Densidades de Variedades Estáveis Forte em Fluxos Anosov.

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  ❲■▲❨ ❙❆❘▼■❊◆❚❖ ❨❯❈❘❆ ❉❊◆❙■❉❆❉❊ ❉❊ ❱❆❘■❊❉❆❉❊❙ ❊❙❚➪❱❊■❙ ❋❖❘❚❊❙ ❊▼ ❋▲❯❳❖❙

  ❆◆❖❙❖❱ ❉✐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✐❛❡✳

  ❱■➬❖❙❆ ▼■◆❆❙ ●❊❘❆■❙ ✲ ❇❘❆❙■▲

  ✷✵✶✼

  ❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛♦s ♠❡✉s ♣❛✐s✱ ❉✐♦♠❡❞❡s ❙❛r♠✐❡♥t♦ ❡ ▼❛r✐❛ ❨✉❝r❛✳

  ✐✐

  ❉❡♣♦✐s ❞❡ ❡s❝❛❧❛r ✉♠❛ ♠♦♥t❛♥❤❛ ♠✉✐t♦ ❛❧t❛✱ ❞❡s❝♦❜r✐♠♦s q✉❡ ❤á ♠✉✐t❛s ♦✉tr❛s ♠♦♥t❛♥❤❛s ♣♦r ❡s❝❛❧❛r✳

  ◆➱▲❙❖◆ ▼❆◆❉❊▲❆

  ✐✐✐

  ❆❣r❛❞❡❝✐♠❡♥t♦s

  ❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ ♣❡❧♦s ❞♦♥s q✉❡ ♠❡ ❢♦r❛♠ ❞❛❞♦s✳ ❉♦♥s q✉❡ ♥ã♦ tr❛❜❛❧❤❡✐ ♣❛r❛ ♦❜t❡r✱ ♠❛s ❧✉t❡✐ ♣❛r❛ ❝✉❧t✐✈❛r✳ ❊ ♣♦r t❡r ♠❡ ❞❛❞♦ s❛ú❞❡ ❡ ❢♦rç❛s ♣❛r❛ ❛❧❝❛♥ç❛r ♠❡✉s ♦❜❥❡t✐✈♦s✱ ❛❧é♠ ❞❡ s✉❛ ✐♥✜♥✐t❛ ❜♦♥❞❛❞❡ ❡ ❛♠♦r✳

  ❙♦✉ ♠✉✐tíss✐♠♦ ❣r❛t♦ ❛♦s ♠❡✉s ♣❛✐s✱ ❉✐♦♠❡❞❡s ❙❛r♠✐❡♥t♦ ❡ ▼❛r✐❛ ❨✉❝r❛✱ q✉❡ ❞❡r❛♠ t♦❞♦ s❡✉s ❡s❢♦rç♦s ♣❛r❛ q✉❡ ❛❣♦r❛ ❡st❡❥❛ ❝✉❧♠✐♥❛♥❞♦ ❡st❛ ❡t❛♣❛ ❞❡ ♠✐♥❤❛ ✈✐❞❛✳ P❡❧♦ ❡①❡♠♣❧♦ ❞❡ s✉♣❡r❛çã♦✱ ❛♠♦r ❡ ♠♦t✐✈❛çã♦ ❝♦♥st❛♥t❡ q✉❡ ♠❡ ❤á ♣❡r♠✐t✐❞♦ s❡r ✉♠❛ ♣❡ss♦❛ ❞❡ ❜❡♠✱ ✈♦❝ês sã♦ ♠✐♥❤❛ ✈✐❞❛✳

  ❯♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❊♥♦❝❤ ❍✉♠❜❡rt♦ ❆♣❛③❛ ❈❛❧❧❛ ❡ ❛ ♠❡✉ ❝♦♦r✐❡♥t❛❞♦r ❇❡r♥❛r❞♦ ▼❡❧♦ ❞❡ ❈❛r✈❛❧❤♦ ♣❡❧❛s s✉❛s ❡✜❝✐❡♥t❡s ♦r✐❡♥t❛çõ❡s✱ ♣❛❝✐ê♥❝✐❛✱ ❜♦❛ ✈♦♥t❛❞❡✱ ❛♠✐③❛❞❡ ❡ ♣❡❧❛s ❝♦rr❡çõ❡s ❡ ✐♥❝❡♥t✐✈♦ ❛♦ ❧♦♥❣❡ ❞❡st❡ tr❛❜❛❧❤♦✳

  ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ✐r♠ã♦s✿ ▼❛r❝♦✱ ❋r♦✐❧❛♥✱ ❨♦♥✐✱ ❘❛✉❧✱ ❏✉❛♥❛ ❡ ▼❛r✐❜❡❧✱ q✉❡ ♠❡ ❞❡r❛♠ ❝♦r❛❣❡♠ ❡ ✐♥❝❡♥t✐✈♦ ♣❛r❛ q✉❡ ♣✉❞❡ss❡ ❡♥❝❛r❛r ❡st❡ ❞❡s✜♦✳

  ❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ❲❛❧t❡r ❍✉❛r❛❝❛ ❡ ❇✉❧♠❡r ▼❡❥í❛ ♣♦r s✉❛s ❡♥✉♠❡rá✈❡✐s ❡ ✈❛❧✐♦s❛s ❝♦♥tr✐❜✉✐çõ❡s ❞✉r❛♥t❡ ♦ ♣❡rí♦❞♦ ❞❡ ♣❡sq✉✐s❛✳

  ❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ ♣❡❧❛ ❛♠✐③❛❞❡✱ ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦ ❡ ❞❡ ❡st✉❞♦s✳ ❱♦❝ês ✜③❡r❛♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❡ ✈ã♦ ❝♦♥t✐♥✉❛r ♣r❡s❡♥t❡s ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❝♦♠ ❝❡rt❡③❛✳

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

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

  ✐✈

  ❙✉♠ár✐♦

  ✈✐

  

  ✈✐✐

  

  ✶

  

  ✹

   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

  ✷✻

   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

  ✺✽

  ❘❡s✉♠♦

  ❨❯❈❘❆✱ ❲✐❧② ❙❛r♠✐❡♥t♦✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❥✉❧❤♦ ❞❡ ✷✵✶✼✳ ❉❡♥s✐❞❛❞❡ ❞❡ ❱❛r✐❡❞❛❞❡s ❊stá✈❡✐s ❋♦rt❡s ❡♠ ❋❧✉①♦s ❆♥♦s♦✈✳ ❖r✐❡♥t❛❞♦r✿ ❊♥♦❝❤ ❍✉♠❜❡rt♦ ❆♣❛③❛ ❈❛❧❧❛✳ ❈♦♦r✐❡♥t❛❞♦r✿ ❇❡r♥❛r❞♦ ▼❡❧♦ ❞❡ ❈❛r✈❛❧❤♦✳ ◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ ♣r♦✈❛r❡♠♦s q✉❡ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈ φ : M ×R −→ M ❞❡ r

  (r ≥ 1) ❝❧❛ss❡ C ✱ ♦♥❞❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛✱ s✉❛✈❡ ❡ t❛❧ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ♥ã♦ ❡rr❛♥t❡s s❡❥❛ ✐❣✉❛❧ ❛ M✱ ❡①✐st❡♠ ❡①❛t❛♠❡♥t❡ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿ q✉❡ ❝❛❞❛ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❢♦rt❡ ❡ ✐♥stá✈❡❧ ❢♦rt❡ é ❞❡♥s❛ ❡♠ M t

  ♦✉ φ é ❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❆♥♦s♦✈ ❞❡ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ 1 ❝♦♠♣❛❝t❛ C ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠ ❡♠ M✳

  ✈✐

  ❆❜str❛❝t

  ❨❯❈❘❆✱ ❲✐❧② ❙❛r♠✐❡♥t♦✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❏✉❧②✱ ✷✵✶✼✳ ❉❡♥s✐t② ♦❢ ❙tr♦♥❣ ❙t❛❜❧❡ ▼❛♥✐❢♦❧❞s ✐♥ ❆♥♦s♦✈ ❋❧♦✇s✳ ❆❞✈✐s❡r✿ ❊♥♦❝❤ ❍✉♠❜❡rt♦ ❆♣❛③❛ ❈❛❧❧❛✳ ❈♦✲❛❞✈✐s❡r✿ ❇❡r♥❛r❞♦ ▼❡❧♦ ❞❡ ❈❛r✈❛❧❤♦✳ r

  (r ≥ 1) ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ✇✐❧❧ ♣r♦✈❡ t❤❛t ❢♦r ❛ ✢♦✇ φ : M × R −→ M ♦❢ ❝❧❛ss❡ C ✱ ✇❤❡r❡ M ✐s ❛ s♠♦♦t❤ ❝♦♠♣❛❝t ❝♦♥♥❡❝t❡❞ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ ❛♥❞ s✉❝❤ t❤❛t t❤❡ s❡t ♦❢ ♥♦♥✇❛♥❞❡r✐♥❣ ♣♦✐♥ts ✐s ❡q✉❛❧ t♦ M✱ t❤❡r❡ ❛r❡ ❡①❛❝t❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ t

  ❡❛❝❤ str♦♥❣ st❛❜❧❡ ❛♥❞ ❡❛❝❤ str♦♥❣ ✉♥st❛❜❧❡ ♠❛♥✐❢♦❧❞ ✐s ❞❡♥s❡ ✐♥ M✱ ♦r φ ✐s 1 t❤❡ s✉s♣❡♥s✐♦♥ ♦❢ ❛♥ ❆♥♦s♦✈ ❞✐✛❡♦♠♦r♣❤✐s♠ ♦❢ ❛ ❝♦♠♣❛❝t C s✉❜♠❛♥✐❢♦❧❞ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ M✳

  ✈✐✐

  ■♥tr♦❞✉çã♦

  P❛r❛ ❝♦♠❡ç❛r ❛ ❡st✉❞❛r s✐st❡♠❛s ❞✐♥â♠✐❝♦s ♣r❡❝✐s❛r❡♠♦s ❞❡ três ✐♥❣r❡❞✐❡♥t❡s✿ ✭✐✮ ❯♠ ❡s♣❛ç♦ ❞❡ ❢❛s❡ M ❛r❜✐trár✐♦✱ q✉❡ é ♦ ❝♦♥❥✉♥t♦ ♦♥❞❡ ✈❛♠♦s ❛ t❡r ❛

  ❞✐♥â♠✐❝❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛✱ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ ❡ ✉♠❛ ❞❛s ♠❛✐s ✐♥t❡r❡ss❛♥t❡s q✉❡ ❞á ♣r♦♣r✐❡❞❛❞❡s ❢✉♥❞❛♠❡♥t❛r❡s s♦❜r❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s ❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐❛❧✳ ◆❡st❡ tr❛❜❛❧❤♦ M ✈❛✐ s❡r ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛ ❡ s✉❛✈❡✱ ❝✉❥♦s ❡❧❡♠❡♥t♦s ♦✉ ♣♦♥t♦s r❡♣r❡s❡♥t❛♠ ♦s ♣♦ssí✈❡✐s ❡st❛❞♦s ❞♦ s✐st❡♠❛✳

  ✭✐✐✮ ❯♠ t❡♠♣♦✱ q✉❡ ♣♦❞❡ s❡r ❞✐s❝r❡t♦ ♦✉ ❝♦♥t✐♥✉♦✳ ✭✐✐✐✮ ❆ ❧❡✐ ❞♦ ♠♦✈✐♠❡♥t♦✱ ❡st❛ ❧❡✐ é ✉♠❛ r❡❣r❛ q✉❡ ♣❡r♠✐t❡ ❞❡t❡r♠✐♥❛r ♦ ❡st❛❞♦

  ❞❡ s✐st❡♠❛ ❡♠ ❝❛❞❛ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ t ❛ ♣❛rt✐r ❞♦s ❡st❛❞♦s ❞♦ s✐st❡♠❛ ❡♠ t♦❞♦s ♦s ✐♥st❛♥t❡s ❞❡ t❡♠♣♦ ❛♥t❡r✐♦r❡s✳ ❉✐t❛ ❧❡✐ ♣❛r❛ ♥♦s s✐♠♣❧❡s♠❡♥t❡ ✈❛✐ s❡r ❞❛❞♦ ♣♦r ✉♠ ✢✉①♦ φ✳

  ❖ ❡st✉❞♦ ♠♦❞❡r♥♦ ❞❛ ❞✐♥â♠✐❝❛ ❞♦s ✢✉①♦s ❢♦✐ ✐♥✐❝✐❛❞❛ ❛ ✜♥❛✐s ❞♦ sé❝✉❧♦ ❳■❳ ❡ ✐♥í❝✐♦s ❞♦ ❳❳✱ ♣♦r ▲✐❛♣✉♥♦✈✱ ❇✐r❦❤♦✛ ❡ P♦✐♥❝❛ré q✉❡ ✐♥tr♦❞✉③ ❛ ✐❞❡✐❛ ❞❡ ❞❡s❝r❡✈❡r q✉❛❧✐t❛t✐✈❛♠❡♥t❡ ❛s s♦❧✉çõ❡s ❞❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s q✉❡ ♥ã♦ ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞❛s ❛♥❛❧✐t✐❝❛♠❡♥t❡✳

  ❯♠ ❞♦s ♣r❡❝✉rs♦r❡s ❞❛ t❡♦r✐❛ ❤✐♣❡r❜ó❧✐❝❛ ❡♠ s✐st❡♠❛s ❞✐♥â♠✐❝♦s é ❉✳ ❱✳ ❆♥♦s♦✈✱ q✉❡ ❡♠ 1967 ❡st✉❞❛ ♦s U − systems✱ ❛❣♦r❛ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ F luxos Anosov

  ✱ q✉❡ ♣♦r ❞❡✜♥✐çã♦ é q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❞❡s❝♦♠♣♦s✐çã♦ ❝♦♥t✐♥✉❛ ❞♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡ M s♦❜r❡ t♦❞❛ ❛ ✈❛r✐❡❞❛❞❡ ❡♠ ✉♠ s✉❜✜❜r❛❞♦ ❝♦♥tr❛t♦r✱ ✉♠ s✉❜✜❜r❛❞♦ ❡①♣❛♥s♦r ❡ ❛ ❞✐r❡çã♦ ❞♦ ✢✉①♦✳ ❖ ✢✉①♦ ❆♥♦s♦✈ ❞❡s❡♠♣❡♥❤♦✉ ✉♠ ♣❛♣❡❧ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♥❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s ❞✐❢❡r❡♥❝✐á✈❡✐s✱ s❡♥❞♦ ❡st❡ ❡str✉t✉r❛❧♠❡♥t❡ ❡stá✈❡❧ ❡ ❣❧♦❜❛❧♠❡♥t❡ ❤✐♣❡r❜ó❧✐❝♦✳ ■st♦ é ✉♠ ❞♦s ♠♦t✐✈♦s q✉❡ ✐♥❝❡♥t✐✈❛♠ ♦ ❡st✉❞♦ ❞❡ ✢✉①♦s ❆♥♦s♦✈✳

  ❯♥s ❞♦s ♣r✐♥❝✐♣❛✐s ❡①❡♠♣❧♦s ❞❡ ✢✉①♦s ❆♥♦s♦✈ sã♦ ♦s ✢✉①♦s ❣❡♦❞és✐❝♦s ♥♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ✉♥✐tár✐♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛ ❞❡ ❝✉r✈❛t✉r❛ ♥❡❣❛t✐✈❛ ❡ ❛s s✉s♣❡♥sõ❡s ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s ❞❡ ❆♥♦s♦✈ ✭✈❡r ❊st❡s ✢✉①♦s s❡rã♦ tr❛♥s✐t✐✈♦s✱ ♥♦ ♣r✐♠❡✐r♦ ❝❛s♦ s❡ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ é ❞❡✜♥✐❞♦ ♥✉♠ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ✉♥✐tár✐♦ ♥✉♠❛ s✉♣❡r✜❝✐❡ ❢❡❝❤❛❞❛ ❞❡ ❝✉r✈❛t✉r❛ ❣❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡ ❡ ✐❣✉❛❧ ❛ −1 ✳

  ❆✳ ❱❡r❥♦✈s❦② ❝♦♥❥❡t✉r♦✉ q✉❡ t♦❞♦ ✢✉①♦ ❆♥♦s♦✈ ❞❡ ❝♦❞✐♠❡♥sã♦ 1 ♥✉♠❛ t ) = ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ❞❡ ❞✐♠❡♥sã♦ ♠❛✐♦r ✐❣✉❛❧ ❛ 3 é tr❛♥s✐t✐✈♦✱ ✐st♦ é✱ q✉❡ Ω(φ

  ✶

  ✷ M

  ✳ ❖ q✉❡ ♥ã♦ é ✈á❧✐❞♦ ❡♠ ❞✐♠❡♥sã♦ ✸ ♣♦✐s ❏✳ ❋r❛♥❦ ❡ ❘✳ ❲✐❧❧✐❛♠s ❝♦♥str✉✐r❛♠ ❡♠ ✶✾✽✵ ♦ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦ ❞❡ ✢✉①♦s ❞❡ ❆♥♦s♦✈ ♥ã♦ tr❛♥s✐t✐✈♦s ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❋❧✉①♦s ❞❡ ❆♥♦s♦✈ ❆♥ô♠❛❧♦s ❊♠ 1974✱ ❆✳ ❱❡r❥♦✈s❦② ♣r♦✈♦✉ q✉❡ t♦❞♦ ✢✉①♦ ❆♥♦s♦✈ ❞❡ ❝♦❞✐♠❡♥sã♦ 1 ♥✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ❞❡ ❞✐♠❡♥sã♦ ♠❛✐♦r ❛ 3 é tr❛♥s✐t✐✈♦✱ ♦✉tr❛ ❞❡♠♦♥str❛çã♦ q✉❡ é ✉♠❛ s✐♠♣❧✐✜❝❛çã♦ ❞❡ ❆✳ ❱❡r❥♦✈s❦② é ❞❛❞❛ ♣♦r ❚✳ ❇❛r❜♦t ♥♦ q✉❛❧ ❡①♣❧✐❝❛ ♣♦r q✉❡ ❝♦♥❥❡t✉r❛ ❢❛❧❤❛ ❡♠ ❞✐♠❡♥sã♦ 3✳ t ) = M (x) (x) u s

  ❙❡ Ω(φ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈✱ ❡♥tã♦ W ❡ W sã♦ ❞❡♥s♦s ❡♠ M ♣❛r❛ ❝❛❞❛ x ∈ M ✳ ▼❛s✱ ♥❡♠ s❡♠♣r❡ é ♦ ❝❛s♦ ❞❡ q✉❡ ❝❛❞❛ ✉♠❛ ❞❛s ❢♦❧❤❛s ❞❡ uu ss F t

  ♦✉ ❞❡ F é ❞❡♥s♦ ❡♠ M✳ ■st♦ ♥ã♦ ♦❝♦rr❡✱ ♣♦r ❡①❡♠♣❧♦✱ s❡ φ é ❛ s✉s♣❡♥sã♦ uu ❞❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❆♥♦s♦✈✳ ◆❡st❡ ❝❛s♦✱ ❝❛❞❛ ❢♦❧❤❛ ❞❡ F ❡ ❝❛❞❛ ❢♦❧❤❛ ❞❡ ss F s✐t✉❛♠✲s❡ ♥✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ❞❡ ❝♦❞✐♠❡♥sã♦ 1 ❡♠ M✳

  ❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❛ ❞✐s❡rt❛çã♦ é ♠♦str❛r ✉♠ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ ♣♦r ❏♦s❡♣❤ ❋✳ P❧❛♥t❡ ❡♠ ❈✉❥♦ ❡♥✉♥❝✐❛❞♦ é ♦ s❡❣✉✐♥t❡✿

  ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧✳ ❙❡❥❛♠ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛✱ r (r ≥ 1)

  ❝♦♥❡①❛✱ s✉❛✈❡ ❡ φ : M × R −→ M ✉♠ ✢✉①♦ ❆♥♦s♦✈ ❞❡ ❝❧❛ss❡ C t❛❧ t ) = M q✉❡ Ω(φ ✳ ❊♥tã♦ ❡①✐st❡♠ ❡①❛t❛♠❡♥t❡ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿ ✐✮ ❈❛❞❛ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❢♦rt❡ ❡ ✐♥stá✈❡❧ ❢♦rt❡ é ❞❡♥s♦ ❡♠ M✱ ♦✉ t

