, s, j = 1, 2✮✱ t❛✐s q✉❡ s s

  DomΠ = S \ (C ∪ C ) u u 1 2 ∪ C ImΠ = S \ (C ). 1 2

  ❖♥❞❡ Π ❞❡♥♦t❛ ❛ ❢✉♥çã♦ ❞❡ Pr✐♠❡✐r♦ ❘❡t♦r♥♦ ❞❡ P♦✐♥❝❛ré✳ j s ❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ❝♦♥str✉✐r C ✱ (j = 1, 2) ❡ ❛♣❧✐❝❛♥❞♦ ♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ u ❛♦ ❝❛♠♣♦ −X ❝♦♥str✉✐♠♦s C i ✳

  ✻✺ ✸✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ❙❡❥❛ O ❛ ór❜✐t❛ ❤✐♣❡r❜ó❧✐❝❛ t✐♣♦ s❡❧❛ q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛ S✳ ❋✐①❡ ✉♠ ♣♦♥t♦ ss ss p ∈ O (p) (p)

  ✱ ❡♥tã♦ W ❡ W sã♦ ✉♥✐❞✐♠❡♥s✐♦♥❛✐s✳ γ γ > 0 t (W (p)) = W (p)

  ❉❡♥♦t❡ ♣♦r t ♦ ♣❡rí♦❞♦ ❞❡ O✱ t❡♠♦s q✉❡ X ♣❛r❛ s γ = uu, ss (p) t

  ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ W é ✉♠ ❝✐❧✐♥❞r♦ ✭s❡ X ♣r❡s❡r✈❛ ❛ ♦r✐❡♥t❛çã♦✮✱ ♦✉ t u (p)

  ✉♠❛ ❢❛✐①❛ ❞❡ ▼ö❡❜✐✉s ✭s❡ X ✐♥✈❡rt❡ ❛ ♦r✐❡♥t❛çã♦✮✳ ❖ ♠❡s♠♦ ♦❝♦rr❡ ❝♦♠ W ✳ ss (p)

  ❯♠ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ ❡♠ W é ❝❤❛♠❛❞♦ ❞♦♠í♥✐♦ ❢✉♥❞❛♠❡♥t❛❧ s❡ ♦s ♣♦♥t♦s t t 2t (a) (a) = b

  ❞♦ ❜♦r❞♦ sã♦ a ❡ b = X ✭❝❛s♦ X ♣r❡s❡r✈❛ ❛ ♦r✐❡♥t❛çã♦ ✮ ♦✉ a ❡ X t ✭❝❛s♦ X ♥ã♦ ♣r❡s❡r✈❛ ❛ ♦r✐❡♥t❛çã♦✮✳ ss

  (p) ❚♦♠❡ ✉♠ ❞♦♠í♥✐♦ ❢✉♥❞❛♠❡♥t❛❧ I ❞❡ W s✉✜❝✐❡♥t❡♠❡♥t❡ ♣ró①✐♠♦ ❞❛ ór❜✐t❛

  ❞❡ p ✳ P♦r ❤✐♣ót❡s❡✱ ∀q ∈ I t❡♠♦s q✉❡ ❛ ór❜✐t❛ ❞❡ q ✐♥t❡rs❡❝t❛ S✳ ❉❡♥♦t❛r❡♠♦s ♣♦r π ❛ ♣r✐♠❡✐r❛ ✐♥t❡rs❡çã♦ ❞❡ ❝❛❞❛ ♣♦♥t♦ ❞❡ I ❡♠ S ♣❡❧♦ ❝❛♠♣♦ X✳

  ❆✜r♠❛çã♦✿ π(a) = π(b) 1 ∈ R t (b) 1 ❙❡❥❛ t t❛❧ q✉❡ X é ❛ ♣r✐♠❡✐r❛ ✐♥t❡rs❡çã♦ ❡♠ S✳ b = X (a) ⇒ X (b) = X (a) ∈ S. t t 1 t +t 1

  • t N (a) ∈ S ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡ ✉♠ N ∈ R t❛❧ q✉❡ N < t 1 ❡ X ✳ a = X (b) ⇒ X N (a) = X N (b) ∈ S < t −t −t
  • 1 ✱ N − t ✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦ ♣♦✐s t 1

      é ❛ ♣r✐♠❡✐r❛ ✐♥t❡rs❡çã♦ ❞❛ ór❜✐t❛ ❞❡ b ❡♠ S✳ ▲♦❣♦✱ π(a) = π(b)✳ t (p) s ❖❜s❡r✈❡ q✉❡ ♥♦ ❝❛s♦ q✉❡ X ♣r❡s❡r✈❛ ❛ ♦r✐❡♥t❛çã♦ ✭♦✉ s❡❥❛✱ W é ✉♠ ss

      (p) ❝✐❧✐♥❞r♦✮✱ ♣♦❞❡♠♦s t♦♠❛r ✉♠ ❞♦♠í♥✐♦ ❢✉♥❞❛♠❡♥t❛❧ ❞✐❢❡r❡♥t❡ ❞❡ I ❡♠ W s t❛❧ q✉❡ ❡st❡ ❞♦♠í♥✐♦ ♣r♦❞✉③ ✉♠❛ ❝✉r✈❛ s✐♠♣❧❡s ❢❡❝❤❛❞❛ ❞✐st✐♥t❛ ❞❡ C 1 ✭❜❛st❛

      ❝♦♥s✐❞❡r❛r ❛ ♣r✐♠❡✐r❛ ✐♥t❡rs❡çã♦ ❝♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡✮✳ s s s s s ∪ C ⊂ W (p)

      ∪ C ❈♦♠♦ C 1 2 t❡♠♦s q✉❡ ♣❛r❛ ♦ ❢✉t✉r♦ ♦s ♣♦♥t♦s ❞❡ C s s 1 2 s❡

      ∪ C ❛❝✉♠✉❧❛♠ ❡♠ O ❧♦❣♦ DomΠ = S \ C ✳ ❆❧é♠ ❞✐ss♦✱ ♦s ♣♦♥t♦s q✉❡ ♥ã♦ 1 2 s

      (p) r❡t♦r♥❛♠ ❛ S ❡stã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝♦♥t✐❞♦s ❡♠ W ✳ ❖❜s❡r✈❛çã♦ ✸✳✶✳✶✵✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M ❡ ✉♠ ❝❛♠♣♦ Y ❡ t♦❞❛s ❛s ❤✐♣ót❡s❡s ❞❡s❝r✐t❛s ♥♦ ❡♥✉♥❝✐❛❞♦ ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ t❡♠♦s s s u u

      ⊂ T ∩ W (O) ⊂ T ∩ W (O) 1 1 q✉❡ ❡①✐st❡♠ ❝✉r✈❛s ❢❡❝❤❛❞❛s s✐♠♣❧❡s C j ❡ C j ✭j = 1, 2✮ t❛✐s q✉❡ s s

      DomΠ = T \ (C ∪ C ) 1 u u 1 2 ImΠ = T \ (C ∪ C ). 2 1 2 1 → T 2 ❖♥❞❡ Π : T ❞❡♥♦t❛ ❛ ❢✉♥çã♦ ❞❡ P♦✐♥❝❛ré✳ ◆❡st❡ ❝❛s♦✱ ❡st❛♠♦s 1 2

      ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛ ❢✉♥çã♦ ❞❡ P♦✐♥❝❛ré ❡♥tr❡ ❞✉❛s tr❛♥s✈❡rs❛✐s✿ T ❡ T ✳ s (O)

      ◆♦ ❝❛s♦ q✉❡ W é ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠ ❝✐❧✐♥❞r♦ t❡♠♦s q✉❡ ❞✉❛s ❝✉r✈❛s s u s (O)

      ❢❡❝❤❛❞❛s s✐♠♣❧❡s C 1 ❡ C 2 ❡✱ ♥♦ ❝❛s♦ q✉❡ W é ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠❛ ❢❛✐①❛ s ❞❡ ▼ö❡❜✐✉s t❡r❡♠♦s ✉♠❛ ú♥✐❝❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ s✐♠♣❧❡s C u 1 ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣❛r❛

      W (O) ✳

      ✻✻ ✸✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ s s j ⊂ T ∩ W (O) 1 u u s u ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛s ❤✐♣ót❡s❡s ❞❡✈❡♠♦s t❡r q✉❡ ❛s ❝✉r✈❛s C ❡ C ⊂ T ∩ W (O) (O) (O) j ✭j = 1, 2✮ ❡stã♦ ♥♦ ❜♦r❞♦ ❞❡ W ❡ W ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ 1 s s

      (O) (O)

      ◆♦ ❝❛s♦ ❡♠ q✉❡ W é ✉♠ ❞❡ ❝✐❧✐♥❞r♦ t❡r❡♠♦s q✉❡ ♦ ❜♦r❞♦ ❞❡ W ✐♥t❡rs❡❝t❛ 1 s s ♦ T ✱ ♣r♦❞✉③✐♥❞♦ ❞✉❛s ❝✉r✈❛s ❢❡❝❤❛❞❛s s✐♠♣❧❡s C s 1 ❡ C 2 ✱ ❝❛s♦ ❝♦♥trár✐♦✱ s❡ ❡①✐st✐ss❡ s 1 (O) ∩ W

      (O) ✉♠ ♣♦♥t♦ x ∈ T t❛❧ q✉❡ ❡st❡ ♣♦♥t♦ ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ ❜♦r❞♦ ❞❡ W s

      (O) ✭✈❡r ❋✐❣✉r❛ ❝♦♠♦ W é ✐♥✈❛r✐❛♥t❡ ♣❡❧♦ ✢✉①♦✱ ❛ ór❜✐t❛ ♣❛r❛ ♦ ❢✉t✉r♦ ❞❡ s x

      (O) 1 s❡ ❛♣r♦①✐♠❛ ❞❡ O ❡ ❛ ór❜✐t❛ ♣❛r❛ ♦ ♣❛ss❛❞♦ ❞❡ x ♣❡rt❡♥❝❡ ❛ W ✱ T é ❜♦r❞♦ 1 ❞❡ M ❛ss✐♠ ❛ ór❜✐t❛ ❞❡ x ❞❡✈❡r✐❛ s❡r t❛♥❣❡♥t❡ ❛ T ✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ❛ ❤✐♣ót❡s❡ 1

      ❞♦ ❝❛♠♣♦ s❡r tr❛♥s✈❡rs❛❧ T ✳ x T 1 O u i ❋✐❣✉r❛ ✸✳✻✳ ❚❛♥❣ê♥❝✐❛ (O) (O)

      ❖ ♠❡s♠♦ ♦❝♦rr❡ ❝♦♠ W ✱ ❡ ♣❛r❛ ♦ ❝❛s♦ q✉❡ W ✱ ✭i = u, s✮ sã♦ ❢❛✐①❛s i ❞❡ ▼ö❡❜✐✉s t❡♠♦s q✉❡ ❛ ❝✉r✈❛ C 1 ✭i = u, s✮ é ♦ ❜♦r❞♦ ❞❛ ❢❛✐①❛ ❞❡ ▼ö❡❜✐✉s✳ ❖❜s❡r✈❛çã♦ ✸✳✶✳✶✶✳ ◆❛ ♦❜s❡r✈❛çã♦ ❛♥t❡r✐♦r ❞❡s❝r❡✈❡♠♦s ♦ ❞♦♠í♥✐♦ ❡ ❛ ✐♠❛❣❡♠ 1 → T 2

      ❞❛ ❢✉♥çã♦ Π : DomΠ ⊂ T ✳ ❚❡♠♦s q✉❡✿ s s \ (C ∪ C

      DomΠ = T 1 ) u u 1 2 s s u u ImΠ = T \ (C ∪ C ). 2 1 2 ⊂ T ∩ W (O) ⊂ T ∩ W (O) 1 1

      ❖♥❞❡ C j ❡ C j ✭j = 1, 2✮✳ ❆❧é♠ ❞✐ss♦✱ Π é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✱ ❡♥tã♦ DomΠ ❡ ImΠ sã♦ ❤♦♠❡♦♠♦r❢♦s✳ s