  ✐✐✮ φ é ❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❆♥♦s♦✈ ❞❡ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ 1 ❝♦♠♣❛❝t❛ C ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠ ❡♠ M✳

  ■st♦ é✱ ♦s ✢✉①♦s ❆♥♦s♦✈ tr❛♥s✐t✐✈♦s q✉❡ ♥ã♦ sã♦ s✉s♣❡♥sõ❡s t❡♠ ❢♦❧❤❡❛çõ❡s ❡stá✈❡✐s ❡ ✐♥stá✈❡✐s ❢♦rt❡s ❞❡♥s❛s ❡♠ M✳ ❆❣♦r❛✱ ❞❛q✉✐ s✉r❣❡ ✉♠❛ ♣❡r❣✉♥t❛ ♥❛t✉r❛❧✱ ❡①✐st❡♠ ✢✉①♦s ❆♥♦s♦✈ tr❛♥s✐t✐✈♦s q✉❡ ♥ã♦ s❡❥❛♠ s✉s♣❡♥sã♦❄ ❈✳ ❇♦♥❛tt✐ ❡ ❘✳ ▲❛♥❣❡✈✐♥ r❡s♣♦♥❞❡♠ ❡st❛ ♣❡r❣✉♥t❛ ❡♠ ❞❡ ♠❛♥❡✐r❛ ♣♦s✐t✐✈❛✳

  ❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡st❛ ♦r❣❛♥✐③❛❞♦ ❝♦♠♦ s❡❣✉❡✿ ◆♦ ❝❛♣✐t✉❧♦ 1✱ ❛♣r❡s❡♥t❛♠♦s ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❝♦♠♦ ✈❛r✐❡❞❛❞❡s

  ❞✐❢❡r❡♥❝✐á✈❡✐s✱ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✱ ❢♦❧❤❡❛çõ❡s✱ ❡ ♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ✢✉①♦s ❆♥♦s♦✈ ❥✉♥t♦ ❝♦♠ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❢✉♥❞❛♠❡♥t❛✐s ❡ ❡♥tr❡ ♦✉tr♦s ❢❛t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✳

  ◆♦ ❝❛♣✐t✉❧♦ 2✱ ❛♣r❡s❡♥t❛♠♦s ❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣♦s✐çõ❡s ❡ ❧❡♠❛s ❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ♠♦str❛r ♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧✳ ❆ ❡str❛té❣✐❛ ♣❛r❛ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧✱ ❛ ❣r♦ss♦ ♠♦❞♦✳ s❡rá ❛ s❡❣✉✐♥t❡✳ ❙✉♣♦r q✉❡ ♥ã♦ ❛❝♦♥t❡❝❡ ♦ ✐t❡♠ (i)✱ ❡♥tã♦ ❡①✐st❡ ✉♠ uu ss

  (x) (x) ♣♦♥t♦ x ∈ M t❛❧ q✉❡ W ♦✉ W ♥ã♦ é ❞❡♥s❛ ❡♠ M✳ ▲♦❣♦✱ ♦ ▲❡♠❛ t (p) uu ❣❛r❛♥t❡ ❛ ❡①✐st❡ ❞❡ ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ p ❞❡ φ t❛❧ q✉❡ W ♥ã♦ é ❞❡♥s❛ ❡♠ M t

  ✳ ❉❛q✉✐✱ ❞❛ Pr♦♣♦s✐çã♦ s❡❣✉❡ q✉❡ φ é ❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ φ | uu r W ✱ ♦♥❞❡ r é ♦ ♣❡rí♦❞♦ ❞❡ p✳ (p) uu

  (p) ❆ss✐♠✱ ♣❛r❛ ❝♦♥❝❧✉✐r ❜❛st❛ ♠♦str❛r q✉❡ W s❡❥❛ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ M✳ uu ss

  (x) (x) ❈♦♠ ❡ss❡ ♣r♦♣ós✐t♦ ♦❜s❡r✈❡♠♦s q✉❡✱ ❝♦♠♦ W ♦✉ W ♥ã♦ é ❞❡♥s❛ ❡♠ M✱ uu ss s❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ q✉❡ F ❡ F sã♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✱ ❞❛q✉✐ ❞❛ u s

  ⊕ E Pr♦♣♦s✐çã♦ s❡❣✉❡ q✉❡ E é ✐♥t❡❣rá✈❡❧✳ ▲♦❣♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ s✉❜❡s♣❛ç♦ 1 u s

  ⊕ E ✐♥t❡❣rá✈❡❧✱ ❡①✐st❡ ✉♠❛ ❢♦❧❤❡❛çã♦ F ❞❡ ❝❧❛ss❡ C t❛♥❣❡♥t❡ ❛ E ✳ ❙❡❥❛ L ✉♠❛

  ✸ uu (p)

  ❢♦❧❤❛ ❞❡ F✳ ❙❡ ♠♦str❛♠♦s q✉❡ L = W s❡ ❝♦♥❝❧✉✐r✐❛ ❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛✱ uu 1 (p)

  ♣♦✐s t❡rí❛♠♦s q✉❡ W é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ ❝❧❛ss❡ C ✱ ❥á q✉❡ L ∈ F é ✉♠❛ 1 s✉❜✈❛r✐❡❞❛❞❡ ❞❡ ❝❧❛ss❡ C ❞❡ M✳ ❆❧é♠ ❞✐ss♦ ❝♦♠♣❛❝t❛✱ ❞❡s❞❡ q✉❡ M é ❝♦♠♣❛❝t❛ uu (p) ⊂ M t

  ❡ W ❢❡❝❤❛❞❛✳ ❊ ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠ ❡♠ M ❞❡s❞❡ q✉❡ φ é ❛ s✉s♣❡♥sã♦ r | uu ❞❡ φ W ✳ (p) uu uu

  (p) (p)) P❛r❛ ✐st♦✱ ❜❛st❛ ♦❜s❡r✈❛r q✉❡ L é ❞❡♥s♦ ❡♠ W ✭✐st♦ é✱ L = W ❡ uu

  (p) q✉❡ L é ❢❡❝❤❛❞♦ ✭✐st♦ é✱ L = L)✱ ♣♦✐s ❝❧❛r❛♠❡♥t❡ ❞❛q✉✐ t❡rí❛♠♦s q✉❡ L = W ✳

  ❈❛♣ít✉❧♦ ✶ Pr❡❧✐♠✐♥❛r❡s

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

  ❆s ♣r✐♥❝✐♣❛✐s r❡❢❡r❡♥❝✐❛s sã♦✿

  ✶✳✶ ❚❡♦r✐❛ ❜ás✐❝❛ ❞❛ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛

  ■♥✐❝✐❛r❡♠♦s ♦ tr❛❜❛❧❤♦ ❢❛③❡♥❞♦ ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ ❞❡ ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐❛✐s ❡ ❣❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✱ ❢♦❝❛♥❞♦ ❡♠ ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ ❥✉❧❣❛♠♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ ❞✐ss❡rt❛çã♦✳ P❛r❛ ✐st♦ s❡❣✉✐♠♦s ♦ ❧✐✈r♦ ❞♦ ♣r♦❢❡ss♦r ▼❛♥❢r❡❞♦

  ❱❛r✐❡❞❛❞❡s ❉✐❢❡r❡♥❝✐á✈❡✐s

  ❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛ M ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❯♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛s ❧♦❝❛✐s ♦✉ ❝❛rt❛ ❧♦❝❛❧ ❡♠ M é ♦ ♣❛r (U, ϕ) ♦♥❞❡ ϕ : U −→ ϕ(U) é ✉♠ m ❤♦♠❡♦♠♦r✜s♠♦ ❞❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ U ⊂ M sô❜r❡ ✉♠ ❛❜❡rt♦ ϕ(U) ⊂ R ✳ ❉❡✜♥✐çã♦ ✶✳✷✳ ❯♠ ❛❧t❛s ❞❡ ❞✐♠❡♥sã♦ m sô❜r❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ M é ✉♠❛ m ❝♦❧❡çã♦ A ❞❡ s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ϕ : U −→ R ❡♠ M✱ ❝✉❥♦s ❞♦♠í♥✐♦s U

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

  ❉❡✜♥✐çã♦ ✶✳✸✳ ❉❛❞♦s ♦s s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ϕ : U −→ R ❡ m ψ : V −→ R −1 ♥♦ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ M✱ t❛✐s q✉❡ U ∩ V 6= ∅✳ ❖ ❤♦♠❡♦♠♦r✜s♠♦ ψ ◦ ϕ

  : ϕ(U ∩ V ) −→ ψ(U ∩ V ) é ❝❤❛♠❛❞♦ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛✳

  ✺ ✶✳✶✳ ❚❡♦r✐❛ ❜ás✐❝❛ ❞❛ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ❋✐❣✉r❛ ✶✳✶✿ ❈❛rt❛ ▲♦❝❛❧✳

  ❋✐❣✉r❛ ✶✳✷✿ ▼✉❞❛♥ç❛ ❞❡ ❈♦♦r❞❡♥❛❞❛✳ r ❯♠ ❛t❧❛s A é ❞✐t♦ ❞❡ ❝❧❛ss❡ C ✱ 1 ≤ r ≤ ∞✱ s❡ t♦❞❛s ❛s ♠✉❞❛♥ç❛s ❞❡ r

  ❝♦♦r❞❡♥❛❞❛s ❞♦ ❛t❧❛s sã♦ ❞❡ ❝❧❛ss❡ C ✳ m ❯♠ s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ϕ : U −→ R ❞❡ M ❞✐③✲s❡ ❛❞♠✐ssí✈❡❧ r r❡❧❛t✐✈❛♠❡♥t❡ ❛ ✉♠ ❛t❧❛s A ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C ✱ r > 0✱ ❞❡ M s❡ ♣❛r❛ m

  ❝❛❞❛ ψ ∈ A ❝♦♠ U ∩ V 6= ∅✱ ♦♥❞❡ ψ : V −→ R t❡♠✲s❡ q✉❡ ❛s ♠✉❞❛♥ç❛s ❞❡ −1 −1 r ❝♦♦r❞❡♥❛❞❛s ψ ◦ ϕ ❡ ϕ ◦ ψ sã♦ ❞❡ ❝❧❛ss❡ C ✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ s❡ A ∪ {ϕ} r é ❛✐♥❞❛ ✉♠ ❛t❧❛s ❞❡ ❝❧❛ss❡ C ❡♠ M✳ r ❉❡✜♥✐çã♦ ✶✳✹✳ ❯♠ ❛t❧❛s A ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C ✱ r > 0✱ ❞❡ M é ❝❤❛♠❛❞♦ ♠á①✐♠♦ s❡ ❝♦♥té♠ t♦❞♦s ♦s s✐st❡♠❛s ❝♦♦r❞❡♥❛❞❛s q✉❡ sã♦ ❛❞♠✐ssí✈❡✐s ❡♠ r❡❧❛çã♦

  ✻ ✶✳✶✳ ❚❡♦r✐❛ ❜ás✐❝❛ ❞❛ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ ❛ A✳ r

  ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ t♦❞♦ ❛t❧❛s ❞❡ ❞✐♠❡♥sã♦ m ❡ ❞❡ ❝❧❛ss❡ C ✱ r > 0✱ r ❞❡ M✱ ♣♦❞❡ 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 >

  ✱ é ✉♠ ♣❛r (M, A) ♦♥❞❡ M é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❞❡ ❍❛✉s❞♦r✛✱ ❝♦♠ ❜❛s❡ r ❡♥✉♠❡rá✈❡❧ ❡ A é ✉♠ ❛t❧❛s ♠á①✐♠♦ ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C ✳ r ❊①❡♠♣❧♦ ✶✳✶✳ ❚♦❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C ✱ é ✉♠❛ ✈❛r✐❡❞❛❞❡ r ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C ✱ ❝♦♠ ♦ ❛t❧❛s A ❢♦r♠❛❞♦ ♣❡❧♦s s✐st❡♠❛s m r

  ❞❡ ❝♦♦r❞❡♥❛❞❛s ϕ : U −→ R q✉❡ sã♦ ✐♥✈❡rs❛s ❞❛s ♣❛r❛♠❡tr✐③❛çõ❡s ❞❡ ❝❧❛ss❡ C ✳ ❉❡✜♥✐çã♦ ✶✳✻✳ ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ❞✐♠❡♥sã♦ m ❡ n r❡s♣❡t✐✈❛♠❡♥t❡✳ ❉✐③❡♠♦s q✉❡ f : M −→ N é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ s❡ f é ❜✐❥❡t♦r❛✱ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ ❝♦♠ ✐♥✈❡rs❛ ❞✐❢❡r❡♥❝✐á✈❡❧✳

  ❙❡ ❡①✐st❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❡♥tr❡ ❞✉❛s ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s M ❡ N ❞✐③❡♠♦s q✉❡ ❡❧❡s sã♦ ❞✐❢❡♦♠♦r❢❛s✳ ❉❡✜♥✐çã♦ ✶✳✼✳ ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ❞✐♠❡♥sã♦ m ❡ n r❡s♣❡t✐✈❛♠❡♥t❡✳ ❯♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ f : M −→ N é ✉♠❛ s✉❜♠❡rsã♦ s❡ Df (p) : T p M −→ T f N (p) p é s♦❜r❡❥❡t♦r❛ ♣❛r❛ t♦❞♦ p ∈ M✱ ✐st♦ é✱ ♦ ♣♦st♦ ❞❛ ♠❛tr✐③ ❥❛❝♦❜✐❛♥❛ (Jf) é ✐❣✉❛❧ ❛ n ♣❛r❛ t♦❞♦ p ∈ M✳

  ❱❛❧❡ r❡ss❛❧t❛r q✉❡ s❡ f : M −→ N é ✉♠❛ s✉❜♠❡rsã♦ ❡♥tã♦ m ≥ n✳ ❚❡♦r❡♠❛ ✶✳✶✳ ✭❋♦r♠❛ ▲♦❝❛❧ ❞❛s ❙✉❜♠❡rsõ❡s✮✳

  ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ❞✐♠❡♥sã♦ m ❡ n r❡s♣❡t✐✈❛♠❡♥t❡ r ❡ f : M −→ N ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❝❧❛ss❡ C ✱ r ≥ 1 q✉❡ é ✉♠❛ m s✉❜♠❡rsã♦ ♥✉♠ ♣♦♥t♦ p ∈ M✳ ❊♥tã♦ ❡①✐st❡♠ ❝❛rt❛s ❧♦❝❛✐s ϕ : U −→ R ✱ p ∈ U n m n m −n

  × R = R

  ❡ ψ : V −→ R ✱ q = f(p) ∈ V ❡ ✉♠❛ ❞❡s❝♦♠♣♦s✐çã♦ R t❛❧ q✉❡ −1 f (U ) ⊂ V (x, y) = x ❡ ψ ◦ f ◦ ϕ ✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ f é ❧♦❝❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡

  ❛ ♣r♦❥❡çã♦ (x, y) 7→ x✳ ❉❡✜♥✐çã♦ ✶✳✽✳ ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ❞✐♠❡♥sã♦ m ❡ n r❡s♣❡t✐✈❛♠❡♥t❡✳ ❯♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ f : M −→ N é ✉♠❛ ✐♠❡rsã♦ p M −→ T f N (p) s❡ Df(p) : T é ✐♥❥❡t♦r❛ ♣❛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❡♥❝✐❛ n − m é ❝❤❛♠❛❞❛ ❞❡ ❝♦❞✐♠❡♥sã♦ ❞❛ ✐♠❡rsã♦ f✳ ❉❡✜♥✐çã♦ ✶✳✾✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❉❡✜♥✐♠♦s ♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡ M ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦✿

  T M M } = {(x, v) : x ∈ M, v ∈ T x

  ✼ ✶✳✷✳ ◆♦çõ❡s ❞❡ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s

  ❱❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛

  ❉❡✜♥✐çã♦ ✶✳✶✵✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❯♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ✭♦✉ ❡str✉t✉r❛ ❘✐❡♠❛♥♥✐♥❛ ✮ ❡♠ M é ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ q✉❡ p p M ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦ p ∈ M✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ <, > ♥♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ T ✱ n

  −→ M q✉❡ ✈❛r✐❛ ❞❡ ❢♦r♠❛ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ✐st♦ é✱ s❡ X : U ⊂ R é ✉♠ s✐st❡♠❛ 1 , ..., x n ) = p ∈ X(U ) ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ❡♠ t♦r♥♦ ❞❡ ✉♠ ♣♦♥t♦ p✱ ❝♦♠ X(x ∂ ∂ ∂ (p) (p) (p)

  , > , ..., x = dx p (0, ..., 1, ...0) p = g ij (x n ) 1

  ❡ ∂x ✱ ❡♥tã♦ < ∂x ∂x é ✉♠❛ ❢✉♥çã♦ i i j ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ U✳

  ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ♠✉♥✐❞❛ ❞❡ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐♥❛ ❝❤❛♠❛✲s❡ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✳ ❉❡✜♥✐çã♦ ✶✳✶✶✳ ❉✐③✲s❡ q✉❡ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ g ♥✉♠❛ ✈❛r✐❡❞❛❞❡ r ❞✐❢❡r❡♥✐❝✐á✈❡❧ M ❞❡ ❞✐♠❡♥sã♦ m é ❞❡ ❝❧❛ss❡ C ✱ r > 0✱ s❡✱ ♣❛r❛ ❝❛❞❛ ❝❛rt❛ m x m m r x

  : U ⊂ M −→ x(U ) ⊂ R : x(U ) × R × R −→ R ❛ ❢✉♥çã♦ g é ❞❡ ❝❧❛ss❡ C ✱ x r 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✳ r Pr♦♣♦s✐çã♦ ✶✳✶✳ ❚♦❞❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M ❞❡ ❝❧❛ss❡ C ✱ r > 0✱ ❛❞♠✐t❡ r −1 ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡ ❝❧❛ss❡ C ✳

  ❉❡♠♦♥str❛çã♦✿ ❱❡r ❡♠ ♣á❣✐♥❛ 210✳ 1 2 ❉❡✜♥✐çã♦ ✶✳✶✷✳ ❉✉❛s ✈❛r✐❡❞❛❞❡s M ❡ M sã♦ tr❛♥s✈❡rs❛✐s ❡♠ M s❡ ♣❛r❛ t♦❞♦ 1 ∩ M p M p M p M 2 1 2

  ♣♦♥t♦ p ∈ M t❡♠♦s q✉❡ ♦s ❡s♣❛ç♦s t❛♥❣❡♥t❡s ❞❡ T ❡ T ❣❡r❛♠ T ✳

  ✶✳✷ ◆♦çõ❡s ❞❡ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s

  ◆❡st❡ s❡çã♦ s❡rã♦ ❞❛❞♦s ♦s ❝♦♥❝❡✐t♦s✱ ♥♦t❛çõ❡s ❡ r❡s✉❧t❛❞♦s ❡ss❡♥❝✐❛✐s s♦❜r❡ ✢✉①♦s✱ ❢♦❧❤❡❛çõ❡s ❡ ✢✉①♦s ❆♥♦s♦✈✳ ❉❛q✉✐ ❡♠ ❞✐❛♥t❡ M ❞❡♥♦t❛✱ s❛❧✈♦ ♠❡♥çã♦ ❡♠ ❝♦♥trár✐♦✱ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛ ✱❝♦♥❡①❛ ❡ s✉❛✈❡✳ ❆s ♣r✐♥❝✐♣❛✐s r❡❢❡r❡♥❝✐❛s ♣❛r❛ ❡st❡ s❡ç❛♦ ❢♦r❛♠

  ❚❡♦r✐❛ ❞❡ ✢✉①♦s r

  (r ≥ 1) ❉❡✜♥✐çã♦ ✶✳✶✸✳ ❯♠ ✢✉①♦ ❞❡ ❝❧❛ss❡ C ❡♠ M é ✉♠❛ ❛♣❧✐❝❛çã♦ r φ

  : M × R −→ M (r ≥ 1) ❞❡ ❝❧❛ss❡ C t❛❧ q✉❡✿

  ✐✮ φ(x, 0) = x✳ ✐✐✮ φ(x, t + s) = φ(φ(x, t), s) ♣❛r❛ t♦❞♦ s, t ∈ R✳

  ✽ t (x) t t : M −→ ✶✳✷✳ ◆♦çõ❡s ❞❡ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s

  P♦❞❡♠♦s ❞❡♥♦t❛r φ ❛ φ(t, x)✱ ❞❡♥♦♠✐♥❛r φ ❛♦ ✢✉①♦ ❡ ❡s❝r❡✈❡r φ M

  ✳ t ❉❡✜♥✐çã♦ ✶✳✶✹✳ ❆ ór❜✐t❛ ❞❡ ✉♠ ♣♦♥t♦ x ∈ M ❝♦♠ r❡s♣❡✐t♦ ❛♦ ✢✉①♦ φ é ♦ ❝♦♥❥✉♥t♦ O (x) = {φ t (x) : t ∈ R}.