      (0) ❆ss✐♠✱ ♥♦ ❝❛s♦ q✉❡ W é ✉♠ ❝✐❧✐♥❞r♦ ♦❜t❡♠♦s ❞✉❛s ❝✉r✈❛s ❢❡❝❤❛❞❛s s✐♠♣❧❡s s s C 1 ❡ C 2 q✉❡ ♥ã♦ ♣❡rt❡♥❝❡♠ ❛♦ ❞♦♠í♥✐♦ ❞❛ Π ❡✱ ♥ã♦ sã♦ ❤♦♠♦tó♣✐❝❛s ❛ ✉♠ ♣♦♥t♦ ❡♠ T 1 ✭♣♦✐s✱ O ♥ã♦ é ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦ ❡♠ M ✮✱ ♥❡ss❛s ❝♦♥❞✐çõ❡s t❡♠♦s q✉❡ DomΠ é ❤♦♠❡♠♦r❢♦ ❛ ❞♦✐s ❝✐❧✐♥❞r♦s ✭♦✉ ❛♥é✐s✮✳ ❈♦♠♦ DomΠ ❡ ImΠ sã♦ 2 (O) u

      ❤♦♠❡♦♠♦r❢♦s t❡♠♦s q✉❡ ImΠ é ❤♦♠❡♦♠♦r❢❛ ❛ ❞♦✐s ❝✐❧✐♥❞r♦s ❡♠ T ✳ ▲♦❣♦✱ W u u 2 ❞❡✈❡rá ♣r♦❞✉③✐r ❞✉❛s ❝✉r✈❛s C u 1 ❡ C 2 ❡♠ T q✉❡ ♥ã♦ ♣❡rt❡♥❝❡♠ ❛ ✐♠❛❣❡♠ ❞❛ Π✱

      (O) ❡♥tã♦ W é ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠ ❝✐❧✐♥❞r♦✳ s

      (O) P❛r❛ ♦ ❝❛s♦ q✉❡ W é ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠❛ ❢❛✐①❛ ❞❡ ▼ö❡❜✐✉s ♣r♦❞✉③✐♠♦s s 1

      ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ s✐♠♣❧❡s C 1 1 ❡♠ T q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ ❞♦♠í♥✐♦ ❞❛ Π ❡ ♥ã♦ ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦ ❡♠ T ✱ ❞❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ O ♥ã♦ s❡r ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦ ❡♠ M ✱ ❛ss✐♠ DomΠ é ❤♦♠❡♦♠♦r❢♦ ❛ ✉♠ ❝✐❧✐♥❞r♦✱ ❧♦❣♦ ImΠ s❡rá ❤♦♠❡♦♠♦r❢❛ u u

      (O) 2 ❛ ✉♠ ❝✐❧✐♥❞r♦✳ P♦rt❛♥t♦✱ W ❞❡✈❡rã♦ ♣r♦❞✉③✐r ✉♠ ❝✉r✈❛ C u u 1 ❡♠ T t❛❧ q✉❡

      \ C ImΠ = T 2 (O) 1 ✱ ❡♥tã♦ W ❞❡✈❡rá s❡r ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠❛ ❢❛✐①❛ ❞❡ ▼ö❡❜✐✉s✳ s u

      (O) (O) ❈♦♥❝❧✉í♠♦s q✉❡✱ ♥❛s ❤✐♣ót❡s❡s ❞♦ ▲❡♠❛ W ❡ W sã♦ ❛♠❜❛s

      ❤♦♠❡♦♠♦r❢❛s ❛ ❝✐❧✐♥❞r♦s ♦✉ ❢❛✐①❛s ❞❡ ▼ö❡❜✐✉s✳

      ✻✼ ✸✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ 1 → T 2 ❉❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ❙❡❥❛ Π : DomΠ ⊂ T ❛ ♣r✐♠❡✐r❛ ❢✉♥çã♦ i

      (O) r❡t♦r♥♦ ❞❡ P♦✐♥❝❛ré✳ ❱❛♠♦s ❛♥❛❧✐s❛r ♦ ❝❛s♦ q✉❡ W ✱ i = u, s✱ sã♦ ❝✐❧✐♥❞r♦s s s u u s (O) (O)

      ❡✱ ♣♦r s✐♠♣❧✐❝✐❞❛❞❡✱ ❞❡♥♦t❛r❡♠♦s W ♣♦r A ❡ W ♣♦r A ✳ ❙❡❥❛♠ C j ✱ s u u j = 1, 2 j ♦s ❜♦r❞♦s ❞❡ A ❡ C ✱ j = 1, 2 ♦s ❜♦r❞♦s ❞❡ A ✱ ❡♥tã♦ s s

      \ (C ∪ C DomΠ = T 1 ) u u 1 2 s ImΠ = T \ (C ∪ C ). 2 1 2 s

      C j 1 j ✱ j = 1, 2 ♥ã♦ sã♦ ❤♦♠♦tó♣✐❝❛s ❛ ✉♠ ♣♦♥t♦ ❡♠ T ✱ ♣♦✐s C ❡ O sã♦ ❧✐✈r❡♠❡♥t❡

      ❤♦♠♦tó♣✐❝♦s ✭✈❡r ❡①❡♠♣❧♦ ❡✱ O ♥ã♦ é ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦ ❡♠ M ✱ s s ❛ss✐♠ C j ♥ã♦ é ❤♦♠♦tó♣✐❝♦ ❛ ✉♠ ♣♦♥t♦ ❡♠ M ✱ ❡♠ ♣❛rt✐❝✉❧❛r C j ♥ã♦ é ❤♦♠♦tó♣✐❝♦ u 1 j ⊂ M ❛ ✉♠ ♣♦♥t♦ ❡♠ T ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ C ✱ j = 1, 2 ♥ã♦ é ❤♦♠♦tó♣✐❝♦ ❛ ✉♠ 2