  ❯♠❛ ór❜✐t❛ ❢❡❝❤❛❞❛ ❞❡ ✉♠ ♣♦♥t♦ x ∈ M é q✉❛♥❞♦ O(x) é ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦✳ t

  ❉❡✜♥✐çã♦ ✶✳✶✺✳ ❯♠ ♣♦♥t♦ x ∈ M é ❝❤❛♠❛❞♦ ♣♦♥t♦ ✜①♦ ♣❛r❛ ✉♠ ✢✉①♦ φ s❡ φ t (x) = x

  ♣❛r❛ t♦❞♦ t ∈ R✳ ❉❡✜♥✐çã♦ ✶✳✶✻✳ ❯♠ ♣♦♥t♦ p ∈ M é ❝❤❛♠❛❞♦ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ♣❛r❛ ✉♠ ✢✉①♦ φ t T (p) = p t (p) 6= p s❡ ❡①✐st❡ T > 0 t❛❧ q✉❡ φ ❡ φ ✱ ♣❛r❛ t♦❞♦ t ∈ (0, T )✳

  ❆ ór❜✐t❛ ❞❡ ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ é ❝❤❛♠❛❞❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛✳ ❉❡♥♦t❛♠♦s t ) ♣♦r P er(φ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s✳ t ❉❡✜♥✐çã♦ ✶✳✶✼✳ ❯♠ ♣♦♥t♦ p ∈ M é ❞✐t♦ ♥ã♦✲❡rr❛♥t❡ ♣❛r❛ ♦ ✢✉①♦ φ s❡✱ ♣❛r❛ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ U ⊂ M ❞❡ p ❡ q✉❛❧q✉❡r ♥✉♠❡r♦ r❡❛❧ T > 0✱ ❡①✐st❡ |t| > T t❛❧ t (U ) ∩ U 6= ∅ q✉❡ φ ✳ ❈❛s♦ ❝♦♥tr❛r✐♦ p é ❞✐t♦ ♣♦♥t♦ ❡rr❛♥t❡✳ t t )

  ❖ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ♥ã♦ ❡rr❛♥t❡s ❞❡ ✉♠ ✢✉①♦ φ s❡rá ❞❡♥♦t❛❞♦ ♣♦r Ω(φ ♦✉ s✐♠♣❧❡s♠❡♥t❡ Ω✳ ❆♦ ❧♦♥❣❡ ❞❡st❡ tr❛❜❛❧❤♦ ❛ss✉♠✐♠♦s q✉❡ Ω = M✳ t ❊①❡♠♣❧♦ ✶✳✷✳ ❙❡ x ∈ M é ✉♠ ♣♦♥t♦ ✜①♦ ❞❡ ✉♠ ✢✉①♦ φ ✱ ❡♥tã♦ x é ✉♠ ♣♦♥t♦ t ♥ã♦ ❡rr❛♥t❡ ♣❛r❛ φ ✱ ♣♦✐s ♣❛r❛ t♦❞❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ x ❡ q✉❛❧q✉❡r t❡♠♣♦ T > 0 t (U ) ∩ U t❡♠♦s x ∈ φ ✳ ❊①❡♠♣❧♦ ✶✳✸✳ ❙❡ p ∈ M é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❝♦♠ ♣❡rí♦❞♦ r > 0 ♣❛r❛ ✉♠ ✢✉①♦ φ t t

  ✱ ❡♥tã♦ p é ✉♠ ♣♦♥t♦ ♥ã♦ ❡rr❛♥t❡ ♣❛r❛ φ ✱ ♣♦✐s ❡①✐st❡ ✉♠ ♥✉♠❡r♦ ♥❛t✉r❛❧ n t❛❧ nr (p) = p q✉❡ nr > 0 ❡ φ ✱ ❞❡ ♦♥❞❡ t❡♠♦s q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ U ❞❡ p φ t (U ) ∩ U 6= ∅

  ✳ ❉❡✜♥✐çã♦ ✶✳✶✽✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ Λ ⊂ M é ❞✐t♦ ✐♥✈❛r✐❛♥t❡ ❝♦♠ r❡s♣❡✐t♦ t t (Λ) = Λ ❛♦ ✢✉①♦ φ ✱ s❡ φ ♣❛r❛ t♦❞♦ t ∈ R✳ t ❊①❡♠♣❧♦ ✶✳✹✳ ❙❡ φ é ✉♠ ✢✉①♦ s♦❜r❡ M✱ ❡♥tã♦ ∅ ❡ M sã♦ ✐♥✈❛r✐❛♥t❡s ❝♦♠ t r❡s♣❡✐t♦ ❛ φ ✳ t ) ⊂ M t

  ❊①❡♠♣❧♦ ✶✳✺✳ ❖ ❝♦♥❥✉♥t♦ Ω(φ é ✐♥✈❛r✐❛♥t❡ ❝♦♠ r❡s♣❡✐t♦ ❛♦ ✢✉①♦ φ ✳ t (Ω(φ t )) = Ω(φ t ), ∀t ∈ R ❉❡ ❢❛t♦✿ Pr♦✈❛r❡♠♦s q✉❡ φ ✳ t (Ω(φ t )) ⊂ Ω(φ t )

  ✐✮ ▼♦str❡♠♦s q✉❡✱ φ ✳ t (Ω(φ t )) t ) t (x) ❙❡❥❛ y ∈ φ ❝♦♠ t ∈ R✱ ❛ss✐♠ ❡①✐st❡ x ∈ Ω(φ t❛❧ q✉❡ y = φ ✳ t ) ❆❣♦r❛ ❝♦♠♦ x ∈ Ω(φ t❡♠♦s q✉❡ ♣❛r❛ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ V ❞❡ x ❡ q✉❛❧q✉❡r

  | > T ♥✉♠❡r♦ r❡❛❧ T > 0✱ ❡①✐st❡ |t t❛❧ q✉❡✿

  φ t (V ) ∩ V 6= ∅.

  ✾ ✶✳✷✳ ◆♦çõ❡s ❞❡ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s

  ❆ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ é ❞❡ ❢á❝✐❧ ✈❡r✐✜❝❛çã♦ ❡ s✉❛ ♣r♦✈❛ ♣♦❞❡ s❡r ❛❝❤❛❞❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ Pr♦♣♦s✐çã♦ ✶✳✷✳ ✐✮ ❖ ❝♦♠♣❧❡♠❡♥t♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ q✉❡ é ✐♥✈❛r✐❛♥t❡ ❝♦♠ r❡s♣❡✐t♦ ❛ ✉♠ ✢✉①♦ é ✐♥✈❛r✐❛♥t❡ ❝♦♠ r❡s♣❡✐t♦ ❛♦ ✢✉①♦✳

  P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ✈❡r ❉❡✜♥✐çã♦ ✶✳✶✾✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C

  ❆ ❞❡s❝♦♠♣♦s✐çã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M ❡♠ s✉❜✈❛r✐❡❞❛❞❡s ✐♠❡rs❛s✱ t♦❞❛s ❞❛ ♠❡s♠❛ ❞✐♠❡♥sã♦✱ ❞á ♦r✐❣❡♠ ❛ ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❛ ✈❛r✐❡❞❛❞❡ M✳ ❯♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✱ ❛ ❣r♦ss♦ ♠♦❞♦✱ é ❛ ❞❡s❝♦♠♣♦s✐çã♦ ❞❡ M ♥✉♠❛ ✉♥✐ã♦ ❞❡ s✉❜✈❛r✐❡❞❛❞❡s ❝♦♥❡①❛s✱ ❞✐s❥✉♥t❛s ❡ ❞❡ ♠❡s♠❛ ❞✐♠❡♥sã♦ ❝❤❛♠❛❞❛s ❢♦❧❤❛s✱ ❛s q✉❛✐s s❡ ❛❝✉♠✉❧❛♠ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♦ ❛s ❢♦❧❤❛s ❞❡ ✉♠ ❧✐✈r♦✳

  ■♥tr♦❞✉③✐♠♦s ♥❡st❛ s❡çã♦ ❛ ♥♦çã♦ ❞❡ ❢♦❧❤❡❛çã♦ ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛✐s ❡❧❡♠❡♥t❛r❡s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞❛s ♥♦ r❡st❛♥t❡ ❞♦ tr❛❜❛❧❤♦✳ ❱❡r❡♠♦s t❛♠❜é♠ ❛❧❣✉♥s ❡①❡♠♣❧♦s q✉❡ ✐❧✉str❛♠ ♦ ❝♦♥❝❡✐t♦✳

  ❋♦❧❤❡❛çõ❡s

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

  ✐✐✮ ❆ ✐♥t❡rs❡çã♦ ❞❡ q✉❛❧q✉❡r ❝♦❧❡çã♦ ❞❡ ❝♦♥❥✉♥t♦s q✉❡ sã♦ ✐♥✈❛r✐❛♥t❡s ❝♦♠ r❡s♣❡✐t♦ ✉♠ ✢✉①♦ é ❛✐♥❞❛ ✐♥✈❛r✐❛♥t❡ ❝♦♠ r❡s♣❡✐t♦ ❛♦ ✢✉①♦✳ ✐✐✐✮ ❆ ✉♥✐ã♦ ❞❡ q✉❛❧q✉❡r ❝♦❧❡çã♦ ❞❡ ❝♦♥❥✉♥t♦s q✉❡ sã♦ ✐♥✈❛r✐❛♥t❡s ❝♦♠ r❡s♣❡✐t♦

  ■st♦ é✱ Ω(φ t ) ⊂ φ t (Ω(φ t )) ♣❛r❛ t♦❞♦ t ∈ R✳ ❆ss✐♠✱ Ω(φ t ) ⊂ M é ✐♥✈❛r✐❛♥t❡ ❝♦♠ r❡s♣❡✐t♦ ❛♦ ✢✉①♦ φ t

  P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❥❛ U ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ y = φ t (x) ✱ ❞❛q✉✐ s❡❣✉❡ q✉❡ x = φ −t (y) ∈ φ −t (U )

  (x) ∈ Ω(φ t ) ✱ ❞❡ ♦♥❞❡ x ∈ φ t (Ω(φ t ))

  ✳ ❆❣♦r❛✱ ❝♦♠♦ φ t (Ω(φ t )) ⊂ Ω(φ t ) ♣❛r❛ t♦❞♦ t ∈ R✱ s❡❣✉❡ q✉❡ φ −t

  ❙❡❥❛ x ∈ Ω(φ t ) ✱ ❧♦❣♦ φ −t (x) ∈ φ −t (Ω(φ t ))

  P♦rt❛♥t♦ y ∈ Ω(φ t ) ❡✱ ❛ss✐♠ φ t (Ω(φ t )) ⊂ Ω(φ t ), ∀t ∈ R ✳ ✐✐✮ ❆❣♦r❛✱ ♣r♦✈❡♠♦s q✉❡ Ω(φ t ) ⊂ φ t (Ω(φ t ))

  φ t (φ t (φ −t (U )) ∩ φ −t (U )) = φ t (φ t (φ −t (U ))) ∩ φ t (φ −t (U )) 6= ∅. ❉❡ ♦♥❞❡✱ φ t (U ) ∩ U 6= ∅.

  é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x✱ ❡♥tã♦✿ φ t (φ −t (U )) ∩ φ −t (U ) 6= ∅. ❆ss✐♠✱

  ✳ ▲♦❣♦ φ −t (U )

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

  ✶✵ ✶✳✷✳ ◆♦çõ❡s ❞❡ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s n m −n 1 × U ⊂ R × R 2 1 2

  ✐✮ ❙❡ (U, ϕ) ∈ F ❡♥tã♦ ϕ(U) = U ✱ ♦♥❞❡ U ❡ U sã♦ ❞✐s❝♦s n m −n ❛❜❡rt♦s ❞❡ R ❡ R r❡s♣❡t✐✈❛♠❡♥t❡✳

  ✐✐✮ ❙❡ (U, ϕ)✱ (V, ψ) ∈ F sã♦ t❛✐s q✉❡ U ∩ V 6= ∅ ❡♥tã♦ ❛ ♠✉❞❛♥ç❛ ❞❡ −1 r : ϕ(U ∩ V ) −→ ψ(U ∩ V )

  ❝♦♦r❞❡♥❛❞❛s ψ ◦ ϕ é ❞❡ ❝❧❛ss❡ C ❡ ❡st❛ ❞❛❞❛ −1 n m −n (x, y) = (h (x, y), h (y)) × R 1 2

  ♣♦r ψ ◦ ϕ ✱ ♦♥❞❡ (x, y) ∈ R ❉✐③❡♠♦s q✉❡ M é ❢♦❧❤❡❛❞❛ ♣♦r F✱ ♦✉ ❛✐♥❞❛ q✉❡ F é ✉♠❛ ❡str✉t✉r❛ ❢♦❧❤❡❛❞❛ r

  ❞❡ ❞✐♠❡♥sã♦ n ❞❡ ❝❧❛ss❡ C s♦❜r❡ M✳ ( ( ( ( ( ( ( ( ❋✐❣✉r❛ ✶✳✸✿ ❋♦❧❤❡❛çã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ m✲ ❞✐♠❡♥s✐♦♥❛❧✳

  ❆s ❝❛rt❛s ❞❡ (U, ϕ) ∈ F sã♦ ❝❤❛♠❛❞❛s ❝❛rt❛s ❢♦❧❤❡❛❞❛s✳ r ❉❡✜♥✐çã♦ ✶✳✷✵✳ ❙❡❥❛♠ F ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❝❧❛ss❡ C ❞❡ ❞✐♠❡♥sã♦ n✱ ❝♦♠ 0 < m n m −n n < m 1 ×U ⊂ R ×R 2

  ✱ ❞❡ M ❡ (U, ϕ) ✉♠❛ ❝❛rt❛ ❧♦❝❛❧ ❞❡ F t❛❧ q✉❡ ϕ(U) = U ✳ −1 × c), c ∈ U

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

  ❋✐①❛♥❞♦ c ∈ U ✱ ❛ ❛♣❧✐❝❛çã♦ g = ϕ é ✉♠ ♠❡r❣✉❧❤♦ ❞❡ r ❝❧❛ss❡ C ✱ ♣♦rt❛♥t♦ ❛s ♣❧❛❝❛s sã♦ s✉❜✈❛r✐❡❞❛❞❡s ❝♦♥❡①❛s ❞❡ ❞✐♠❡♥sã♦ n ❞❡ ❝❧❛ss❡ r C

  ❞❡ M✳ ❆❧é♠ ❞✐ss♦✱ s❡ α ❡ β sã♦ ♣❧❛❝❛s ❞❡ U✱ ❡♥tã♦ α ∩ β 6= ∅ ♦✉ α = β✳ , ..., α k

  ❉❡✜♥✐çã♦ ✶✳✷✶✳ ❯♠ ❝❛♠✐♥❤♦ ❞❡ ♣❧❛❝❛s ❞❡ F é ✉♠❛ s❡q✉ê♥❝✐❛ α j ∩ α j 6= ∅ +1 1 ❞❡ ♣❧❛❝❛s t❛❧ q✉❡ α ✱ ♣❛r❛ t♦❞♦ j ∈ {1, ..., k − 1}✳

  ❈♦♠♦ M é r❡❝♦❜❡rt❛ ♣❡❧❛s ♣❧❛❝❛s ❞❡ F✱ ❞❡✜♥✐♠♦s ❡♠ M ❛ r❡❧❛çã♦ ❞❡ 1 , ..., α k 1 ❡q✉✐✈❛❧ê♥❝✐❛✿ “pRq s❡ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ ❞❡ ♣❧❛❝❛s α ❝♦♠ p ∈ α ❡ q ∈ α ” k

  ✳ ❆s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ r❡❧❛çã♦ R sã♦ ❝❤❛♠❛❞♦s ❢♦❧❤❛s ❞❡ F✳ ❙❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ q✉❡ ✉♠❛ ❢♦❧❤❛ ❞❡ F é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ M ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s✳

  ✶✶ m n r ✶✳✷✳ ◆♦çõ❡s ❞❡ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s

  −→ N ❊①❡♠♣❧♦ ✶✳✻✳ ❙❡❥❛ f : M ✉♠❛ s✉❜♠❡rsã♦ ❞❡ ❝❧❛ss❡ C ✳ ❊♥tã♦ ❛s −1 r

  (c) ❝✉r✈❛s ❞❡ ♥í✈❡❧ f ✱ c ∈ N✱ sã♦ ❢♦❧❤❛s ❞❡ ✉♠❛ ❢♦❧❤❡❛çã♦ F ❞❡ ❝❧❛ss❡ c ❞❡ M✳

  ❉❡ ❢❛t♦✱ ✉t✐❧✐③❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❛ ❋♦r♠❛ ▲♦❝❛❧ ❞❛s ❙✉❜♠❡rsõ❡s t❡♠♦s q✉❡ ❞❛❞♦s✱ x ∈ M ❡ q = f(x) ∈ N✱ ❡①✐st❡♠ ❝❛rt❛s ❧♦❝❛✐s (U, ϕ) ❡♠ M✱ (V, ψ) ❡♠ N t❛✐s q✉❡ x ∈ U✱ q ∈ V t❡♠✲s❡ q✉❡✿ m −n n

  × U ⊂ R × R ✶✮ ϕ(U) = U 1 2 2 ⊃ U 2

  ✷✮ ψ(V ) = V −1 = π 2

  ✸✮ ψ ◦ f ◦ ϕ ✱ ♦♥❞❡ π é ❛ ♣r♦❥❡çã♦ t❛❧ q✉❡ (x, y) 7→ y ❆s ❝❛rt❛s ❞❛❞❛s ♣❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❋♦r♠❛ ▲♦❝❛❧ ❞❛s ❙✉❜♠❡rsõ❡s ❞❡ M ❞❡✜♥❡♠

  ✉♠❛ ❢♦❧❤❡❛çã♦ F ❞❡ M✳ ❈♦♠ ❡❢❡✐t♦✱ ♦ ✐t❡♠ (i) ❞❛ ❉❡✜♥✐çã♦ é s❛t✐s❢❡✐t♦ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r ❛❝✐♠❛✳ P❛r❛ ♠♦str❛r ♦ ✐t❡♠ (ii) ❞❛ ❉❡✜♥✐çã♦ ❜❛st❛ ♠♦str❛r q✉❡ ❛ ❝♦♠♣♦s✐çã♦ ❞♦ ✐t❡♠ (ii) é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ x✳ ❙❡❥❛♠ (U, ϕ) ❡ (U, ϕ) ❝❛rt❛s ❞❡ M ❢♦r♥❡❝✐❞♦s ♣❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❋♦r♠❛ ▲♦❝❛❧ ❞❛s ❙✉❜♠❡rsõ❡s✳ ▼♦str❛r❡♠♦s −1 1 q✉❡✱ ϕ ◦ ϕ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ x ∈ U ✳ −1 −1 −1

  π 2 ◦ ϕ ◦ ϕ = ψ ◦ f ◦ ϕ ◦ ϕ ◦ ϕ −1 = ψ ◦ f ◦ ϕ −1 −1 = ψ ◦ ψ ◦ ψ ◦ f ◦ ϕ −1 = ψ ◦ ψ ◦ π 2

  ❡♥tã♦✱ −1 −1 π ◦ (ϕ ◦ ϕ ) = (ψ ◦ ψ ) ◦ π . 2 2 1 ❉❛q✉✐✱ ❛ ❝♦♠♣♦s✐çã♦ ❞♦ ✐t❡♠ (ii) ❞❛ ❉❡✜♥✐çã♦ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ x ∈ U ✳ r

  ■st♦ ♣r♦✈❛ q✉❡✱ F é ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❝❧❛ss❡ C ❞❡ M✳ P♦r ❞❡✜♥✐çã♦ ❛s ♣❧❛❝❛s ❞❡ F ❡stã♦ ❝♦♥t✐❞❛s ♥❛s ❝✉r✈❛s ❞❡ ♥í✈❡❧ ❞❡ f✳ ■st♦ ♣r♦✈❛ q✉❡✱ ❛s ❢♦❧❤❛s ❞❡ F sã♦ ♣r❡❝✐s❛♠❡♥t❡ ♦s ❝♦♥❥✉♥t♦s ❞❡ ♥í✈❡❧ ❞❡ f ❡ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳

  ❯♠ ❡①❡♠♣❧♦ ♠❛✐s ❡s♣❡❝✐✜❝♦ é ♦ s❡❣✉✐♥t❡✳ 3 −→ R

  ❊①❡♠♣❧♦ ✶✳✼✳ ❙❡❥❛ f : R ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡✜♥✐❞❛ ♣♦r 2 z 2 2 2 f (x, y, z) = α(r )e , = x + y

  ♦♥❞❡ r ✱ ❡ ∞ ′ α : R −→ R (t) < 0

  é ✉♠❛ ❛♣❧✐❝❛çã♦ C t❛❧ q✉❡ α(0) = 1✱ α(1) = 0 ❡ α ✱ ♣❛r❛ t♦❞♦ t > ✳ ❊♥tã♦ f é ✉♠❛ s✉❜♠❡rsã♦✱ ♦♥❞❡ ❛s ❢♦❧❤❛s sã♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ❞❛s −1

  (c) s✉♣❡r❢í❝✐❡s ❞❡ ♥í✈❡❧ f ♦♥❞❡ c ∈ R✳

  • y
  • 2 = 1 ✱ ♦ q✉❡ &ea
  • y
  • 2 = 1 ✳ ❆q✉✐ ❛s ❝✉r✈❛s ❞❡ ♥í✈❡❧

      ❡✱ ♣♦r t❛♥t♦✱ x 2

      (x 2 ) .x = 0 ⇒ x = 0.

      2α(x 2 ) α

      ❉❛í✱ z = −

      = −ln(α(r 2 )).

      ◗✉❛♥❞♦ c = 1 t❡♠♦s✱ z = −ln(α(r 2 )). ❖ ❣rá✜❝♦ ❞❛ ❝✉r✈❛ ❛❝✐♠❛ ♥♦ ♣❧❛♥♦ y = 0 é ❞❛❞♦ ♣♦r z

      ) > 0 ✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ z = ln(c) − ln(α(r 2 )).

      )e z = c > 0 ✳ ❆ss✐♠ α(r 2

      ✳ ✐✐✮ ❙❡ c > 0✱ ❡♥tã♦ α(r 2

      ❝♦rr❡s♣♦♥❞❡♠ ❛♦ ❝✐❧✐♥❞r♦ ❞❡ r❛❞✐♦ 1 q✉❡ é r❡♣r❡s❡♥t❛❞♦ ♣♦r f −1 (0)

      ✐✮ ❙❡ c = 0✱ ❡♥tã♦ α(r 2 ) = 0

      ✶✷ ✶✳✷✳ ◆♦çõ❡s ❞❡ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s

      α (r 2 )e z = c.

      (x, y, z) = c, ♦✉ s❡❥❛✱

      ❢♦❧❤❛s ❞❡ ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠ ❡ ❞❡ ❝❧❛ss❡ C ❞❡ M✳ ❆s ❢♦❧❤❛s ❞❡st❛ ❢♦❧❤❡❛çã♦ sã♦ ❞❡s❝r✐t❛s ♣♦r f

      ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ▲♦❣♦ f é ✉♠❛ s✉❜♠❡rsã♦ ❡ ❝♦♠♦ ✈✐st♦ ♥♦ ❊①❡♠♣❧♦ ❛s ❝✉r✈❛s ❞❡ ♥í✈❡❧ sã♦

      P♦rt❛♥t♦✱ x = y = 0✱ ❛❧é♠ ❞✐ss♦✱ ❞❡ α(r 2 ) = 0 t❡♠♦s q✉❡ x 2

      ❉❛í✱ x = y = 0 ❡ α(r 2 ) = 0

      (2α (r 2 )xe z , 2α (r 2 )ye z , α (r 2 )e z ) = (0, 0, 0) ✳

      ∇f (x, y, z) = 0, ♦✉ s❡❥❛✱

      ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛♠♦s✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ f ♥ã♦ s❡❥❛ ✉♠❛ s✉❜♠❡rsã♦✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ♣♦♥t♦ (x, y, z) t❛❧ q✉❡

      ❊♥tã♦ x = 0 é ♦ ú♥✐❝♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ z✳ ❚❡♠♦s z → ∞ q✉❛♥❞♦ x → 1 + ♦✉ 1 ✳ ❖ ❣rá✜❝♦ ❞❡ z é ✉♠❛ ♣❛rá❜♦❧❛✳

      ✶✸ ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈

      ◆♦ ❝❛s♦ c < 0 ❛ ❛♥á❧✐s❡ é s✐♠✐❧❛r✳ ❖ ❣rá✜❝♦ ❞❛s ❢♦❧❤❛s ❞❡ F ❡stã♦ r❡♣r❡s❡♥t❛❞♦s ♥♦ ❣rá✜❝♦ ❋✐❣✉r❛ ✶✳✹✿ ❊①❡♠♣❧♦ ❞❡ ❋♦❧❤❡❛çã♦✳ R ❊①❡♠♣❧♦ ✶✳✽✳ ❯♠ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❞✐♠❡♥sã♦ 1 é ❛ ❢♦❧❤❡❛çã♦ ❞❡ 2 1 2−1 2−1

      = R × R ♦♥❞❡ ❛s ❢♦❧❤❛s sã♦ r❡t❛s ❞❛ ❢♦r♠❛ R × {c} ❝♦♠ c ∈ R ✳

      ◆❛ t❡♦r✐❛ ❞❡ ❢♦❧❤❡❛çõ❡s é ✉s✉❛❧ ❝♦♥s✐❞❡r❛r t❛♠❜é♠ ❛s ❢♦❧❤❡❛çõ❡s ❞❡✜♥✐❞❛s✿ −1 (b)

      ✐✮ P♦r ✉♠❛ ✜❜r❛çã♦ (E, π, B, F )✱ ♦♥❞❡ ❛s ✜❜r❛s✱ π ✱ b ∈ B✱ ❞❡✜♥❡♠ ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ E✱ ❝✉❥❛s ❢♦❧❤❛s sã♦ ✐s♦♠♦r❢❛s às ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ❞❡ F ✳

      ✐✐✮ P♦r ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s X s❡♠ s✐♥❣✉❧❛r✐❞❛❞❡s✱ ♦♥❞❡ ❛s ❢♦❧❤❛s sã♦ ❛s ❝✉r✈❛s dx = X(x) s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ✳ dt

      ✶✳✸ ❋❧✉①♦s ❆♥♦s♦✈

      ◆❡st❛ s❡çã♦ ❞❡✜♥✐r❡♠♦s ✉♠ ❞♦s ❝♦♥❝❡✐t♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♥❡st❛ ❞✐ss❡rt❛çã♦✱ ✢✉①♦s ❆♥♦s♦✈ ❡ ❛❜♦r❞❛r❡♠♦s ❛❧❣✉♠❛s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ❞❛ t❡♦r✐❛ ❞❡ ✢✉①♦s ❆♥♦s♦✈✳ ❆s ♣r✐♥❝✐♣❛✐s r❡❢❡r❡♥❝✐❛s ♣❛r❛ ❡st❡ s❡çã♦ ❢♦r❛♠ t : M −→ M ❉❡✜♥✐çã♦ ✶✳✷✷✳ ❯♠ ✢✉①♦ φ ✱ (t ∈ R) é ✉♠ ✢✉①♦ ❞❡ ❆♥♦s♦✈ r t

      ❞❡ ❝❧❛ss❡ C ✱ (r ≥ 1) s❡ ❡①✐st❡ ✉♠❛ ❞❡s❝♦♠♣♦s✐çã♦ ❝♦♥t✐♥✉❛ ❡ φ ✲✐♥✈❛r✐❛♥t❡ ❞♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡ M ❡♠ três s✉❜✜❜r❛❞♦s✱ ✐st♦ é✱ s T u T T x M = E ⊕ E ⊕ E , ∀x ∈ M. x x x t u ❖♥❞❡ E é ♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡ ❞✐♠❡♥sã♦ 1 ♣❛r❛ ♦ ✢✉①♦ φ ♥ã♦✲s✐♥❣✉❧❛r ❡ E ❡ s E s❛t✐s❢❛③❡♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ u

      ✐✮ ❊①✐st❡♠ ❝♦♥st❛♥t❡s A > 0✱ µ > 1 t❛❧ q✉❡ ♣❛r❛ t♦❞♦ t ∈ R✱ v ∈ E ✐♠♣❧✐❝❛ t ||Dφ ||v||. t (x).v|| ≥ Aµ

      ✶✹ ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈ s

      ✐✐✮ ❊①✐st❡♠ ❝♦♥st❛♥t❡s B > 0✱ λ < 1 t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ t ∈ R✱ v ∈ E ✐♠♣❧✐❝❛ t ||Dφ ||v|| t (x).v|| ≤ Bλ

      s u

      ❋✐❣✉r❛ ✶✳✺✿ ❋❧✉①♦ ❆♥♦s♦✈✿ E ❝♦♥tr❛❡ ❡ E ❡①♣❛♥❞❡ ◆♦t❡♠♦s✱ q✉❡ ❞❡s❞❡ q✉❡ M ❡ ❝♦♠♣❛❝t❛✱ s❡ ❡s❝♦❧❤❡ss❡ ♦✉tr❛ ♠étr✐❝❛ 1

      ❘✐❡♠❛♥♥✐❛♥❛ q✉❡ ❞❡t❡r♠✐♥❡ ❛ ♥♦r♠❛ |.| ✱ s❡ t❡r✐❛ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ k > 0 −1 ≤ |v| ≤ k |v| M x t❛❧ q✉❡ k|v| 1 1 ✱ ♣❛r❛ t♦❞♦ x ∈ T ✱ ❡ ♣♦rt❛♥t♦ (i) ❡ (ii) s❡ ✈❡r✐✜❝❛

      ✭ ❝♦♠ ❝♦♥st❛♥t❡s ❞✐❢❡r❡♥t❡s ❛ A ❡ B✮✳ ▲♦❣♦ ❛ ❞❡✜♥✐çã♦ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡s❝♦❧❤✐❞❛✳

      ❖s ♣r✐♥❝✐♣❛✐s ❡①❡♠♣❧♦s ❞❡ ✢✉①♦s ❆♥♦s♦✈ sã♦ ✢✉①♦s ❣❡♦❞és✐❝♦s ♥♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ✉♥✐tár✐♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛ ❞❡ ❝✉r✈❛t✉r❛ ♥❡❣❛t✐✈❛ ✭✈❡r ❡ s✉s♣❡♥sõ❡s ❞❡ ❞✐❡❢❡♦♠♦r✜s♠♦ ❆♥♦s♦✈✳ ❊st❡ ú❧t✐♠♦ ❡①❡♠♣❧♦ é ❞❡s❝r✐t♦ ❝♦♠♦ s❡❣✉❡✿

      ❙❡❥❛ f : N −→ N ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❆♥♦s♦✈ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ u s N

      ⊕ E ✭✐st♦ s✐❣♥✐✜❝❛ q✉❡ ❡①✐st❡ ✉♠ f✲✐♥✈❛r✐❛♥t❡ ❞❡❝♦♠♣♦s✐çã♦ T N = E ✱ ❛s q✉❛✐s s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s ❛♥á❧♦❣❛s ❛♦ (i) ❡ (ii) ❞❛ ❞❡✜♥✐çã♦ ❈♦♥s✐❞❡r❡

      ♦ ✢✉①♦ η t : N × R −→ N × R

      (x, s) 7−→ η(x, s) = (x, s + t) t ❛ s✉s♣❡♥sã♦ ❞❡ f é ♦ ✢✉①♦ ✐♥❞✉③✐❞♦ ♣♦r η ♥❛ ✈❛r✐❡❞❛❞❡ ♦❜t✐❞♦ ❞❡ N × R t♦r♥❛♥❞♦ ❛s ✐❞❡♥t✐✜❝❛çõ❡s (x, s) ∼ (f(x), s + 1)✳

      ❉❡✜♥✐çã♦ ✶✳✷✸✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ s✉❛✈❡ ❡ E ⊂ T M ✉♠ s✉❜✜❜r❛❞♦ ❝♦♥tí♥✉♦ ❞♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡✳ E é ❝❤❛♠❛❞♦ ✐♥t❡❣rá✈❡❧ s❡ é ♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡ ✉♠❛ 1

      ❢♦❧❤❡❛çã♦ C ✭♦✉ s❡❥❛ ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡t❡r♠✐♥❛❞❛ ♣♦r ✉♠❛ ❝❛rt❛ ❝♦♦r❞❡♥❛❞♦ ❞❡ 1 ❝❧❛ss❡ C ✮✳

      ✶✺ ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈

      ✐♥stá✈❡❧✱ ❡stá✈❡❧✱ ✐♥stá✈❡❧ ❢♦rt❡✱ ❡stá✈❡❧ ❢♦rt❡ ❞❡ x✱ r❡s♣❡t✐✈❛♠❡♥t❡✳ ❉❡ ♠❛♥❡✐r❛ ♠❛✐s ♣r❡❝✐s❛✳

      ✐♥✈❛r✐❛♥t❡ t❡♠♦s A = B = 1✳ ❊st❡ ♣r❡ss✉♣♦st♦ ♥❛ ♠étr✐❝❛ ✐♠♣❧✐❝❛ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s s♦❜r❡ ❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ d✳

      W s (x) = [ t ∈R φ t (W ss (x)) . P♦❞❡♠♦s ❡s❝♦❧❤❡r ❛ ♠étr✐❝❛ ❡♠ M✱ ❞❡ ♠♦❞♦ q✉❡ ♣❛r❛ ❛ ❞❡❝♦♠♣♦s✐çã♦

      ❝♦♠♦ s❡♥❞♦ r❡s♣❡t✐✈❛♠❡♥t❡ ♦s ❝♦♥❥✉♥t♦s W u (x) = [ t ∈R φ t (W uu (x))

      ❉❡✜♥❛♠♦s ❛ ✈❛r✐❡❞❛❞❡s ❡stá✈❡❧ ❡ ✐♥stá✈❡❧ ❞❡ ✉♠ ♣♦♥t♦ x ∈ M ♣❛r❛ ♦ ✢✉①♦ φ t

      ❙ã♦ ❝❤❛♠❛❞♦s r❡s♣❡t✐✈❛♠❡♥t❡ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❡ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧ ❢♦rt❡ ❞♦ ♣♦♥t♦ x ♣❛r❛ ♦ ✢✉①♦ φ t ✳ ❖♥❞❡ d ❞❡♥♦t❛ ❛ ❢✉♥çã♦ ❞✐st❛♥❝✐❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ M✳

      W uu (x) = {y ∈ M : d(φ t (x), φ t (y)) → 0, quando, t → −∞}

      W ss (x) = {y ∈ M : d(φ t (x), φ t (y)) → 0, quando, t → +∞} ❡

      ▼❛✐♦r❡s ❞❡t❛❧❤❡s ❞❛ ❖❜s❡r✈❛çã♦ ❉❡✜♥✐çã♦ ✶✳✷✹✳ ❖s ❝♦♥❥✉♥t♦s✿

      (x), W s (x), W uu (x), W ss (x) ❡ sã♦ ❝❤❛♠❛❞❛s ❛s ✈❛r✐❡❞❛❞❡s

      ❖❜s❡r✈❛çã♦ ✶✳ ❛✮ E u ❡ E s sã♦ ❝❤❛♠❛❞♦s ♦s s✉❜✲✜❜r❛❞♦s ❞❡ ❡①♣❛♥sã♦ ❡ ❝♦♥tr❛çã♦ ❞❡ T M✱ r❡s♣❡t✐✈❛♠❡♥t❡✳

      ❞✮ ❙❡ x ∈ M✱ ❡♥tã♦ ❛s r❡s♣❡t✐✈❛s ❢♦❧❤❛s ❞❡st❛s s♦❧✉çõ❡s q✉❡ ❝♦♥tê♠ x sã♦ ❞❡♥♦t❛❞❛s ♣♦r W u

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

      ✱ E s ⊕ E T

      ❝✮ ❖s ✜❜r❛❞♦s E u ⊕ E T

      E u ⊕ E s ✱ ❛❧é♠ ❞✐ss♦ L é F uu ✲s❛t✉r❛❞♦ ❡ F ss ✲s❛t✉r❛❞♦✳

      E u ⊕ E s é t❛♥❣❡♥t❡ ❛ F✳ ❙❡ L é ✉♠❛ ❢♦❧❤❛ ❞❡ F ❡♥tã♦ L é t❛♥❣❡♥t❡ ❛

      ❢♦r ✐♥t❡❣rá✈❡❧ ❡♥tã♦✱ ❡①✐st❡ ✉♠❛ ❢♦❧❤❡❛çã♦ F ❞❡ ❝❧❛ss❡ C 1 t❛❧ q✉❡ ♦ s✉❜❡s♣❛ç♦

      ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ é ✐♥t❡❣rá✈❡❧✳ ❙❡ E u ⊕ E s

      ❜✮ ❖ s✉❜❡s♣❛ç♦ E u ⊕ E s

      ✭❛✮ ❙❡ x ❡ y ❡st✐✈❡r❡♠ ♥❛ ♠❡s♠❛ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❡♥tã♦ d (φ t (x), φ t (y)) ≤ d(x, y), ♣❛r❛ t ≥ 0

      ✶✻ ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈

      ❋✐❣✉r❛ ✶✳✻✿ ❱❛r✐❡❞❛❞❡s ❡stá✈❡❧ ❡ ✐♥stá✈❡❧ ❡ s❡ x ❡ y ❡st✐✈❡r❡♠ ♥❛ ♠❡s♠❛ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧ ❡♥tã♦ d (φ t (x), φ t (y)) ≥ d(x, y),

      ♣❛r❛ t ≥ 0. ✭❜✮ ❙❡ x ❡ y ❡st✐✈❡r❡♠ ♥❛ ♠❡s♠❛ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❢♦rt❡ ❡♥tã♦ t d (φ t (x), φ t (y)) ≤ λ d (x, y),

      ♣❛r❛ t ≥ 0 ❡ s❡ x ❡ y ❡st✐✈❡r❡♠ ♥❛ ♠❡s♠❛ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧ ❢♦rt❡ ❡♥tã♦ t d d

      (φ t (x), φ t (y)) ≥ u (x, y), ♣❛r❛ t ≥ 0

      ♦♥❞❡ λ < 1 ❡ µ > 1 sã♦ ♦s ❞❛❞♦s ♣❛r❛ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❤✐♣❡r❜ó❧✐❝♦ ❝♦♠ r❡s♣❡✐t♦ ❛ ♠étr✐❝❛✳ ❯♠❛ ♠étr✐❝❛ s❛t✐s❢❛③❡♥❞♦ ❡st❛s ❝♦♥❞✐çõ❡s é ❞✐t❛ ✧❛❞❛♣t❛❞❛✧♥♦ r❡st♦ ❞❡st❛

      ❞✐ss❡rt❛çã♦✱ ♥♦s ❛ss✉♠✐♠♦s q✉❡ ❛ ♠étr✐❝❛ s❛t✐s❢❛③ (a) ❡ (b)✳ u , d s , d uu , d ss ❆❣♦r❛✱ s❡ d ❞❡♥♦t❛ ❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛ ♣♦r d s♦❜r❡ ❛s ❢♦❧❤❛s ❞❡ u s uu ss