      ♣♦♥t♦ ❡♠ T ✳ s s 1 P♦❞❡♠♦s ✜①❛r ❞✉❛s ✈✐③✐♥❤❛♥ç❛s ❛♥❡❧❛r❡s ♣❡q✉❡♥❛s ❡ ❞✐s❥✉♥t❛s R s s s s 1 ✱ R 2 ❡♠ T 1 ) ⊂ DomΠ \ (R ∪ R ❞❡ C 1 ❡ C 2 ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ s❛t✐s❢❛③❡♥❞♦ T s s u u 1 2 ❡ Π(T \ (R ∪ R )) = T \ (R ∪ R ). s s u u 1 1 2 2 1 2

      , C } , C } 1 ❈♦♠♦ ❛s ❝✉r✈❛s {C 1 2 ❡ {C 1 2 ♥ã♦ sã♦ ❤♦♠♦tó♣✐❝❛s ❛ ✉♠ ♣♦♥t♦ ❡♠ T ❡ s s u u s s

      T 2 ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦✱ ❡♥❝♦❧❤❡♥❞♦ {R ❡ {R ✱ ♦s ❛♥é✐s {R ❡ u u 1 , R } , R } , R } 2 1 2 1 2 {R , R } 1 2 ♣♦❞❡♠ s❡r ❡s❝♦❧❤✐❞♦s ❝♦♠♦ ♥ã♦ ❝♦♥tr❛t❡✐s ✭✐st♦ é✱ ♥ã♦ t❡♠ ♦ ♠❡s♠♦ t✐♣♦ s s 1 2 , R }

      ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ ✉♠ ♣♦♥t♦✮ ❡♠ T ❡ T ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❣✉❡ q✉❡ {R ❡ u u 1 2 {R , R } 1 2 sã♦ ♣❛r❛❧❡❧♦s ❡♠ T ❡ T ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ 1 2

      ❉❡✈✐❞♦ à ❤✐♣❡r❜♦❧✐❝✐❞❛❞❡ ❞❛ ór❜✐t❛ O ❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦ ❝❛♠♣♦ t❡♠♦s q✉❡ s s u ∪ R

      ❛ ✉♥✐ã♦ ❞❛ s❛t✉r❛çã♦ ❞❡ R 1 2 ✭♣❡❧♦ ✢✉①♦ ❞❡ Y ✮ ❡ ♦ ❛♥❡❧ A é ✉♠ ❤♦♠❡♦♠♦r❢♦ ❛ t♦r♦ só❧✐❞♦ ST q✉❡ ❝♦♥té♠ O ✭♣♦r ❡①❡♠♣❧♦✱ ♥❛ ❋✐❣✉r❛ ❊st❡ t♦r♦ só❧✐❞♦ s s u u

      , R , R , R } t❡♠ ♦✐t♦ ❛♥é✐s ♥❛ ❢r♦♥t❡✐r❛✱ ❝♦rr❡s♣♦♥❞❡♥❞♦ ❛ q✉❛tr♦ ❛♥é✐s {R ♥❛ 1 2 1 2 1 , A , A , A } 2 3 4 ❢r♦♥t❡✐r❛ ❞❡ M ❡ ♦✉tr♦s q✉❛tr♦ ♥♦ ✐♥t❡r✐♦r {A ✭✈❡r ❋✐❣✉r❛ s s

      R 1 C 1 A 2 A u u 1 R R 2 u 1 C 1 O u C 2 A 3 s s A 4 C 2 R 2

      ❋✐❣✉r❛ ✸✳✼✳ ❙❛t✉r❛çã♦ ♣❡❧♦ ❋❧✉①♦

      ✻✽ ✸✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❤✐♣ót❡s❡✱ ♦s ❛♥é✐s ❞♦ ✐♥t❡r✐♦r sã♦ ✐♥❝♦♠♣r❡ssí✈❡✐s ❡♠ M ✱

      ♣♦✐s s❡✉s ❜♦r❞♦s ♥ã♦ sã♦ ❤♦♠♦tó♣✐❝♦s ❛ ✉♠ ♣♦♥t♦ ❡♠ M ✭♦s ❜♦r❞♦s ❞❡ss❡s ❛♥é✐s s s u u sã♦ ❢♦r♠❛❞♦s ♣❡❧♦s ❜♦r❞♦s ❞❡ R ✱ R ✱ R ✱ R ✱ q✉❡ ♥ã♦ sã♦ ❤♦♠♦tó♣✐❝♦s ❛ ✉♠ 1 2 1 2 ♣♦♥t♦ ❡♠ M ✮✱ ❡♥tã♦ ❡ss❡s ❛♥é✐s sã♦ ✐♥❝♦♠♣r❡ssí✈❡✐s ❡♠ M ✳ 1

      ❖ r❡st❛♥t❡ ❞❡ T sã♦ ❞♦✐s ❛♥é✐s ❝✉❥❛ s❛t✉r❛çã♦ ❞❡ ❝❛❞❛ ✉♠ é ❤♦♠❡♦♠♦r❢❛ ❛ ✉♠ t♦r♦ só❧✐❞♦✳ ❆ss✐♠✱ ❛ ✈❛r✐❡❞❛❞❡ M ❢♦✐ ❞✐✈✐❞✐❞❛ ❡♠ três t♦r♦s só❧✐❞♦s ❝♦❧❛❞♦s ♣❡❧♦ ❜♦r❞♦✱ ❝✉❥❛ ✐♥t❡rs❡çã♦ sã♦ ❛♥é✐s ✐♥❝♦♠♣r❡ssí✈❡s ❡♠ M ✳ ❈❛❞❛ t♦r♦

      − S só❧✐❞♦ é ✐rr❡❞✉tí✈❡❧✱ ❛ss✐♠ M ✱ ♦♥❞❡ S ❞❡♥♦t❛ ❛ ✉♥✐ã♦ ❞❡ss❡s ❛♥é✐s ❞✐s❥✉♥t♦s ❡ ✐♥❝♦♠♣r❡ssí✈❡✐s✱ é ✐rr❡❞✉tí✈❡❧✱ ❡♥tã♦ ♣❡❧❛ Pr♦♣♦s✐çã♦ t❡♠✲s❡ q✉❡ M é ✐rr❡❞✉tí✈❡❧✳ s u