      ❢♦❧❤❡❛çõ❡s F ✱ F ✱ F ✱ F ✱ r❡s♣❡t✐✈❛♠❡♥t❡✱ ❞❡✜♥❡✲s❡ ♣❛r❛ ❝❛❞❛ x ∈ M✱ δ > 0✳ u u B δ (x) = {y ∈ M : d(x, y) < δ} B δ s s (x) = {y ∈ W (x) : d u (x, y) < δ}

      B uu uu δ (x) = {y ∈ W (x) : d s (x, y) < δ} B δ ss ss (x) = {y ∈ W (x) : d uu (x, y) < δ}

      B δ (x) = {y ∈ W (x) : d ss (x, y) < δ}. ❚❡♦r❡♠❛ ✶✳✷✳ ✭ ❚❡♦r❡♠❛ ❞❛ ❱❛r✐❡❞❛❞❡ ❊stá✈❡❧ ♣❛r❛ ❋❧✉①♦s✮✳ ❙❡❥❛ Λ ⊂ M t ✉♠ ❝♦♥❥✉♥t♦ ❤✐♣❡r❜ó❧✐❝♦ ✐♥✈❛r✐❛♥t❡ ♣❛r❛ ✉♠ ✢✉①♦ φ ✳ ❊♥tã♦ ❡①✐st❡ ǫ > 0 t❛❧ q✉❡ ss uu

      (p) (p) ♣❛r❛ ❝❛❞❛ ♣♦♥t♦ p ∈ Λ ❡①✐st❡♠ ❞♦✐s ❜♦❧❛s ♠❡r❣✉❧❤❛❞❛s B δ ❡ B δ ♦s q✉❛✐s p p s u sã♦ t❛♥❣❡♥t❡s ❛ E ❡ E ✱ r❡s♣❡t✐✈❛♠❡♥t❡✳

      ❉❡♠♦♥str❛çã♦✿ ❱❡r ❡♠

      ✶✼ ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈

      

    ❚❡♦r❡♠❛ ❞❡ ❱✐③✐♥❤❛♥ç❛ Pr♦❞✉t♦ ▲♦❝❛❧ ♣❛r❛ ❋❧✉①♦s ❞❡

    ❆♥♦s♦✈ t ) = M t >

      ❚❡♦r❡♠❛ ✶✳✸✳ ❙❡❥❛ Ω(φ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈ φ ❡♥tã♦ ❡①✐st❡ δ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ x ∈ M t❛❧ q✉❡ ♣❛r❛ 0 < δ ≤ δ ❛s ❛♣❧✐❝❛çõ❡s s uu

      G : B (x) × B (x) −→ M δ δ s uu (y, z) 7−→ G(y, z) = B (z) ∩ B (y) 2δ 2δ

      ❡ ss u H

      : B (x) × B (x) −→ M δ δ ss u (y, z) 7−→ H(y, z) = B (z) ∩ B (y) 2δ 2δ sã♦ ❞❡✜♥✐❞❛s ✉♥✐✈♦❝❛♠❡♥t❡ ❡ sã♦ ❤♦♠❡♦♠♦r✜s♠♦s s♦❜r❡ s✉❛s r❡s♣❡t✐✈❛s ✐♠❛❣❡♥s✳

      ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r♦ ♠♦str❛r❡♠♦s q✉❡✱ s❡ y, z ∈ Ω ❡ d(y, z) < δ ♣❛r❛ ❛❧❣✉♠ δ ❡♥tã♦ ♣❛r❛ s uu s uu

      (z) ∩ B (y) 6= ∅ (z) ∩ B (y) t♦❞♦ δ ≤ δ t❡♠✲s❡ q✉❡ B ✱ ✐st♦ é✱ ❡①✐st❡ p ∈ B ✳ 2δ 2δ 2δ 2δ s T u ⊕ E

      ❊st❡ ❢❛t♦ s❡❣✉❡ ❞❛ tr❛♥s✈❡rs❛❧✐❞❛❞❡ ✉♥✐❢♦r♠❡ ❞❡ E ❡ E ✳ ❆❧é♠ ❞✐ss♦✱ s uu (z) ∩ B (y) ⊂ M = Ω

      ❝❧❛r❛♠❡♥t❡ p ∈ Ω ❞❡s❞❡ q✉❡ B ✳ 2δ 2δ ❛❣♦r❛✱ ♣❛ss❛♠♦s ❛ ♣r♦✈❛r q✉❡ p é ú♥✐❝♦✳ s uu

      (z) ∩ B (y) ❆✜r♠❛çã♦ ✶✳✶✳ p ∈ B 2δ 2δ é ú♥✐❝♦✳ s uu

      ∈ B (z)∩B (y) ❉❡ ❢❛t♦✱ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ ♦✉tr♦ ♣♦♥t♦ p 2δ 2δ ♠♦str❛r❡♠♦s

      = p q✉❡ p ✳ s uu s (z) ∩ B (y) (z)

      ❈♦♠♦ p ∈ B 2δ 2δ t❡♠♦s q✉❡ p ∈ B 2δ ✱ ❡♥tã♦ d s (p, z) < 2δ. ′ ′ s uu s ✭✶✳✶✮ ∈ B (z) ∩ B (y) ∈ B (z)

      P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ p 2δ 2δ t❡♠♦s q✉❡ p 2δ ✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ d , z s (p ) < 2δ.

      ✭✶✳✷✮ ❆❣♦r❛✱ ❞❡ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r t❡♠♦s q✉❡✱ ′ ′ d , p , z s (p ) < d s (p ) + d s (z, p)

      < 2δ + 2δ = 4δ , p ⇒ d s (p ) < 4δ. ❉❡ ♦♥❞❡✱ s p ∈ B 4δ ✭✶✳✸✮ (p).

      ✶✽ ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈

      ❉❡ ♠❛♥❡✐r❛ s✐♠✐❧❛r ♣r♦✈❡♠♦s q✉❡✱ uu p ∈ B (p). s uu uu ✭✶✳✹✮

      (z) ∩ B (y) (y) P♦✐s✱ ❞❡s❞❡ q✉❡ p ∈ B 2δ 2δ t❡♠♦s q✉❡ p ∈ B 2δ ✱ ❡♥tã♦ d (p, y) < 2δ. uu ′ s uu ′ uu ✭✶✳✺✮

      ∈ B (z) ∩ B (y) ∈ B (y) P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ p t❡♠♦s q✉❡ p ✱ ❞❡ ♦♥❞❡ 2δ 2δ 2δ s❡❣✉❡ q✉❡✱ d , y uu (p ) < 2δ.

      ✭✶✳✻✮ ❉❛q✉✐✱ ❞❡ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ t❡♠♦s q✉❡✱ ′ ′ d , p , z uu (p ) < d uu (p ) + d uu (z, p)

      < 2δ + 2δ = 4δ , p ⇒ d uu (p ) < 4δ. ❉❡ ♦♥❞❡✱ uu p ∈ B (p). s uu

      ∈ B (p) ∩ B (p) = {p} ▲♦❣♦✱ ❞❡ t❡♠♦s q✉❡ p 4δ 4δ ✳ ❉❛q✉✐✱ s❡❣✉❡

      ∈ {p} q✉❡ p ✳ ❊ ❛ss✐♠✱ p = p.

      ❋✐❣✉r❛ ✶✳✼✿ ❱✐③✐♥❤❛♥ç❛ Pr♦❞✉t♦ ▲♦❝❛❧

      ✶✾ ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈

      ❚❡♦r❡♠❛ ❞❡ ❆♥♦s♦✈ t : M −→ M

      ❉❡✜♥✐çã♦ ✶✳✷✺✳ ❙❡❥❛♠ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❡ φ ✉♠ ✢✉①♦✱ δ, T > 0 ❛ ❛♣❧✐❝❛çã♦ c : R −→ M é ❝❤❛♠❛❞❛ ✉♠❛ δ ✲♣s❡✉❞♦ ór❜✐t❛ s❡✿ d (φ t (c(τ )), c(τ + t)) ≤ δ ♣❛r❛ t♦❞♦ | t |≤ T ❡ ∀τ ∈ R✳ ❉❡✜♥✐çã♦ ✶✳✷✻✳ ❯♠❛ ❝✉r✈❛ c : R −→ M é ǫ ✲s♦♠❜r❡❛❞♦ ♣❡❧❛ ór❜✐t❛ ❞❡ x ∈ M d s

      − 1| < ǫ t (x)) < ǫ s❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ s : R −→ R ❝♦♠ | t❛❧ q✉❡ d(c(s(t)), φ dt

      ♣❛r❛ t♦❞♦ t ∈ R✳ ▲❡♠❛ ✶✳✶✳ ✭❙♦♠❜r❡❛♠❡♥t♦ ♣❛r❛ ✢✉①♦s✮✳ t

      ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐♥❛✱ φ ✉♠ ✢✉①♦ ❞✐❢❡r❡♥❝✐❛✈❡❧ ❡ Λ ✉♠ ❝♦♥❥✉♥t♦ t ❤✐♣❡r❜ó❧✐❝♦ ❝♦♠♣❛❝t❛ ♣❛r❛ φ ✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U(Λ) ⊃ Λ t❛✐s q✉❡ ♣❛r❛ t♦❞♦ ǫ > 0 ❡①✐st❡✱ δ > 0 t❛❧ q✉❡ t♦❞❛ ❛ δ ✲♣s❡✉❞♦ ♦r❜✐t❛ é ǫ ✲ s♦♠❜r❡❛❞❛ ♣♦r ✉♠❛ t

      ór❜✐t❛ ❞❡ φ ✳ t ❈♦r♦❧ár✐♦ ✶✳✹✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐♥❛✱ φ ✉♠ ✢✉①♦ ❞✐❢❡r❡♥❝✐❛✈❡❧ ❡ Λ t ✉♠ ❝♦♥❥✉♥t♦ ❤✐♣❡r❜ó❧✐❝♦ ❝♦♠♣❛❝t❛ ♣❛r❛ φ ✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U(Λ) ⊃ Λ t❛✐s q✉❡ ∀ǫ > 0, ∃δ > 0 t❛❧ q✉❡ t♦❞❛ ❛ δ ✲♣s❡✉❞♦ ór❜✐t❛ é ǫ ✲ s♦♠❜r❡❛❞❛ ♣♦r ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛✳

      ❉❡♠♦♥str❛çã♦✿ ❱❡r ❆ ❞❡♠♦♥str❛çã♦ ❞♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ t❛♠❜é♠ ❡♠ ❄❄ t ) = M

      ❚❡♦r❡♠❛ ✶✳✺✳ ✭❚❡♦r❡♠❛ ❞❡ ❆♥♦s♦✈ ✮✳ ❙❡ Ω(φ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈ t ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s ❞❡ φ é ❞❡♥s♦ ❡♠ Ω✳

      ❉❡♠♦♥str❛çã♦✿ t ) = Ω ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ P er(φ ✳ ❉❡ ❢❛t♦✿ t ) ⊂ M t ) ⊂ Ω

      ✐✮ P er(φ ❂Ω ❡♥tã♦ P er(φ é ✐♠❡❞✐❛t♦✱ ❛ss✐♠✱ ❜❛st❛ ♠♦str❛r q✉❡✱ t ) ✐✐✮ Ω ⊂ P er(φ ✳ t )

      ■st♦ é✱ ❞❛❞♦ x ∈ Ω ♠♦str❛r❡♠♦s q✉❡ x ∈ P er(φ ✳ t ) x x t ) 6= ∅ ∩ per(φ ▲❡♠❜r❡♠♦s q✉❡ x ∈ P er(φ s❡✱ s♦♠❡♥t❡ s❡✱ ∀V ✱ V ✐st♦ é✱ s❡

      , ∀V x ∃p ∈ V x ∩ per(φ t )

      ✳ ǫ t ❉❛❞♦s x ∈ Ω ❡ ǫ > 0 ❛r❜✐trár✐♦✱ s❡❥❛ V x ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x✱ ❈♦♠♦ φ é δ (x)

      ✉♠ ✢✉①♦ ❆♥♦s♦✈✱ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ♣r♦❞✉t♦ N ❞❡ x ♦♥❞❡ δ > 0 ❝♦♠ ǫ N δ (x) ⊂ V x ✳

      ✷✵ ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈

      ▲♦❣♦✱ ❝♦♠♦ x ∈ Ω ❡♥tã♦ ∀t > 0, ∀V x = V, ∃T ≥ t t❛❧ q✉❡ φ T (V ) ∩ V 6= ∅ ✳ ❈♦♠♦ ✐st♦ é ✈á❧✐❞♦ ♣❛r❛ q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ ❞❡ x✱ ❡♠ ♣❛rt✐❝✉❧❛r ♣❛r❛ ❛ ✈✐③✐♥❤❛♥ç❛ ♣r♦❞✉t♦✱ ❛ss✐♠ ♣♦❞❡♠♦s t♦♠❛r V = N δ (x)

      ❉❡ ❛q✉✐✱ ❡①✐st❡ q ∈ φ T (V ) ∩ V ✳ ❆ss✐♠✱

      φ −T (q) ∈ V ∩ φ −T (V ). ❆❣♦r❛✱ ❞❡♥♦t❡♠♦s z = φ −T (q) ✱ ❞❛q✉✐ s❡❣✉❡ q✉❡ ❡①✐st❡ z ∈ V ∩ φ −T (V ) ❡ ❛❧é♠

      ❞✐ss♦ φ T (z) ∈ φ T (V ) ∩ V ✳ ■st♦ é✱ d (φ T (z), z) < δ.

      ✭✶✳✼✮ ❆✜r♠❛çã♦ ✶✳✷✳ ❊①✐st❡ ✉♠❛ δ− ♣s❡✉❞♦ ór❜✐t❛ c : R −→ M ❞❡ φ t ♣❛r❛ |t| ≤ T ✳

      ❉❡ ❢❛t♦✱ s❡❥❛ T > 0 ❡✱ c : R −→ M t

      7−→ c(t) = φ tmodT (z) ♣❛r❛ |t| ≤ T ✳

      ✐✮ P❛r❛ |t| < T ✱ t❡♠♦s q✉❡ ❡①✐st❡ δ > 0 t❛❧ q✉❡ d(φ t (c(τ )), c(t + τ )) < δ ✱ ♣❛r❛ t♦❞♦ τ ∈ R✳ P♦✐s✱ ♦❜s❡r✈❡ q✉❡ c

      (τ + t) = φ (τ +t)modT (z) = φ τ modTtmodT (z)) = φ τ modT (φ t (z)) = φ t (φ τ modT (z)) = φ t (c(τ )).

      ❉❡ ♦♥❞❡✱ d (φ t (c(τ )), c(t + τ )) = 0 < δ, ∀τ ∈ R

      ❡ |t| < T ✳ ✐✐✮ P❛r❛ |t| = T ✱ t❡♠♦s q✉❡ d(φ t (c(τ ), c(t + τ )) < δ ✱ ∀τ ∈ R✳

      ❖❜s❡r✈❡ ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡✱ c(T ) = φ T (z), z ∈ W s (z)

      ❡✱ d (φ t (c(τ ), c(t + τ )) = d(φ t (φ τ modT (z)), φ (t+τ )modT (z)) t =T

      = d(φ T (φ τ modT (z)), φ τ modT (φ T modT (z))) = d(φ τ modT (φ T (z)), φ τ modT (z)) < d (φ T (z), z) < δ.

      ✷✶ ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈

      ❆ss✐♠✱ d (φ t (c(τ ), c(t + τ )) < δ, ∀τ ∈ R.

      ❉❡ ♦♥❞❡ s❡ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞❛ ❆✜r♠❛çã♦ ❉❛í✱ ♣❡❧♦ ❝♦r♦❧ár✐♦ ❛ δ− ♣s❡✉❞♦ ór❜✐t❛ é ǫ✲ s♦♠❜r❡❛❞❛ ♣♦r ✉♠❛ ór❜✐t❛

      ♣❡r✐ó❞✐❝❛✳ P❡❧♦ q✉❡ ❡①✐st❡✱ p ∈ per(φ t ) ∩ V ǫ x , ∀V ǫ x

      ✳ ❉❡ ♦♥❞❡✱ x ∈ per(φ t ).

      ❉❛í✱ Ω ⊂ per(φ t ).

      ❆ss✐♠✱ ✜♥❛❧♠❡♥t❡ t❡♠♦s q✉❡ per(φ t ) = Ω ✳ ✭❱❡r ✜❣✉r❛ ❋✐❣✉r❛ ✶✳✽✿ ❚❡♦r❡♠❛ ❞❡ ❆♥♦s♦✈

      ❈♦♥❥✉♥t♦s F σ ✲s❛t✉r❛❞♦ (σ = u, s, uu, ss)

      ❉❡✜♥✐çã♦ ✶✳✷✼✳ ❙❡❥❛ F σ ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✳ ❊♥tã♦ ♦ ❝♦♥❥✉♥t♦ K ⊂ M

      é ❝❤❛♠❛❞♦ F σ ✲s❛t✉r❛❞♦ s❡ é ✉♠❛ ✉♥✐ã♦ ❞❡ ❢♦❧❤❛s ❞❡ F σ ♦♥❞❡ (σ = u, s, uu, ss)

      ✳

      ✷✷ σ ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈

      ▲❡♠❛ ✶✳✷✳ ❙❡❥❛ F ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✳ ❊♥tã♦ ♦ ❝♦♥❥✉♥t♦ σ σ K

      ⊂ M (x) ⊂ K

      é F ✲s❛t✉r❛❞♦ s❡✱ s♦♠❡♥t❡✱ s❡ ♣❛r❛ t♦❞♦ x ∈ K t❡♠✲s❡ q✉❡ W ✱ ♦♥❞❡ (σ = u, s, uu, ss)✳

      ❉❡♠♦♥str❛çã♦✿ σ ❙✉♣♦♥❤❛♠♦s q✉❡ K ⊂ M é F ✲s❛t✉r❛❞♦✱ ❡♥tã♦ ♠♦str❛r❡♠♦s q✉❡ ❞❛❞♦ x ∈ K σ

      (x) ⊂ K ❛r❜✐trár✐♦ t❡♠✲s❡ q✉❡ W ✳

      ❉❡ ❢❛t♦✿ σ ❈♦♠♦ K ⊂ M é F ✲s❛t✉r❛❞♦✱ t❡♠♦s ♣❡❧❛ ❞❡✜♥✐çã♦ q✉❡✿ [ σ K = W (x). x ∈K σ

      (x) ⊂ K ❉❛❞♦ x ∈ K q✉❡r❡♠♦s ♠♦str❛r q✉❡ W ✳ σ σ [ σ

      W (x) (x) ⊂ (x)

      ❙❡❥❛ y ∈ W ✱ ❧♦❣♦ ❝♦♠♦ W x ∈K t❡♠♦s q✉❡✱ [ σ y ∈ W x ∈K (x) = K.

      ■st♦ é✱ y ∈ K. σ (x) ⊂ K σ ❆❣♦r❛✱ s✉♣♦♥❤❛♠♦s q✉❡ ❞❛❞♦ x ∈ K ❛r❜✐trár✐♦✱ s❡ W ❡♥tã♦ K é

      F ✲s❛t✉r❛❞♦✳ ■st♦ é q✉❡✱ [ σ

      K W [ σ = (x). x ∈K W

      (x) ⊂ K ✐✮ Pr♦✈❡♠♦s q✉❡✱ ✳ [ σ x ∈K σ

      W (x) (x) ❙❡❥❛ y ∈ ❡♥tã♦ y ∈ W ♣❛r❛ ❛❧❣✉♠ x ∈ K✳ x ∈K σ

      (x) ⊂ K ▲♦❣♦✱ ❝♦♠♦ W s❡❣✉❡ q✉❡ y ∈ K✳ [ σ

      W (x) ✐✐✮ Pr♦✈❡♠♦s q✉❡✱ K ⊂ ✳ x ∈K σ σ [

      (y) ⊂ W (x) ❙❡❥❛ y ∈ K✱ ❧♦❣♦ ❝♦♠♦ y ∈ W t❡♠♦s q✉❡✱ [ x ∈K σ y ∈ W x ∈K (x).