      (O) (O) ❖ s❡❣✉♥❞♦ ❝❛s♦✱ ❡♠ q✉❡ W ❡ W sã♦ ❢❛✐①❛s ❞❡ ▼ö❡❜✐✉s✱ é ♠✉✐t♦ s✐♠✐❧❛r ❛♦ ♣r✐♠❡✐r♦✳ ◆❡st❡ ❝❛s♦ t❡♠♦s q✉❡ s

      DomΠ = T \ C 1 u 1 \ C s s u ImΠ = T 2 . 1 1 u ❙❡❥❛♠ R ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛♥❡❧❛r ❞❡ C 1 ❡♠ T ❡ R ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛♥❡❧❛r ❞❡

      C 1 ❡♠ T 2 ❞❡ t❛❧ ❢♦r♠❛ q✉❡✿ s u s u Π(T \ R ) = T \ R . 1 2 ❆❧é♠ ❞✐ss♦ C ❡ C ♥ã♦ sã♦ ❤♦♠♦tó♣✐❝❛s ❛ ✉♠ ♣♦♥t♦ ❡♠ M ❡✱ 1 1 1 2

      ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❡♠ T ❡ T ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦✐s O é ♥ã♦ ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ 1 2 s 3 4 ♣♦♥t♦ ❡♠ M ✳ ❉❡♥♦t❡ ♣♦r C ❡ C ♦s ❜♦r❞♦s ❞♦ ❛♥❡❧ R ❡ ♣♦r C ❡ C ♦s u s u 1

      ❜♦r❞♦s ❞♦ ❛♥❡❧ R ✱ ❝♦♠♦ C 1 ❡ C 1 ♥ã♦ sã♦ ❤♦♠♦tó♣✐❝❛s ❛ ✉♠ ♣♦♥t♦ ❡♠ T ❡ T j 2

      ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ s❡❣✉❡ q✉❡ C ✱ j = 1, 4 ♥ã♦ sã♦ ❤♦♠♦tó♣✐❝❛s ❛ ✉♠ ♣♦♥t♦ ♥♦ ❜♦r❞♦ ✭❡✱ t❛♠❜é♠ ❡♠ M ✮✳ s u

      (O) ❆ s❛t✉r❛çã♦ ♣❡❧♦ ✢✉①♦ ❞♦ ❛♥❡❧ R ✉♥✐ã♦ ❝♦♠ W é ❤♦♠❡♦♠♦r❢♦ ❛ ✉♠ t♦r♦ s u

      } , R , A 1 , A 2 1 só❧✐❞♦✱ ST ✱ ❝♦♠ q✉❛tr♦ ❛♥é✐s ♥♦ ❜♦r❞♦ {R ✱ ♦♥❞❡ A é ♦ ❛♥❡❧ ❝✉❥♦s s u 2

      ❜♦r❞♦s sã♦ ✉♠ ❜♦r❞♦ ❞♦ ❛♥❡❧ R ❡ ✉♠ ❜♦r❞♦ ❞♦ ❛♥❡❧ R ❡ A é ✉♠ ❛♥❡❧ ❝✉❥♦ s u ❜♦r❞♦ é ♦ ♦✉tr♦ ❜♦r❞♦ ❞♦ ❛♥❡❧ R ❡ ♦ ♦✉tr♦ ❜♦r❞♦ ❞♦ ❛♥❡❧ R ✱ ♣♦r ❡①❡♠♣❧♦✱ ♦s 1 1 3 2 2 4

      ❜♦r❞♦s ❞♦ ❛♥❡❧ A sã♦ C ❡ C ❡ ♦s ❜♦r❞♦s ❞♦ ❛♥❡❧ A sã♦ C ❡ C ✳ s 1 \ R 1 1 2 ❊✱ t❛♠❜é♠ T é ✉♠ ❛♥❡❧ ❡♠ T ❝✉❥♦s ❜♦r❞♦s sã♦ C ❡ C ✱ ❛ss✐♠ ❝♦♠♦ u

      T \ R 2 2 3 4 s é ✉♠ ❛♥❡❧ ❡♠ T ❝✉❥♦s ❜♦r❞♦s sã♦ C ❡ C ✳ ❆ s❛t✉r❛çã♦ ♣❡❧♦ ✢✉①♦ ❞❡ T \ R 1 1 s❡rá ❤♦♠❡♦♠♦r❢♦ ✉♠ t♦r♦ só❧✐❞♦✱ ST ✱ ❝♦❧❛❞♦ ♣❡❧♦ ❜♦r❞♦ ❝♦♠ ♦ t♦r♦ só❧✐❞♦

      ST ) ∩ ∂(ST ) = A ∪ A ✱ ♦♥❞❡ ∂(ST 1 1 2 ✳ ❖s ❛♥é✐s A 1 ❡ A 2 sã♦ ✐♥❝♦♠♣r❡ssí✈❡✐s ❡♠

      M 1 2 3 4 ✱ ✉♠❛ ✈❡③ q✉❡ s❡✉s ❜♦r❞♦s ✭C ✱ C ✱ C ❡ C ✮ ♥ã♦ sã♦ ❤♦♠♦tó♣✐❝♦s ❛ ✉♠ ♣♦♥t♦

      ❡♠ M ✳ ❆ss✐♠✱ ❛ ✈❛r✐❡❞❛❞❡ M ❢♦✐ ❞✐✈✐❞✐❞❛ ❡♠ ❞♦✐s t♦r♦s só❧✐❞♦s ❝♦❧❛❞♦s ♣❡❧♦ ❜♦r❞♦✱

      ❝✉❥❛ ✐♥t❡rs❡çã♦ sã♦ ❛♥é✐s ✐♥❝♦♠♣r❡ssí✈❡✐s ❡♠ M ✳ ❯s❛♥❞♦ ♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ ❞♦ ♣r✐♠❡✐r♦ ❝❛s♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ M é ✐rr❡❞✉tí✈❡❧✳