      ✷✸ ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈

      σ σ

      ❖❜s❡r✈❛çã♦ ✷✳ ❛✮ ❙❡ K ⊂ M é F ✲s❛t✉r❛❞♦ ❡♥tã♦ K ✭♦ ❢❡❝❤♦ ❞❡ K✮ é F ✲ s❛t✉r❛❞♦✱ ♦♥❞❡ (σ = u, s, uu, ss)✳ ❉❡ ❢❛t♦✿ σ

      (x) ⊂ K ❙❡❥❛ x ∈ K ♠♦str❛r❡♠♦s q✉❡ W ✳ n n ∈N ) ∈ K ❈♦♠♦ x ∈ K✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s (x t❛❧ q✉❡ σ σ x lim n = x

      (x n ) ⊂ K n −→∞ ✱ ❛❣♦r❛ ❝♦♠♦ K é F ✲s❛t✉r❛❞♦ t❡♠✲s❡ q✉❡ W ✳ σ σ (x)

      P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❥❛ y ∈ W ✳ ❉❛í✱ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛s ❢♦❧❤❛s W ❡ ❞♦ σ x n = x k ) k ∈N r (y) ∩ W (x n ) 6= ∅ ❢❛t♦ q✉❡ lim ✱ ❡①✐st❡ (n t❛❧ q✉❡ B k k ✱ ♦♥❞❡✱ n −→∞ σ

      B > r (y) k k ∈ B r (y) ∩ W (x n ) k k k é ✉♠❛ ❜♦❧❛ ❞❡ y ❡ r❛❞✐♦ r ✳ ■st♦ é✱ ❡①✐st❡ y σ σ y = y (y) ∩ W (x ) ⊂ W (x ) ⊂ K k r k n k n k t❛❧ q✉❡ lim ✳ ▲♦❣♦ ❝♦♠♦ B t❡♠♦s k ∈ K k ) k ∈ K y k = y k −→∞ ∈N q✉❡✱ y ✳ ❆ss✐♠✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ (y t❛❧ q✉❡ lim k −→∞

      ❞❛í✱ s❡❣✉❡ q✉❡ y ∈ K✳ σ c σ ❜✮ ❙❡ K ⊂ M é F ✲s❛t✉r❛❞♦✱ ❡♥tã♦ K ✭♦ ❝♦♠♣❧❡♠❡♥t♦ ❞❡ K✮ é F ✲s❛t✉r❛❞♦✱

      ♦♥❞❡ (σ = u, s, uu, ss)✳ ❉❡ ❢❛t♦✿ c σ c

      (y) 6⊂ K P♦r ❝♦♥tr❛❞✐çã♦✱ s✉♣♦♥❤❛♠♦s q✉❡ ❞❛❞♦ y ∈ K t❡♠✲s❡ q✉❡ W ✳ σ c

      (y) ❊♥tã♦ ❞❛❞♦ z ∈ W t❡♠♦s q✉❡ z 6∈ K ✱ ❧♦❣♦ z ∈ K ❛❣♦r❛✱ ❝♦♠♦ K é σ σ σ F (z) ⊂ K (y)

      ✲s❛t✉r❛❞♦ s❡❣✉❡ q✉❡ W ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ z ∈ W ❡♥tã♦ σ c y ∈ W (z) ❞❛í✱ y ∈ K ✐st♦ é✱ y 6∈ K ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳

      ❆❣♦r❛ ❞❡♠♦s ✉♠ ❡①❡♠♣❧♦ ♠❛✐s ❡s♣❡❝í✜❝♦✳ u u (x)

      ❊①❡♠♣❧♦ ✶✳✾✳ W é F ✲s❛t✉r❛❞♦✳ u u u (x) (y) ⊂ W (x)

      ❉❡ ❢❛t♦✱ ❞❛❞♦ y ∈ W q✉❡r❡♠♦s ♠♦str❛r q✉❡ W ✳ u u u u (y) (x) (y) = W (x)

      ❙❡❥❛ z ∈ W ✱ ❝♦♠♦ y ∈ W t❡♠♦s q✉❡ W ✱ ❞❡ ♦♥❞❡ u z ∈ W (x)

      ✳ u u (x)

      ❖❜s❡r✈❛çã♦ ✸✳ W é F ✲s❛t✉r❛❞♦✳ ❊st❡ ❢❛t♦ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❊①❡♠♣❧♦

       ❯♠❛ ❈♦♥s❡q✉ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❆♥♦s♦✈ u s ❆❣♦r❛✱ ♠♦str❛♠♦s ✉♥s ❞♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❡st❛ s❡ç❛♦✱ q✉❡ ❛✜r♠❛ q✉❡

      W (x) (x)

      ❡ W sã♦ ❞❡♥s♦s ❡♠ M ♣❛r❛ ❝❛❞❛ x ∈ M✳ ▼❛s ♣♦st❡r✐♦r♠❡♥t❡ ♥♦ ❝❛♣✐t✉❧♦ 2 ♣r♦✈❛r❡♠♦s q✉❡ ♥❡♠ s❡♠♣r❡ é ♦ ❝❛s♦ ❞❡ q✉❡ ❝❛❞❛ ✉♠❛ ❞❛s ❢♦❧❤❛s ❞❡ uu ss F

      ♦✉ ❞❡ F é ❞❡♥s♦ ❡♠ M✳ t ) = M (x) (x) u s ❚❡♦r❡♠❛ ✶✳✻✳ ❙❡ Ω(φ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈ ❡♥tã♦ W ❡ W sã♦ ❞❡♥s♦s ❡♠ M ♣❛r❛ ❝❛❞❛ x ∈ M.

      ✷✹ u ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈

      (x) = M ❉❡♠♦♥str❛çã♦✿ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ W ✱ ♣❛r❛ t♦❞♦ x ∈ M✳ ❉❡ ❢❛t♦✿ ❙❡❥❛ x ∈ M✱ ❝♦♠♦ M é ❝♦♥❡①♦ ♥ã♦ ❛❞♠✐t❡ ♦✉tr❛ ❝✐sã♦ ❛❧é♠ ❞❛ tr✐✈✐❛❧ ✐st♦ é✿ ❙❡ M = A ∪ B ♦♥❞❡ A ❡ B sã♦ ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s ❡ ❛❜❡rt♦s ❡♥tã♦ A = ∅ ♦✉

      B = ∅ ✳ u u

      (x) (x) ❉❡♥♦t❡♠♦s A = W ❡ B = M − W ✳ u u

      M (x) ∪ − W (x)

      ❡ A 6= ∅✱ ❛❧é♠ ❞✐ss♦ B é ❛❜❡rt♦✱ s✉♣♦♥❞♦ ❈♦♠♦ M = W u u

      (x) = ∅ (x) = M q✉❡ A é ❛❜❡rt♦ t❡rí❛♠♦s q✉❡ B = M − W ❡ ❞❛í W ♦ q✉❡ q✉❡rí❛♠♦s ♣r♦✈❛r✳ u

      (x) ❆ss✐♠✱ só ❢❛❧t❛r✐❛ ♠♦str❛r q✉❡ W é ❛❜❡rt♦ ♣❛r❛ t♦❞♦ x ∈ M✳ u

      (x) ❆✜r♠❛çã♦ ✶✳✸✳ W é ❛❜❡rt♦ ♣❛r❛ t♦❞♦ x ∈ M✳ u

      (x) u ❉❛❞♦ z ∈ W ✱ ♣r♦✈❛r❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ N t❛❧ q✉❡ z ∈ N ⊂ W

      (x) ✳ u

      (x) ❙❡❥❛ z ∈ W ✱ ❡♥tã♦ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ✈✐③✐♥❤❛♥ç❛ ♣r♦❞✉t♦ ❧♦❝❛❧✱ ❡①✐st❡ ✉♠❛ δ (z)

      ✈✐③✐♥❤❛♥ç❛ ♣r♦❞✉t♦ N = N ❝♦♥t❡♥❞♦ z✱ ♦♥❞❡ δ > 0✳ u (x)

      ❉❛q✉✐✱ ❜❛st❛ ♠♦str❛r q✉❡ N ⊂ W ✳ u (x)

      ❖✉ s❡❥❛✱ ❞❛❞♦ q ∈ N✱ ❛r❜✐trár✐♦ q✉❡r❡♠♦s ♠♦str❛r q✉❡ q ∈ W ✳ ■st♦ é✱ q ∈ V q ∩ W (x) u ♣❛r❛ t♦❞♦ V ✭✈✐③✐♥❤❛♥ç❛ ❞❡ q✮ ❡①✐st❡ p ✳

      ❈♦♠ ❡ss❡ ✜♠✱ ❡♥✉♥❝✐❛♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❝✉❥❛ ♣r♦✈❛ s❡rá ❢❡✐t♦ ❞❡♣♦✐s ❞❡ u (x)

      ♠♦str❛r q✉❡ N ⊂ W ✳ u (x)

      ❘❡s✉❧t❛❞♦ ✶✳✶✳ N ∩ P er(φ) ⊂ W ✳ u (x)

      ❆❣♦r❛✱ ♣r♦✈❡♠♦s q✉❡ N ⊂ W ✳ t ) ❉❡ ❢❛t♦✱ s❡❥❛ q ∈ N✱ ❧♦❣♦ ♣❡❧♦ ❚❡♦r❡♠❛ t❡♠♦s q✉❡ M = P er(φ ❡ ❝♦♠♦

      N ⊂ M t ) q q ∩ per(φ t ) ✱ s❡❣✉❡ q✉❡✱ q ∈ P er(φ ✱ ✐st♦ é✱ ♣❛r❛ t♦❞♦ V ❡①✐st❡ p ∈ V ✳

      P♦r ♦✉tr♦ ❧❛❞♦✱ ♦❜s❡r✈❡✲s❡ q✉❡✿ q ⊂ N t ) ✐✮ ❙❡ V t❡♠♦s q✉❡✱ p ∈ N ∩ per(φ ✳ q 6⊆ N

      ✐✐✮ ❙❡ V ✱ ♣❡❧❛ ❞❡♥s✐❞❛❞❡ ❞❡ ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s ❡①✐st❡ ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ t ) ∩ V q ∩ N t ) ∩ N t❛❧ q✉❡ p ∈ per(φ ✐st♦ é p ∈ per(φ ✳ t ) ∩ N ■st♦ é✱ ❡♠ q✉❛❧q✉❡r ❝❛s♦ t❡♠♦s q✉❡ p ∈ per(φ ✳ u

      (x) ▲♦❣♦ ♣❡❧♦ ❘❡s✉❧t❛❞♦ t❡♠♦s q✉❡ N ∩ P er(φ) ⊂ W ❞❡ ♦♥❞❡ s❡❣✉❡ u

      (x) p q✉❡ p ∈ W ✱ ❞❛í ♣❛r❛ t♦❞❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ p q✉❡ ❞❡♥♦t❛♠♦s ♣♦r V ❡①✐st❡ u p ∈ V ∩ W p (x)

      ✷✺ p p ⊂ V q ✶✳✸✳ ❋❧✉①♦s ❆♥♦s♦✈

      ▲♦❣♦✱ ❝♦♠♦ ✐st♦ s❡ ❝✉♠♣r❡ ♣❛r❛ t♦❞♦ V ✱ ❡♠ ♣❛rt✐❝✉❧❛r s❡ ❝✉♠♣r❡ ♣❛r❛ V ✱ ′ u ∈ V q ∩ W (x) q

      ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ p ♣❛r❛ t♦❞♦ V ✳ ▲♦❣♦ u V ∩ W . q (x) 6= ∅, ∀V q ❉❛í✱ s❡❣✉❡ q✉❡ u q

      ∈ W (x). Pr♦✈❛♥❞♦ ❛ss✐♠ q✉❡✱ u N ⊂ W (x).

      ❉❡♠♦♥str❛çã♦✿ ❉♦ ❘❡s✉❧t❛❞♦ u u (x)

      Pr✐♠❡✐r♦✱ ♦❜s❡r✈❡♠♦s q✉❡ ♣❡❧❛ ♦❜s❡r✈❛çã♦ W é F ✲s❛t✉r❛❞♦ ❡ ❝♦♠♦ u z ∈ W (x) t❡♠♦s q✉❡✱ u u

      W (z) ⊂ W (x). t ) ✭✶✳✽✮

      ❆❣♦r❛✱ ❞❡s❞❡ q✉❡ per(φ é ❞❡♥s♦ ❡♠ M t❡♠♦s q✉❡ é ❞❡♥s♦ ❡♠ N✱ ❡①✐st❡ p ∈ N ∩ P er(φ) s uu ✱ ❧♦❣♦ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ✈✐③✐♥❤❛♥ç❛ ♣r♦❞✉t♦ ❧♦❝❛❧ ❡①✐st❡ w ∈ W

      (p) ∩ W (z) 2δ 2δs s (p) ⊂ W (p)

      ▲♦❣♦✱ ❝♦♠♦ w ∈ W 2δ t❡♠♦s✱ φ lim t (w) = p. i t i →∞ uu uu ✭✶✳✾✮

      (z) ⊂ W (z) t (w) ∈ i P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ w ∈ W 2δ t❡♠♦s q✉❡ φ uu u

      φ t (W (z)) ⊂ W (z) i ❞❡ ♦♥❞❡✱ u u u φ t (w) ∈ W (z). i

      (z) ⊂ W (x) ▲♦❣♦✱ ❝♦♠♦ W ♣♦r t❡♠♦s q✉❡ u

      φ t i (w) ∈ W (x) u (x)

      ❞❛q✉✐✱ s❡❣✉❡ ♣♦r q✉❡✱ p ∈ W ✳ ❆ss✐♠✱ ❝♦♥❝❧✉í♠♦s ❝♦♠ ❛ ♣r♦✈❛ ❞♦ ❘❡s✉❧t❛❞♦ ❡ ❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛✳

      ❈❛♣ít✉❧♦ ✷ ❘❡s✉❧t❛❞♦ Pr✐♥❝✐♣❛❧

      ◆❡st❡ ❝❛♣ít✉❧♦ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ❛♣r❡s❡♥t❛r ❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛ q✉❡ ♠♦t✐✈♦✉ ❡st❡ ❡st✉❞♦✱ ❝✉❥♦ ❡♥✉♥❝✐❛❞♦ é ❞❛❞♦ ❛ s❡❣✉✐r✳

      ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧✳ ❙❡❥❛♠ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛✱ r (r ≥ 1)

      ❝♦♥❡①❛✱ s✉❛✈❡ ❡ φ : M × R −→ M ✉♠ ✢✉①♦ ❆♥♦s♦✈ ❞❡ ❝❧❛ss❡ C t❛❧ t ) = M q✉❡ Ω(φ ✳ ❊♥tã♦ ❡①✐st❡♠ ❡①❛t❛♠❡♥t❡ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿ ✐✮ ❈❛❞❛ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❢♦rt❡ ❡ ✐♥stá✈❡❧ ❢♦rt❡ é ❞❡♥s♦ ❡♠ M✱ ♦✉ t

      ✐✐✮ φ é ❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❆♥♦s♦✈ ❞❡ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ 1 ❝♦♠♣❛❝t❛ C ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠ ❡♠ M✳

      P❛r❛ t♦❞♦s ♦s r❡s✉❧t❛❞♦s ❞❡❝❧❛r❛❞♦s ♥❡st❡ ❝❛♣✐t✉❧♦ ♦s t❡r♠♦s ❡stá✈❡❧ ❡ ✐♥stá✈❡❧ ♣♦❞❡♠ s❡r tr♦❝❛❞❛s ❡♥tr❡ s✐✳

      ❆s ♣r✐♥❝✐♣❛✐s r❡❢❡r❡♥❝✐❛s sã♦✿ ❡ ♦✉tr❛s r❡❢❡r❡♥❝✐❛s q✉❡ ♣♦❞❡♠ s❡r ❞❡ ❝♦♠♣❧❡♠❡♥t❛çã♦ ❞❛ t❡♦r✐❛ ❢♦r❛♠

      ✷✳✶ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s t ) = M (x) (x) u s

      ❙❛❜❡♠♦s q✉❡✱ s❡ Ω(φ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈ ❡♥tã♦ W ❡ W sã♦ ❞❡♥s♦s ❡♠ M ♣❛r❛ ❝❛❞❛ x ∈ M ✭✈❡r ❚❡♦r❡♠❛ ▼❛s ♥❡♠ s❡♠♣r❡ é ♦ ❝❛s♦ uu ss ❞❡ q✉❡ ❝❛❞❛ ✉♠❛ ❞❛s ❢♦❧❤❛s ❞❡ F ♦✉ ❞❡ F é ❞❡♥s♦ ❡♠ M✳ ■st♦ ♥ã♦ ♦❝♦rr❡✱ t ♣♦r ❡①❡♠♣❧♦✱ s❡ φ é ❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❆♥♦s♦✈✳ ◆❡st❡ ❝❛s♦✱ uu ss ❝❛❞❛ ❢♦❧❤❛ ❞❡ F ❡ ❝❛❞❛ ❢♦❧❤❛ ❞❡ F s✐t✉❛✲s❡ ♥✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ❞❡ ❝♦❞✐♠❡♥sã♦ 1 ❡♠ M✳ ❊st❡ ❢❛t♦ é ✉♠ ❝❧❛r♦ r❡s✉❧t❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✳ Pr♦♣♦s✐çã♦ ✷✳✶✳ ❙❡ p ∈ M é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈ φ t❛❧ uu

      (p) q✉❡ W ♥ã♦ é ❞❡♥s♦ ❡♠ M✳ ❊♥tã♦ 1 uu (p)

      ✐✮ M é ✉♠ ✜❜r❛❞♦ s♦❜r❡ S ❝♦♠ ✜❜r❛ W ✱ ❡

      ✷✼ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      (p) é ❞❡♥s♦ ❡♠ M ❡ [ 0≤t≤r

      ❞❡✜♥✐❞♦ ♣♦r x 7−→ φ r (x) é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✱ ❞❡s❞❡ q✉❡ φ r é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ✜①❛♥❞♦ r

      P❛r❛ ✐ss♦ ♦❜s❡r✈❡♠♦s q✉❡✱ φ r | W uu (p) : W uu (p) −→ W uu (p)

      é q✉❡ φ t é ❛ s✉s♣❡♥sã♦ ❞❡ φ r | W uu (p) .

      ✳ P❛r❛ ✜♥❛❧✐③❛r ❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛✱ só ❢❛❧t❛ ♠♦str❛r ❛ ♣❛rt❡ (ii)✱ ✐st♦

      ❉❛í q✉❡ M é ✉♠ ✜❜r❛❞♦ s♦❜r❡ S 1 ❝♦♠ ✜❜r❛ W uu (p)

      (p) é ❞✐s❥✉♥t❛✳

      ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❞❡s❞❡ q✉❡ M = [ 0≤t≤r φ t W uu

      ✭♠♦❞ r✮ ♦♥❞❡ x ∈ W uu (p)

      M = [ 0≤t≤r φ t W uu (p) . ◆♦t❡ q✉❡✱ ✉♠❛ ✈❡③ q✉❡ ❛s ✜❜r❛s sã♦ ❞✐s❥✉♥t❛s ❛ ✉♥✐ã♦ ❛❝✐♠❛ é ❞✐s❥✉♥t❛✳ ❆❣♦r❛✱ ♦❜s❡r✈❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ ♣r♦❥❡çã♦ π : M −→ S 1 ❞❛❞❛ ♣♦r φ t (x) 7−→ t

      é ❢❡❝❤❛❞♦ ❡♠ M✱ s❡❣✉❡ q✉❡✱

      φ t W uu (p)

      ❆❣♦r❛✱ ❝♦♠♦ ♣❡❧♦ ❚❡♦r❡♠❛ W u

      ✐✐✮ φ t é ❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ φ r | W uu (p) ✱ ♦♥❞❡ r é ♦ ♣❡rí♦❞♦ ❞❡ p✳ ❆ ♣r♦✈❛r ❞❛ Pr♦♣♦s✐çã♦ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ s❡❣✉✐♥t❡ ▲❡♠❛ ♦♥❞❡ ❛✜r♠❛ q✉❡ ❛s ✜❜r❛s sã♦ ❞✐s❥✉♥t❛s✳

      ⊃ [ t ∈R φ t (W uu (p)) = W u (p).

      φ t W uu (p)

      φ t W uu (p) = [ t ∈R

      0≤t≤r

      ❢♦r♠❛♠ ❞❡ ❢❛t♦ t♦❞❛ ❛ ✈❛r✐❡❞❛❞❡✱ ✐st♦ é✿ M = [ 0≤t≤r φ t W uu (p) . P❛r❛ ❝♦♥❝❧✉✐r ❡st❡ ❢❛t♦✱ ♦❜s❡r✈❡♠♦s q✉❡✿ [

      ❉❡♠♦♥str❛çã♦✿ ❉❛ Pr♦♣♦s✐çã♦ ❈♦♠❡ç❛♠♦s ❛ ♣r♦✈❛ ♠♦str❛♥❞♦ q✉❡ ❛❧é♠ ❞❛s ✜❜r❛s s❡r❡♠ ❞✐s❥✉♥t❛s✱ ❡❧❛s

      ❆ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ s❡rá ❢❡✐t❛ ❞❡♣♦✐s ❞❡ ❝♦♥❝❧✉✐r ❝♦♠ ❛ ♣r♦✈❛ ❞❛ Pr♦♣♦s✐çã♦

      ♣❛r❛ t♦❞♦ t ∈ (0, r)✱ ♦♥❞❡ r é ♦ ♣❡rí♦❞♦ ❞❡ p✳

      W uu (p) ∩W uu (p) = ∅

      (p) ♥ã♦ é ❞❡♥s♦ ❡♠ M✳ ❊♥tã♦ φ t

      ▲❡♠❛ ✷✳✶✳ ❙❡ p ∈ M é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈ φ t❛❧ q✉❡ W uu

      ✳

      ✷✽ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      (p) = W uu (p) ✱ ♦♥❞❡ r é ♦ ♣❡rí♦❞♦ ❞❡ p✳

      ❉❛❞♦ s ∈ R ♣r♦✈❛r❡♠♦s q✉❡ s ∈ A✳ ■st♦ é✱ ♣❛r❛ t♦❞♦ δ > 0 ❡①✐st❡ t 1

      ✐✮ ❝❧❛r❛♠❡♥t❡✱ A ⊂ R✳ ✐✐✮ ❆❣♦r❛✱ ♣r♦✈❡♠♦s q✉❡ R ⊂ A✳

      é ❞❡♥s♦ ❡♠ R✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡✱ A = R✳

      W uu (p) = W uu (p)}

      ✳ ❆✜r♠❛çã♦ ✷✳✶✳ A = {t ∈ R : φ t

      ✱ ♦♥❞❡ r é ♦ ♣❡rí♦❞♦ ❞❡ p✳ ❉❡ ❢❛t♦✿ ❙✉♣♦♥❤❛♠♦s✱ ♣♦r ❝♦♥tr❛❞✐çã♦✱ q✉❡✱ ❡①✐st❡ t ∈ (0, r) t❛❧ q✉❡ φ t W uu (p) ∩ W uu (p) 6= ∅

      (p) ∩ W uu (p) = ∅ ♣❛r❛ t♦❞♦ t ∈ (0, r)

      (p) ♥ã♦ é ❞❡♥s♦ ❡♠ M✳ ❊♥tã♦ φ t W uu

      ❆ ♣r♦✈❛ ❞♦ ▲❡♠❛ ❉❡♠♦♥str❛çã♦✿ ❉♦ ▲❡♠❛ ▼♦str❛r❡♠♦s q✉❡✱ s❡ p ∈ M é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈ φ t❛❧ q✉❡ W uu

      ♣❛r❛ t♦❞♦ t ∈ (0, r)✱ ❡♥tã♦ φ t W uu

      ▲♦❣♦✱ ❞❡s❞❡ q✉❡ M ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♥✐ã♦ ❞✐s❥✉♥t❛ ❞❡ φ t W uu (p)

      ∩ W uu (p) 6= ∅

      W uu (p)

      (p) ♥ã♦ é ❞❡♥s♦ ❡♠ M ❡ φ t

      ▲❡♠❛ ✷✳✷✳ ❙❡ p ∈ M é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈ φ t❛❧ q✉❡ W uu

      ∼ ≃ M. ❆ss✐♠✱ ❝♦♥❝❧✉í♠♦s q✉❡ φ t : M −→ M é ♦ ✢✉①♦ ❞❡ s✉s♣❡♥sã♦✳ ❆❣♦r❛✱ ♠♦str❡♠♦s ♦ ▲❡♠❛ ❝♦♠ ❡ss❡ ✜♠ ❡♥✉♥❝✐❛♠♦s ♦ s❡❣✉❡♥t❡ r❡s✉❧t❛❞♦✳

      ❈❧❛r❛♠❡♥t❡ H é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❡ H(r, x) = φ r (x) = H(0, φ r (x)) ❞❡ ♦♥❞❡ [0, r] × W uu (p)

      ∼ −→ M (t, x) 7−→ φ t (x).

      H : [0, r] × W uu (p)

      (p) ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r (r, x) ∼ (0, φ r (x)) ♠❡❞✐❛♥t❡ ❛ s❡❣✉✐♥t❡ ❛♣❧✐❝❛çã♦✿

      ♦♥❞❡ t ∈ [0, r) ✱ ❡♠ [0, r] × W uu

      (s − δ, s + δ) ∩ A ✳

      ✷✾ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s t W (p) ∩ W (p) 6= ∅ uu uu

      ❙❡❥❛♠ s ∈ R ❡ δ > 0 t♦♠❡♠♦s t ∈ (0, δ) t❛❧ q✉❡ φ ✳ ❊♥tã♦ ♣❡❧♦ ▲❡♠❛ t❡♠✲s❡ q✉❡✱ uu uu φ t W (p) = W (p).

      ❉❛í✱ t❛♠❜é♠ t❡♠✲s❡ q✉❡✱ uu uu φ W (p) = W (p) 2t

      ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ uu uu φ nt W (p) = W (p).

      ❉❡ ♦♥❞❡ s❡❣✉❡ q✉❡✱ nt ∈ A ✱ ∀n ∈ Z✳ ▲♦❣♦✱ ❝♦♠♦ |nt − (n − 1)t| = |nt − nt + t| = |t| < δ✳ ❊①✐st❡ n ∈ Z t❛❧ q✉❡✱ + nt ∈ (s − δ, s + δ).

      P❡❧♦ q✉❡✱ nt ∈ (s − δ, s + δ) ∩ A✳ ❉❛í✱ s❡❣✉❡ q✉❡ (s − δ, s + δ) ∩ A 6= ∅✳ t W (p) = W (p)} uu uu ❆ss✐♠✱ A = {t ∈ R : φ é ❞❡♥s♦ ❡♠ R✳ ❉❡ ♦♥❞❡ s❡ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞❛ ❆✜r♠❛çã♦ uu

      (p) ❆❣♦r❛✱ ♦❜s❡r✈❡♠♦s q✉❡ ❛ ❆✜r♠❛çã♦ ✐♠♣❧✐❝❛ q✉❡ W s❡❥❛ ❞❡♥s♦ ❡♠ M✳

      P♦✐s✱ [ uu uu φ t W (p) T W (p)

      ❉❛❞♦s x ∈ M ❡ δ > 0 ❝♦♠♦ M = ❡♥tã♦ x ∈ φ T (y) (p) 0≤t≤r uu ♣❛r❛ ❛❧❣✉♠ T ∈ [0, r]✱ ❞❛í q✉❡ x = φ ♦♥❞❡ y ∈ W ✳

      ❆❣♦r❛✱ ♦❜s❡r✈❡♠♦s q✉❡ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦ ✢✉①♦ φ ❡①✐st❡ ǫ > 0 t❛❧ q✉❡ s❡ t ∈ [T − ǫ, T + ǫ] ❡♥tã♦✱

      φ t (y) ∈ B δ (x).

      P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ ♣❡❧❛ ❆✜r♠❛çã♦ A é ❞❡♥s♦ ❡♠ R ❡①✐st❡ t (T − ǫ, T + ǫ) ∩ A

      ∈ (T − ǫ, T + ǫ) ❉❛q✉✐✱ ❝♦♠♦ t ❡♥tã♦✱ ∗ uu uu φ t (y) ∈ B δ (x).

      W ∈ A t (p) = W (p)

      ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ t ❡♥tã♦ φ ✳ uu uu uu ∗ ∗ W

      (p) t (y) ∈ φ t (p) = W (p) ▲♦❣♦✱ ❝♦♠♦ y ∈ W t❡♠♦s q✉❡ φ ✳ ❆ss✐♠✱ uu

      φ t (y) ∈ W (p).

      ✸✵ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      (p) s❛t✐s❢❛③✿ ✐✮ W uu

      ✐✐✐✮ φ r (K) = K ✳ ❊ t❛❧ q✉❡✱ ♥❡♥❤✉♠ s✉❜❝♦♥❥✉♥t♦ ♣ró♣r✐♦ ❞❡ K s❛t✐s❢❛③ (i)✱ (ii) ❡ (iii)✳ ❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❛✜r♠❛ q✉❡ M é ✉♥✐ã♦ ❞❡ φ t (K) ♦♥❞❡ t ∈ [0, r]✱ ♠❛s ♥ã♦

      ✐✮ K é ❢❡❝❤❛❞♦ ❡♠ M✳ ✐✐✮ K é F uu ✲s❛t✉r❛❞♦✳

      (p) ✭♥ã♦✲✈❛③✐♦✮ t❛❧ q✉❡✿

      (p) ♥ã♦ é ❞❡♥s♦ ❡♠ M✳ ❊♥tã♦ ❡①✐st❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ K ⊂ W uu

      P❛r❛ ♠♦str❛r ♦ ▲❡♠❛ ♣r❡❝✐s❛r❡♠♦s ❞❡ três ❛✜r♠❛çõ❡s q✉❡ s❡rã♦ ❡♥✉♥❝✐❛❞♦s ❡♠ s❡❣✉✐❞❛ ❡ ♣r♦✈❛❞❛s ❞❡♣♦✐s ❞❡ ♠♦str❛r ❡st❡ ❧❡♠❛✳ ❆✜r♠❛çã♦ ✷✳✷✳ ❙❡ p ∈ M é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈ φ t❛❧ q✉❡ W uu

      ❆❧é♠ ❞✐ss♦✱ ♥❡♥❤✉♠ s✉❜❝♦♥❥✉♥t♦ ♣ró♣r✐♦ ❞❡ W uu (p) s❛t✐s❢❛③ (i) ✱ (ii) ❡ (iii)✳

      (p) = W uu (p) ✳

      é F uu ✲s❛t✉r❛❞♦✳ ✐✐✐✮ φ r W uu

      ✐✐✮ W uu (p)

      (p) é ❢❡❝❤❛❞❛ ❡♠ M✳

      (p) ♥ã♦ é ❞❡♥s♦ ❡♠ M✳ ❊♥tã♦ W uu

      ❞❛í q✉❡✱ φ t

      ❢❛t♦ ❝♦♥❝❧✉✐r❡♠♦s ❝♦♠ ❛ ♣r♦✈❛ ❞♦ ▲❡♠❛ ▲❡♠❛ ✷✳✸✳ ❙❡ p ∈ M é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈ φ t❛❧ q✉❡ W uu

      

    ❈♦♠ ❡ss❡ ✜♠ ♣r♦✈❛r❡♠♦s ♣r✐♠❡✐r♦ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✱ ❧♦❣♦ ✉s❛♥❞♦ ❡ss❡

      ■st♦ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞♦ ▲❡♠❛ P❛r❛ ✜♥❛❧✐③❛r só ❢❛❧t❛ ♠♦str❛r ♦ ▲❡♠❛ q✉❡ ❢♦✐ ✉s❛❞♦ ♣❛r❛ ♣r♦✈❛r ♦ ▲❡♠❛

      (p) é ❞❡♥s♦ ❡♠ M✱ ❝♦♥tr❛❞✐③❡♥❞♦ ❛ ❤✐♣ót❡s❡✳

      W uu (p) = M ✱ ✐st♦ é q✉❡ W uu

      (p) = W uu (p) ✱ ❞❛í q✉❡

      (p) é ❢❡❝❤❛❞♦ ❡♥tã♦ t❡♠♦s q✉❡ W uu

      W uu (p) = M . ▼❛s ❝♦♠♦ W uu

      (p) é ❞❡♥s♦ ❡♠ M✱ ✐✳❡

      W uu (p) ∩ B δ (x) 6= ∅. ❉❡ ♦♥❞❡ W uu

      (y) ∈ W uu (p) ∩ B δ (x) ❧♦❣♦✱

      ❛✜r♠❛ q✉❡ é ✉♥✐ã♦ ❞✐s❥✉♥t❛✱ é ✐♠♣♦rt❛♥t❡ ❞❡st❛❝❛r ✐ss♦✳

      ✸✶ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      ❆✜r♠❛çã♦ ✷✳✸✳ ❙❡ p ∈ M é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈ φ t❛❧ q✉❡ uu W

      (p) ♥ã♦ é ❞❡♥s♦ ❡♠ M ❡ K é ❝♦♠♦ ♥❛ ❆✜r♠❛çã♦ ❡♥tã♦ [

      M φ = t (K) 0≤t≤r t (K)

      P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❛✜r♠❛ q✉❡ ♦ ❝♦♥❥✉♥t♦ ♠✐♥✐♠❛❧ K ❡ φ t (K) ♥ã♦ s❡ ✐♥t❡rs❡❝t❛♠ ♣❛r❛ t♦❞♦ t ∈ (0, r)✱ ❝❧❛r❛♠❡♥t❡ q✉❛♥❞♦ K 6= φ ✳ ❙❡♠ ♠❛✐s ♣r❡â♠❜✉❧♦s ❡♥✉♥❝✐❛♠♦s✳ t (K) 6= ∅ t (K).

      ❆✜r♠❛çã♦ ✷✳✹✳ ❙❡ 0 < t < r ❡ K ∩ φ ❡♥tã♦ K = φ ❆s ❛✜r♠❛çõ❡s ❆ uu

      (p) ✐❞❡✐❛ é ♣r♦✈❛r q✉❡ K = W ❡ ❞❛í ✉s❛♥❞♦ ❛ ❆✜r♠❛çã♦ ♦❜t❡r ♦ r❡s✉❧t❛❞♦✳

      ❉❡♠♦♥str❛çã♦✿ ❉♦ ▲❡♠❛ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡✱ s❡ p ∈ M é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❞❡ ♣❡rí♦❞♦ r ❞❡ ✉♠ uu

      (p) ✢✉①♦ ❆♥♦s♦✈ φ ❡♥tã♦ W s❛t✐s❢❛③✿ uu

      (p) ✐✮ W é ❢❡❝❤❛❞❛ ❡♠ M✳ uu uu

      (p) ✐✐✮ W é F ✲s❛t✉r❛❞♦✳ uu uu r (p) = W (p) W

      ✐✐✐✮ φ ✳ uu (p)

      ❆❧é♠ ❞✐ss♦✱ ♥❡♥❤✉♠ s✉❜❝♦♥❥✉♥t♦ ♣ró♣r✐♦ ❞❡ W s❛t✐s❢❛③ (i)✱ (ii) ❡ (iii)✳ ❉❡ ❢❛t♦✿ P❡❧❛ ❆✜r♠❛çã♦ t❡♠♦s q✉❡✱ [

      M φ [ = t (K). 0≤t≤r φ t (K)

      ❉❛í✱ ❞❛❞♦ p ∈ M = t❡♠✲s❡ q✉❡✱ 0≤t≤r p ∈ φ T (K)

      ✭✷✳✶✮ ♣❛r❛ ❛❧❣✉♠ T ∈ [0, r]✳ −T (p) ∈ K uu

      ❊♥tã♦✱ φ ✳ ▲♦❣♦✱ ♣❡❧❛ ❆✜r♠çã♦ ❝♦♠♦ K é F ✲s❛t✉r❛❞♦ t❡♠♦s uu uu (φ (p)) ⊂ K (W (p)) ⊂ K −T −T q✉❡✱ W ✳ P❡❧♦ q✉❡✱ φ ✳ ❉❡ ♦♥❞❡✱ s❡❣✉❡ q✉❡ uu

      φ (W (p)) ⊂ K −T uu uu ✳ ▲♦❣♦✱ ♣❡❧❛ ❆✜r♠çã♦ ❝♦♠♦ K é ❢❡❝❤❛❞♦ t❡♠♦s q✉❡✱ φ (W (p)) ⊂ K W (p) ⊂ K −T −T

      ✳ ❉❛q✉✐✱ s❡❣✉❡ q✉❡ φ ✳ ❆ss✐♠✱ uu W (p) ⊂ φ T (K).

      ✸✷ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      ❆♥♦s♦✈ φ✱ ❡♥tã♦ ❡①✐st❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ K ⊂ W uu (p)

      ✐✮ µ = \ i ∈N

      ⊃ µ 2 ⊃ µ 3 ⊃ ...... ⊃ ✳✳✳✳ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ℑ ♦❜s❡r✈❡♠♦s q✉❡✱

      ❈♦♥s✐❞❡r❡♠♦s ❡♠ ℑ ❛ ♦r❞❡♠ ♣❛r❝✐❛❧ ✐♥❞✉③✐❞❛ ♣♦r ✐♥❝❧✉sã♦ ❞❡ ❝♦♥❥✉♥t♦✳ ▲♦❣♦✱ ❞❛❞❛ ✉♠❛ s❡q✉ê♥❝✐❛ µ 1

      (p) : µ 6= ∅, µ é ❢❡❝❤❛❞♦✱ µ é F uu ✲s❛t✉r❛❞♦ ❡ φ r (µ) = µ}

      ❊ t❛❧ q✉❡✱ ♥❡♥❤✉♠ s✉❜❝♦♥❥✉♥t♦ ♣ró♣r✐♦ ❞❡ K s❛t✐s❢❛③ (i)✱ (ii) ❡ (iii)✳ ❉❡ ❢❛t♦✿ ❙❡❥❛✱ ℑ ❂ {µ ⊂ W uu

      ✐✐✮ K é F uu ✲s❛t✉r❛❞♦✳ ✐✐✐✮ φ r (K) = K

      ✭♥ã♦✲✈❛③✐♦✮ t❛❧ q✉❡✿ ✐✮ K é ❢❡❝❤❛❞♦ ❡♠ M✳

      ▲❡♠❛ ❉❡♠♦♥str❛çã♦✿ ❉❛ ❆✜r♠❛çã♦ ▼♦str❛r❡♠♦s q✉❡✱ s❡ p ∈ M é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❞❡ ♣❡rí♦❞♦ r ❞❡ ✉♠ ✢✉①♦

      ❉❛q✉✐ ❡ ❞♦ ❢❛t♦ q✉❡ K ⊂ W uu (p)

      K = W uu (p). ▲♦❣♦ ✉s❛♥❞♦ ❛ ❆✜r♠❛çã♦ ✜❝❛ ♣r♦✈❛❞❛✳ ❆❣♦r❛ ♣❛ss❛♠♦s ❛ ♠♦str❛r ❛s três ❛✜r♠❛çõ❡s q✉❡ ❢♦r❛♠ ✉s❛❞♦s ♣❛r❛ ♣r♦✈❛r ♦

      W uu (p) ⊂ K. ❆ss✐♠✱

      (p) ⊂ K ✳ ▲♦❣♦✱ ❝♦♠♦ K é ❢❡❝❤❛❞♦✱ t❡♠✲s❡ q✉❡✱

      ✲s❛t✉r❛❞♦✱ t❡♠♦s q✉❡ W uu (p) ⊂ K. ❉❡ ♦♥❞❡✱ W uu

      K = φ T (K). P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❡ t❡♠♦s q✉❡ p ∈ φ T (K) ❡♥tã♦ p ∈ K✳ ❆❣♦r❛✱ ❝♦♠♦ K é F uu

      K ∩ φ T (K) = K 6= ∅. ▲♦❣♦✱ ❝♦♠♦ ❑ ∩φ T (K) 6= ∅ ✱ ♣❡❧❛ ❆✜r♠❛çã♦ s❡❣✉❡ q✉❡✱

      ♣❡❧❛ ❆✜r♠❛çã♦ t❡♠♦s q✉❡ K ⊂ φ T (K). ❊♥tã♦✱

      µ i 6= ∅ ❞❡s❞❡ q✉❡ µ i 6= ∅ ✱ ∀i ∈ N✳

      ✸✸ \ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      µ = i

      ✐✐✮ µ é ❢❡❝❤❛❞♦✳ i ∈N i P♦✐s✱ µ é ❢❡❝❤❛❞♦ ♣❛r❛ t♦❞♦ i ∈ N✳ ▲♦❣♦✱ ❛ ✐♥t❡rs❡❝çã♦ ❛r❜✐tr❛r✐❛ ❞❡ ❢❡❝❤❛❞♦s é ❢❡❝❤❛❞❛✳ ∗ u [

      µ = i

      ✐✐✐✮ µ é F ✲s❛t✉r❛❞♦✳ i ∈N \ µ

      = i i P♦✐s✱ ❞❛❞♦ z ∈ µ t❡♠♦s q✉❡✱ z ∈ µ ♣❛r❛ t♦❞♦ i ∈ N. ▲♦❣♦✱ ❝♦♠♦ uu i ∈N µ i

      é F ✲s❛t✉r❛❞♦✱ t❡♠♦s q✉❡✱ uu W (z) ⊂ µ , ∀i ∈ N. i ❉❛í q✉❡✱ uu \ W µ .