      ❉❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❙❡❥❛♠ M✱ X✱ T ❡ O ❝♦♠♦ ♥♦ ❡♥✉♥❝✐❛❞♦✳ T = O ❆ss✐♠ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ór❜✐t❛ O ❞❡ X q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛ T ❡♥tã♦ σ ❡✱

      ✻✾ ✸✳✷✳ P❘❖❱❆ ❉❖ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ T ♣♦rt❛♥t♦ σ é ❝♦♥❡①❛✳

      ❙❡❣✉❡ q✉❡ ❞♦ ▲❡♠❛ q✉❡ T ♥ã♦ s❡♣❛r❛ M✱ ❡♥tã♦ ♣❛r❛ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ♣r♦✈❛r q✉❡ ❛ ✈❛r✐❡❞❛❞❡ M ♦❜t✐❞❛ ♣❡❧♦ ❝♦rt❡ ❞❡ M

      ❛♦ ❧♦♥❣♦ ❞❡ T é ✐rr❡❞✉tí✈❡❧✳ P❛r❛ ♣r♦✈❛r q✉❡ M é ✐rr❡❞✉tí✈❡❧✱ ♦❜s❡r✈❛♠♦s q✉❡ M é ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ 1 2

      ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛ ❝✉❥❛ ❢r♦♥t❡✐r❛ ❝♦♥s✐st❡ ❞❡ ❞♦✐s t♦r♦s✱ T ✱ T ✳ ❙❡❥❛ Y ♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ✐♥❞✉③✐❞♦ ♣♦r X ❡♠ M ✱ ♣♦❞❡♠♦s ❡♥①❡r❣❛r Y ❝♦♠♦ s❡♥❞♦ ❛ r❡str✐çã♦ ❞❡ X ❛ M ✳ ❚❡♠♦s q✉❡ O é ❛ ú♥✐❝❛ ór❜✐t❛ ❞❡ Y q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛ ∂M ✳ ❆✜r♠❛çã♦✿ ❖ ❝❛♠♣♦ Y ❡♠ M s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s 1✱ 2 ❡ 3 ❞♦ ▲❡♠❛

      ∼ M − V T T = {x ∈ X t (T ); t ∈ (−ǫ, ǫ)} ❙❡❥❛ M ♦♥❞❡ V ✱ ❝♦♠ ǫ > 0 1 = X ǫ (T ) 2 = X −ǫ (T ) s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✳ ❉❡♥♦t❡ T ❡ T ✱ t❡♠♦s q✉❡ ♦ ❝❛♠♣♦

      X 1 2 é tr❛♥s✈❡rs❛❧ ❛ T ❡ ❛ T ✳ 1 ∈ T 1 1 = X ǫ (q)

      ❙❡ q ❡♥tã♦ q ✱ q ∈ T ✱ ❛ss✐♠ ♣❛r❛ 0 < δ < 2ǫ t❡♠♦s q✉❡ X (q ) / ∈ M −δ 1

      ✳ 1 1 ❊♥tã♦✱ ♦ ❝❛♠♣♦ ❛♣♦♥t❛ ♣❛r❛ ❞❡♥tr♦ ❞❡ T ❡♠ q ✳

      ∈ T = X (p) ❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ p λ (p ) / ∈ M 2 2 2 ❡♥tã♦ p 2 −ǫ ✱ p ∈ T ❡✱ ♣❛r❛ 0 < λ < 2ǫ s❡❣✉❡ q✉❡ X ✳ 2 2

      ▲♦❣♦✱ ♦ ❝❛♠♣♦ ❛♣♦♥t❛ ♣❛r❛ ❢♦r❛ ❞❡ T ❡♠ p ✳ P♦rt❛♥t♦ Y ❛♣♦♥t❛ ♣❛r❛ ❞❡♥tr♦ 1 2 ❡♠ T ❡ ♣❛r❛ ❢♦r❛ ❡♠ T ✱ s❛t✐s❢❛③❡♥❞♦ 1✳

      ❚❡♠♦s q✉❡ O ♥ã♦ é ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦ ❡♠ M ✱ ♣♦✐s O é ♥ã♦ ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦ ❡♠ M✳ ❆❧é♠ ❞✐ss♦✱ O é ❤✐♣❡r❜ó❧✐❝♦ ✭♣♦r ❤✐♣ót❡s❡✮✱ t✐♣♦ s❡❧❛ ✭♣❡❧♦ ▲❡♠❛ ❡ ♣❡r✐ó❞✐❝❛ ❛ss✐♠ ♥ã♦ é ❤♦♠♦tó♣✐❝❛ ❛ ✉♠ ♣♦♥t♦ ❡♠ M ✱ s❛t✐s❢❛③❡♥❞♦

      2 ❡ 3✳

      ❊♥tã♦✱ ♣❡❧♦ ▲❡♠❛ s❡❣✉❡ q✉❡ M é ✐rr❡❞✉tí✈❡❧✳

      ✸✳✷ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧

      ❚❡♦r❡♠❛ ✸✳✷✳✶✳ ❙❡❥❛ T ✉♠ t♦r♦ ♠❡r❣✉❧❤❛❞♦ ❡♠ ✉♠❛ ✸✲✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ M✳ ❙✉♣♦♥❤❛ q✉❡ 1

      ✶✳ T é tr❛♥s✈❡rs❛❧ ❛ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X✱ C ✱ ❡♠ M✳ ✷✳ ❊①✐st❡ ✉♠❛ ú♥✐❝❛ ór❜✐t❛ O ❞❡ X q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛ T ✳ ✸✳ O é ❤✐♣❡r❜ó❧✐❝♦ ❡ ♥ã♦ é ❤♦♠♦tó♣✐❝♦ ❛ ✉♠ ♣♦♥t♦ ❡♠ M✳

      ❊♥tã♦✱ T é ✐♥❝♦♠♣r❡ssí✈❡❧ ❡ M é ✐rr❡❞✉tí✈❡❧✳

      ✼✵ ✸✳✷✳ P❘❖❱❆ ❉❖ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ T ✱ X✱ M✱ O ❝♦♠♦ ♥♦ ❡♥✉♥❝✐❛❞♦✳ P♦r ✉♠ ❧❛❞♦✱ T ✱ M✱ X✱ O s❛t✐s❢❛③❡♠ ❛ ❤✐♣ót❡s❡ ❞♦ ❚❡♦r❡♠❛ ❡♥tã♦ M é ✐rr❡❞✉tí✈❡❧✳

      P♦r ♦✉tr♦ ❧❛❞♦✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ór❜✐t❛ O q✉❡ ♥ã♦ ✐♥t❡rs❡❝t❛ T t❡♠♦s q✉❡ σ T = O T

      ❡✱ ❛ss✐♠✱ σ é ❝♦♥❡①❛✳ ❈♦♠♦ M é ✐rr❡❞✉tí✈❡❧✱ s❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ q✉❡ T é ✐♥❝♦♠♣r❡ssí✈❡❧✳

      ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s

      ❖ ❛rt✐❣♦ ❞❛ r❡❢❡rê♥❝✐❛ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❛r ✉♠❛ r❡s♣♦st❛ ❛ s❡❣✉✐♥t❡ ♣❡r❣✉♥t❛✿ ❙❡❥❛ M é ✉♠❛ 3−✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ❡ ♦r✐❡♥tá✈❡❧ ❝♦♠ ✉♠ t ✢✉①♦ X ❆♥♦s♦✈ tr❛♥s✐t✐✈♦ tr❛♥s✈❡rs♦ ❛ ✉♠ t♦r♦ T ✳ ❊♥tã♦ M é t♦♣♦❧♦❣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❞✐❢❡♦♠♦r✜♠♦ ❞❡ ❆♥♦s♦✈❄

      P❛r❛ r❡s♣♦♥❞❡r ❡ss❛ q✉❡stã♦✱ ♦s ❛✉t♦r❡s ❝♦♥str✉✐r❛♠ ✉♠ ❡①❡♠♣❧♦ ♥✉♠❛ 3−

      ✈❛r✐❡❞❛❞❡ M ❝♦♠ ✉♠ ✢✉①♦ ❆♥♦s♦✈ tr❛♥s✐t✐✈♦ ❛♣r❡s❡♥t❛♥❞♦ ✉♠ t♦r♦ tr❛♥s✈❡rs❛❧ q✉❡ ♥ã♦ é t♦♣♦❧♦❣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❆♥♦s♦✈✳ ❈♦♥str♦❡♠ ✉♠❛ 3−✈❛r✐❡❞❛❞❡ M ❝♦♠♣❛❝t❛ ♦r✐❡♥tá✈❡❧ ❝✉❥♦ ❜♦r❞♦ ❝♦♥s✐st❡ ❞❡ ❞♦✐s t♦r♦s ❡ ❝♦♠ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s Y tr❛♥s✈❡rs❛❧ ❛♦ ❜♦r❞♦ ❞❡ M ❡ ❛ ✈❛r✐❡❞❛❞❡ M✱ ♦❜t✐❞❛ ♣❡❧❛ ✐❞❡♥t✐✜❝❛çã♦ ❞♦s t♦r♦s ❞♦ ❜♦r❞♦ ❞❡ M ✱ s❛t✐s❢❛③ ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛

      ❯♠❛ ú❧t✐♠❛ ♦❜s❡r✈❛çã♦ s♦❜r❡ ✉♠ ❝❛♠♣♦ X ❝♦♠♦ ♥♦ ❡♥✉♥❝✐❛❞♦ ❚❡♦r❡♠❛ é q✉❡ ❡st❡ ❝❛♠♣♦ ♥ã♦ ♣♦ss✉✐ s✐♥❣✉❧❛r✐❞❛❞❡s✳

      ❈❛s♦ p ∈ M é ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ t❡♠♦s q✉❡ p ∈ T ♦✉ p ∈ M − T ✳ ❙❡ p ∈ T p T ❡♥tã♦ 0 = X(p) ∈ T ✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦ ❞❡ T s❡r tr❛♥s✈❡rs❛❧ ❛♦ ❝❛♠♣♦ X✳ X (p) ∩ T = ∅ T ❆❣♦r❛✱ s❡ p ∈ M − T ❡♥tã♦ {p} = O ❧♦❣♦ p ∈ σ ✱ ✉♠ ❛❜s✉r❞♦ ♣♦✐s σ T X (p)

      é ✉♠❛ ú♥✐❝❛ ór❜✐t❛ O ✭O é ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛✮ ❡ O 6= O ✳ P♦rt❛♥t♦✱ X é ✉♠ ❝❛♠♣♦ s❡♠ s✐♥❣✉❧❛r✐❞❛❞❡s✳

      ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

      ❬✶❪ ❆♣❛③❛✱ ❊✳ ❙♦❜r❡ ❆tr❛t♦r❡s ❡ ❈♦♥s❡q✉ê♥❝✐❛s ❚♦♣♦❧ó❣✐❝❛s✳ ❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦ ✭✷✵✵✻✮✱ ❯❋❘❏✳

      ❬✷❪ ❇❛✉t✐st❛✱ ❙✳❀ ▼♦r❛❧❡s ❈✳ ▲❡❝t✉r❡s ♦♥ s❡❝t✐♦♥❛❧ ❆♥♦s♦✈✳ ❉✐s♣♦♥í✈❡❧ ❡♠✿ ❤tt♣✿✴✴✇✇✇✳♣r❡♣r✐♥t✳✐♠♣❛✳❜r✴❙❤❛❞♦✇s✴❙❊❘■❊❴❉✴✷✵✶✶✴✽✻✳❤t♠❧✳ Ú❧t✐♠♦ ❛❝❡ss♦ ❡♠✿ ✺ ❞❡ ❞❡③❡♠❜r♦ ❞❡ ✷✵✶✻✳

      ❬✸❪ ❇♦♥❛tt✐✱ ❈✳❀ ▲❛♥✈❡❣✐♥ ❘✳ ❯♥ ❡①❛♠♣❧❡ ❞❡ ❢♦t ❞✬❆♥♦s♦✈ tr❛♥s✐t✐❢ tr❛♥s✈❡rs❡ à ✉♥ t♦r❡ ❡t ♥♦♥ ❝♦♥❥✉❣✉é à ✉♥❡ s✉s♣❡♥s✐♦♥ ✭❋r❡♥❝❤✮✱ ❊r❣♦❞✐❝ ❚❤❡♦r② ✫ ❉②♥❛♠✳ ❙②s✳ ✶✹ ✭✶✾✾✹✮✱ ✻✸✸✲✻✹✸✳

      ❬✹❪ ❇r✉♥❡❧❧❛✱ ▼✳ ❙❡♣❛r❛t✐♥❣ t❤❡ ❜❛s✐❝ s❡ts ♦❢ ❛ ♥♦♥tr❛♥s✐t✐✈❡ ❆♥♦s♦✈ ✢♦✇✱ ❇✉❧❧✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ✷✺ ✭✶✾✾✸✮✱ ✹✽✼✲✹✾✵✳

      ❬✺❪ ❋❡♥❧❡②✱ ❙✳ ❙✉r❢❛❝❡s tr❛♥s✈❡rs❡ t♦ ♣s❡✉❞♦✲❆♥♦s♦✈ ✢♦✇s ❛♥❞ ✈✐rt✉❛❧ ✜❜❡rs ✐♥ ✸✲♠❛♥✐❢♦❧❞s✱ ❚♦♣♦❧♦❣② ✸✽ ✭✶✾✾✾✮✱ ✽✷✸✲✽✺✾✳

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

      ❬✼❪ ❍❡♠♣❡❧✱ ❏✳ ✸✲▼❛♥✐❢♦❧❞s✱ ❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝s ❙t✉❞✐❡s✱ ✽✻✳ Pr✐♥❝❡t♦♥✱ ◆❏✿ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✼✻✳

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

      ❬✾❪ ❍✐rs❝❤✱ ▼✳ ❲✳ ❉✐✛❡r❡♥t✐❛❧ ❚♦♣♦❧♦❣②✱ ●r❛❞✉❛t❡ ❚❡①ts ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✈✳ ✸✸✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ✶✾✾✹✳

      ❬✶✵❪ ▼❡❧♦✱ ❲✳ ❚♦♣♦❧♦❣✐❛ ❞❛s ❱❛r✐❡❞❛❞❡s✳ ❉✐s♣♣♦♥í✈❡❧ ❡♠✿ ✇✸✳✐♠♣❛✳❜r✴∼ ❞❡♠❡❧♦✴t♦♣♦❧♦❣✐❛❞✐❢❡r❡♥❝✐❛❧✷✵✶✶✳ Ú❧t✐♠♦ ❛❝❡ss♦ ❡♠✿ ✷✵ ❞❡ ♥♦✈❡♠❜r♦ ❞❡ ✷✵✶✻✳

      ❬✶✶❪ ▼♦✐s❡✱ ❊✳ ❊✳ ●❡♦♠❡tr✐❝ ❚♦♣♦❧② ✐♥ ❉✐♠❡♥s✐♦♥s ✷ ❛♥❞ ✸✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✳ ◆❡✇ ❨♦r❦✱ ✶✾✼✼✳

      ❬✶✷❪ ▼♦r❛❧❡s✱ ❈✳ ❆✳ ❆①✐♦♠ ❆ ✢♦✇s ✇✐t❤ ❛ tr❛♥s✈❡rs❡ t♦r✉s✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝ ✸✺✺ ✭✷✵✵✸✮✱ ✼✸✺✲✼✹✺✳

      ❬✶✸❪ ▼♦r❛❧❡s✱ ❈✳ ❆✳ ■♥❝♦♠♣r❡ss✐❜✐❧✐t② ♦❢ ❚♦r✉s tr❛♥s✈❡rs❡ t♦ ✈❡❝t♦r ✜❡❧❞s✳ ❚♦♣♦❧♦❣② Pr♦❝❡❡❞✐♥❣s✱ ❱♦❧✳ ✷✽✱ ◆♦✳ ✶✱ ✷✵✵✹✱ ✷✶✾✲✷✷✽✳

      ✼✸ ❘❊❋❊❘✃◆❈■❆❙ ❇■❇▲■❖●❘➪❋■❈❆❙

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

      ❬✶✺❪ ▲✐♠❛✱ ❊✳ ▲✳ ❱❛r✐❡❞❛❞❡s ❉✐❢❡r❡♥❝✐á✈❡✐s✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ■▼P❆✱ ✶✾✼✼✳ ❬✶✻❪ P❛❧✐s✱ ❏✳❀ ▼❡❧♦✱ ❲✳ ■♥tr♦❞✉çã♦ ❛♦s ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿

      Pr♦❥❡t♦ ❊✉❝❧✐❞❡s✱ ■▼P❆✱ ✶✾✼✽✳ ❬✶✼❪ P❡r❦♦✱ ▲✳ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ❛♥❞ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s✱ ✸r❞ ❡❞✳✱

      ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ✷✵✶✶✳ ❬✶✽❪ ❘♦❜✐♥s♦♥✱ ❈✳ ❉②♥❛♠✐❝❛❧ ❙②st❡♠✳ ❙t❛❜✐❧✐t②✱ ❙②♠❜♦❧✐❝ ❉②♥❛♠✐❝s✱ ❛♥❞

      ❝❤❛♦s✳ ❙❡❝♦♥❞ ❡❞✐t✐♦♥✳ ❙t✉❞✐❡s ✐♥ ❆❞✈❛♥❝❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❈❘❈ Pr❡ss✱ ❇♦❝❛ ❘❛t♦♥✱ ❋▲✱ ✶✾✾✾✳

      ❬✶✾❪ ❘♦❧❢s❡♥✱ ❉✳ ❑♥♦ts ❛♥❞ ▲✐♥❦s✱ ▼❛t❤❡♠❛t✐❝s ▲❡❝t✉r❡ ❙❡r✐❡s✱ ◆♦✳ ✼✳ ❇❡r❦❡❧❡②✱ ❈❛❧✐❢✳✿ P✉❜❧✐s❤ ♦r P❡r✐s❤✱ ■♥❝✳✱ ✶✾✼✻✳

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

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

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