      (z) ⊂ i = µ i ∈N ❆ss✐♠✱ uu W . ∗ ∗ (z) ⊂ µ r (µ ) = µ

      ✐✈✮ φ ✳ P♦✐s✱ \ \ \ ! φ r (µ ) = φ r µ i = φ r (µ i ) = µ i = µ . i i i ∈N ∈N ∈N

      ⊂ µ i ❆❧é♠ ❞✐ss♦✱ µ ✱ ∀i ∈ N✳ ❊♥tã♦ ♣❡❧♦ ❧❡♠❛ ❞❡ ❩♦r♥✁s✱ ℑ ❝♦♥té♠ ✉♠ ❡❧❡♠❡♥t♦ ♠✐♥✐♠❛❧✱ ❝❧❛r❛♠❡♥t❡ s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❛❝✐♠❛✳ ■st♦ ❝♦♥❝❧✉í ❛ ♣r♦✈❛ ❞❛ ❆✜r♠❛çã♦

      ❉❡♠♦♥str❛çã♦✿ ❉❛ ❆✜r♠❛çã♦ [ φ t (K). ◗✉❡r❡♠♦s ♠♦str❛r q✉❡✱ M = 0≤t≤r P❛r❛ ♣r♦✈❛r ❡st❛ ❛✜r♠❛çã♦ ♣r❡❝✐s❛♠♦s ❞❡ ❞♦✐s r❡s✉❧t❛❞♦s q✉❡ ❛♣r❡s❡♥t❛♠♦s

      ❡♠ s❡❣✉✐❞❛✱ ♠❛s ❛♥t❡s ❞❡❧❡ ❞❡♥♦t❡♠♦s✿ [ u 0≤t≤r φ . t (K) = K ❘❡s✉❧t❛❞♦ ✷✳✶✳ K é F ✲s❛t✉r❛❞♦✳ [

      φ = t (K) s (K)

      ❉❡ ❢❛t♦✱ s❡❥❛ z ∈ K ✱ ❡♥tã♦ ❡①✐st❡ s ∈ [0, r] t❛❧ q✉❡ z ∈ φ ✳ 0≤t≤r s (x) ❉❛q✉✐✱ ❡①✐st❡ x ∈ K t❛❧ q✉❡ z = φ ✳

      ✸✹ t (W (z)) ⊂ K uu ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      ❆❣♦r❛✱ ♥♦t❡ q✉❡✱ ❜❛st❛ ♠♦str❛r q✉❡ φ ♣❛r❛ t♦❞♦ t ∈ R✳ ❏á q✉❡ ❞❛q✉✐ t❡rí❛♠♦s q✉❡✱ u uu [ W φ .

      (z) = t (W (z)) ⊂ K t ∈R ❈♦♠ ❡ss❡ ✜♠✱ ♦❜s❡r✈❡♠♦s q✉❡ uu uu

      φ t (W (z)) = W (φ t (z)) uu = W (φ t (φ s (x))) uu = W (φ (x)) t +s uu = φ t (W (x)). +s

      ■st♦ é✱ uu uu φ t (W (z)) = φ t (W (x)). +s

      ▲♦❣♦✱ t♦♠❛♥❞♦ t + s = nr + T ♦♥❞❡ 0 ≤ T ≤ r ❡ n ∈ Z✳ ❚❡♠♦s q✉❡✱ uu uu uu φ t (W (x)) = φ nr (W (x)) = φ T (φ nr (W (x)). +s +T

      ❉❡ ♦♥❞❡✱ uu uu φ t (W (z)) = φ T (φ nr (W (x))). uu uu ✭✷✳✷✮ (x) ⊂ K

      P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ K é F ✲s❛t✉r❛❞♦✱ ♣❛r❛ x ∈ K t❡♠✲s❡ q✉❡ W ✳ ❊♥tã♦✱ uu

      φ r (W (x)) ⊂ φ r (K) = K ♣❡❧♦ q✉❡✱ uu

      φ r (W (x)) ⊂ K 2r (W (x)) ⊂ φ r (K) = K uu ❞❛q✉✐✱ t❛♠❜é♠ t❡♠✲s❡ q✉❡ φ ❞❡ ♦♥❞❡✱ uu

      φ (W (x)) ⊂ K 2r ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ t❡♠♦s q✉❡✱ uu

      φ nr (W (x)) ⊂ K. ❊♥tã♦✱ uu [

      φ φ T (φ nr (W (x))) ⊂ φ T (K) ⊂ t (K) = K 0≤t≤r ■st♦ é✱ uu ∗ φ T (φ nr (W (x))) ⊂ K . T (φ nr (W (x))) = φ t (W (z)). uu uu

      ❆❣♦r❛✱ ❝♦♠♦ ♣♦r t❡♠♦s q✉❡ φ ❈♦♥❝❧✉í♠♦s q✉❡✱ uu φ . t (W (z)) ⊂ K

      ✸✺ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      ❞❡ ♦♥❞❡✱ u W

      (z) ⊂ K ❆ss✐♠✱ ❝♦♥❝❧✉✐♥❞♦ ❛ ♣r♦✈❛ ❞♦ ❘❡s✉❧t❛❞♦

      ❘❡s✉❧t❛❞♦ ✷✳✷✳ K é ❢❡❝❤❛❞♦ ❡♠ M✳ ❉❡ ❢❛t♦✱ t❡♠♦s q✉❡ [ [

      K φ φ = t (K) = (t, K) 0≤t≤r 0≤t≤r [

      φ = (t, x) 0≤t≤r,x∈K = {φ(t, x) : 0 ≤ t ≤ r, x ∈ K} = φ([0, r] × K).

      ■st♦ é✱ K = φ([0, r] × K).

      ❆❣♦r❛✱ ❝♦♠♦ K ⊂ M é ❢❡❝❤❛❞♦ ❡ M ❝♦♠♣❛❝t♦✱ t❡♠♦s q✉❡ K é ❝♦♠♣❛❝t♦✳ ❉❛q✉✐ s❡❣✉❡ q✉❡✱

      [0, r] × K é ❝♦♠♣❛❝t♦. ▲♦❣♦✱ ❝♦♠♦ φ é ❝♦♥t✐♥✉❛✱ s❡❣✉❡ q✉❡ φ([0, r] × K) é ❝♦♠♣❛❝t♦✳

      = φ([0, r] × K) ❆ss✐♠✱ K é ❝♦♠♣❛❝t♦✳ ❉❛í q✉❡✱

      K é ❢❡❝❤❛❞♦. ❉❡ ♦♥❞❡ s❡ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞♦ ❘❡s✉❧t❛❞♦ ❆❣♦r❛✱ ❥á t❡♠♦s ❛s ❢❡rr❛♠❡♥t❛s ♥❡❝❡ssár✐❛s ♣❛r❛ ♠♦str❛r ❛ ❆✜r♠❛çã♦ ∗ ∗ u

      K P♦✐s✱ ❝♦♠♦ ♣❡❧♦ ❘❡s✉❧t❛❞♦ é F ✲s❛t✉r❛❞♦✱ ❞❛❞♦ z ∈ K t❡♠♦s q✉❡✱ u

      W (z) ⊂ K

      ❡♥tã♦✱ u ∗ W . u (z) ⊂ K u (z)

      (z) ▲♦❣♦✱ ❝♦♠♦ W é ❞❡♥s♦ ❡♠ M ♣❡❧♦ ❚❡♦r❡♠❛ t❡♠♦s q✉❡✱ M = W ✳

      ❉❡ ♦♥❞❡✱ u ∗ M .

      = W (z) ⊂ K ∗ ∗ K

      = K ) ❊ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ ♣❡❧♦ ❘❡s✉❧t❛❞♦ é ❢❡❝❤❛❞♦ ✐✳❡ (K t❡♠♦s q✉❡✱ M ⊂ K ✳ ❉❛í✱ ❝♦♠♦ [

      K = φ t (K) ⊂ M 0≤t≤r

      ✸✻ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s t❡♠♦s q✉❡✱ M = K ✳ ■st♦ é✱ [

      M φ = t (K). 0≤t≤r ■st♦ ❝♦♥❝❧✉í ❛ ♣r♦✈❛ ❞❛ ❆✜r♠❛çã♦

      ❉❡♠♦♥str❛çã♦✿ ❉❛ ❆✜r♠❛çã♦ t (K) 6= ∅ t (K). ▼♦str❛r❡♠♦s q✉❡✱ s❡ 0 < t < r ❡ K ∩ φ ❡♥tã♦ K = φ ❉❡ ❢❛t♦✿ t (K)

      ■✮ Pr♦✈❡♠♦s q✉❡✱ K ⊂ φ ✳ t (K) ❖❜s❡r✈❡ q✉❡✱ K ∩ φ s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s (i), (ii) ❡ (iii) ❞❛ ❛✜r♠❛çã♦ ✐✳❡✿ t (K)

      ✐✮ Pr♦✈❡♠♦s q✉❡✱ K ∩ φ é ❢❡❝❤❛❞♦✳ t t (K) P♦✐s✱ ❝♦♠♦ K é ❢❡❝❤❛❞♦ ❡ φ é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ✜①❛♥❞♦ t✱ t❡♠♦s q✉❡ φ t (K) é ❢❡❝❤❛❞♦✳ ▲♦❣♦✱ ❝♦♠♦ ✐♥t❡rs❡çã♦ ❞❡ ❢❡❝❤❛❞♦s é ❢❡❝❤❛❞❛ t❡♠♦s q✉❡ K∩φ é ❢❡❝❤❛❞♦✳ t (K) uu

      ✐✐✮ Pr♦✈❡♠♦s q✉❡✱ K ∩ φ é F ✲s❛t✉r❛❞♦✳ t (K) (x) ⊂ K ∩ φ t (K) uu ❉❛❞♦ x ∈ K ∩ φ q✉❡r❡♠♦s ♠♦str❛r q✉❡✱ W ✳ ❈♦♠♦✱ x

      ∈ K ∩ φ t (K). uu ❚❡♠♦s q✉❡✱ x ∈ K✳ ▲♦❣♦✱ ❝♦♠♦ K é F ✲s❛t✉r❛❞♦ t❡♠♦s q✉❡✿ uu

      W (x) ⊂ K. t (K) (x) ∈ K −t ✭✷✳✸✮ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡ x ∈ φ ❡♥tã♦✱ φ ✳ ❆❣♦r❛✱ ❝♦♠♦ K é uu F

      ✲s❛t✉r❛❞♦ t❡♠♦s q✉❡✱ uu uu W (φ (x)) ⊂ K −t −t (W (x)) ⊂ K ♣❡❧♦ q✉❡✱ φ ✳ ❊♥tã♦✱ uu W (x) ⊂ φ t (K).

      ✭✷✳✹✮ ▲♦❣♦✱ ❞❡ s❡❣✉❡ q✉❡✱ uu

      W t (K) uu (x) ⊂ K ∩ φ t (K). ❆ss✐♠✱ K ∩ φ é F ✲s❛t✉r❛❞♦✳

      ✸✼ r (K ∩ φ t (K)) = K ∩ φ t (K) ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      ✐✐✐✮ Pr♦✈❡♠♦s q✉❡✱ φ ✳ ♣♦✐s✱

      φ r (K ∩ φ t (K)) = φ r (K) ∩ φ r (φ t (K)) = φ (K) ∩ φ (K) r r +t = φ r (K) ∩ φ t (K) +r φ = φ r (K) ∩ φ t (φ r (K)) r (K)=K = K ∩ φ (K). t

      ❉❡ ♦♥❞❡✱ t (K) φ r (K ∩ φ t (K)) = K ∩ φ t (K).

      ❆ss✐♠✱ K ∩ φ é ✉♠ ❝♦♥❥✉♥t♦ q✉❡ s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s (i), (ii) ❡ (iii) ❞❛ ❆✜r♠❛çã♦

      P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❡❧❛ ❆✜r♠❛çã♦ ❝♦♠♦ K é ♦ ❝♦♥❥✉♥t♦ ♠✐♥✐♠❛❧ s❛t✐s❢❛③❡♥❞♦ (i), (ii)

      ❡ (iii) t❡♠♦s q✉❡✱ K t (K) ⊂ K t (K) = K. ⊂ K ∩ φ t (K).

      ❉❛í✱ ❝♦♠♦ K ∩ φ t❡♠♦s q✉❡✱ K ∩ φ ❉❡ ♦♥❞❡✱ t (K) ⊂ K K ⊂ φ (K). t

      ■■✮ ❆❣♦r❛✱ ♣r♦✈❡♠♦s q✉❡ φ ✳ s (K) ◆♦t❡ q✉❡✱ ♣❛r❛ t♦❞♦ s ∈ R✱ φ s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s (i), (ii) ❡ (iii) ❞❛

      ❆✜r♠❛çã♦ ✐✳❡✿ s (K) ✐✮ Pr♦✈❡♠♦s q✉❡✱ φ é ❢❡❝❤❛❞♦✳ t s (K)

      P♦✐s✱ ❝♦♠♦ K é ❢❡❝❤❛❞♦ ❡ φ é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ✜①❛♥❞♦ s✱ t❡♠♦s q✉❡ φ é ❢❡❝❤❛❞♦✳ s (K) uu

      ✐✐✮ Pr♦✈❡♠♦s q✉❡✱ φ é F ✲s❛t✉r❛❞♦✳ s (K) (x) ⊂ φ s (K) uu ❉❛❞♦ x ∈ φ q✉❡r❡♠♦s ♠♦str❛r q✉❡✱ W ✳ ❉❡ ❢❛t♦✿ s (K) (x) ∈ K −s uu ❈♦♠♦ x ∈ φ ❡♥tã♦✱ φ ✳ ▲♦❣♦✱ ❝♦♠♦ K é F ✲s❛t✉r❛❞♦ t❡♠♦s q✉❡✱ uu

      W (φ (x)) ⊂ K −s ♣❡❧♦ q✉❡✱ uu

      φ (W (x)) ⊂ K −s ❞❡ ♦♥❞❡✱ uu

      W (x) ⊂ φ s (K).

      ✸✽ r (φ s (K)) = φ s (K) ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      ✐✐✐✮ Pr♦✈❡♠♦s q✉❡✱ φ ✳ P♦✐s✱ φ r (K)=K

      φ r (φ s (K)) = φ r (K) = φ s (K) = φ s (φ r (K)) = s (K) +s +r φ ❞❡ ♦♥❞❡✱ φ r (φ s (K)) = φ s (K). s (K)

      ❆❣♦r❛✱ ❝♦♠♦ ♣❛r❛ t♦❞♦ s ∈ R✱ φ s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s (i), (ii) ❡ (iii) −t (K) ❞❛ ✜r♠❛çã♦

      ❉❛í✱ ♣❡❧❛ ✜r♠❛çã♦ ❝♦♠♦ K é ✉♠ ❝♦♥❥✉♥t♦ ♠✐♥✐♠❛❧ s❛t✐s❢❛③❡♥❞♦ (i), (ii) ❡ (iii) t❡♠♦s q✉❡✿

      K ⊂ φ (K). −t ❉❛✐✱ s❡❣✉❡ q✉❡

      φ t (K) ⊂ K. ❆ss✐♠✱

      K = φ t (K). ■st♦ ❝♦♥❝❧✉í ❛ ♣r♦✈❛ ❞❛ ❆✜r♠❛çã♦ ❆❣♦r❛ ❥á t❡♠♦s ❛s ❢❡rr❛♠❡♥t❛s ♥❡❝❡ssár✐❛s ♣❛r❛ ♣r♦✈❛r ♦ ▲❡♠❛ ❉❡♠♦♥str❛çã♦✿ ❉♦ ▲❡♠❛ ■st♦ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞♦ ▲❡♠❛ ❏á q✉❡ uu

      (p) = K ♣❡❧♦ ▲❡♠❛ s❡❣✉❡ q✉❡ uu uu φ W t (p) = W (p)

      ♣❛r❛ t♦❞♦ t ∈ (0, r)✳ ◆♦ s❡❣✉✐♥t❡ ❧❡♠❛ ❞❛♠♦s ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ q✉❡ ❛ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧

      ❢♦rt❡ s❡❥❛ ❞❡♥s❛ ❡♠ M ♣❛r❛ t♦❞♦ ♣♦♥t♦ x ∈ M✳ t ) = M (p) uu ▲❡♠❛ ✷✳✹✳ ❙❡❥❛ Ω(φ ♣❛r❛ ✉♠ ✢✉①♦ ❆♥♦s♦✈✳ ❙❡ W é ❞❡♥s♦ ❡♠ M uu

      (x) ♣❛r❛ t♦❞♦s ♦s ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s p ❡♥tã♦ W é ❞❡♥s♦ ❡♠ M ♣❛r❛ t♦❞♦ x ∈ M✳

      ❉❡♠♦♥str❛çã♦✿ uu (x) = M

      ▼♦str❛r❡♠♦s q✉❡ W ♣❛r❛ t♦❞♦ x ∈ M✳ uu uu (x) ⊂ M (x)

      ❈♦♠♦ W só ❢❛❧t❛r✐❛ ♠♦str❛r q✉❡ M ⊂ W ✳ ❉❡ ❢❛t♦✿ uu ❙❡❥❛ W ∈ F ✜①❛❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧ ❢♦rt❡✱ x ✉♠ ǫ (x) ∩ W 6= ∅

      ♣♦♥t♦ ❛r❜✐trár✐♦ ❞❡ M ❡ ǫ > 0 ♣r♦✈❛r❡♠♦s q✉❡ B ✳

      ✸✾ t ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      ❈♦♠♦ x ∈ M ❡ φ é ✉♠ ✢✉①♦ ❆♥♦s♦✈ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ♣r♦❞✉t♦ N r (x)

      ♦♥❞❡ r > 0 ❞❡✜♥✐❞♦ ♣♦r✿ [ s r (x) N B r (x) = (y). y ∈B (x) uu r δ (x) ⊂ N r (x) ▲♦❣♦ ❝♦♠♦ N é ❛❜❡rt♦✱ ❞❛❞♦ r > 0 ❡①✐st❡ δ = δ(r) t❛❧ q✉❡ B

      ♣❛r❛ t♦❞♦ x ∈ M✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ ♦s ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s sã♦ ❞❡♥s♦s ❡♠ M✱ ❡①✐st❡ ✉♠❛ δ (p i ) i ( ) ε

      ❝♦❜❡rt✉r❛ ❞❡ ❜♦❧❛s ❛❜❡rt❛s B 2 ❞❡ M✱ ♦♥❞❡ p é ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❞❡ ♣❡rí♦❞♦ t i r❡s♣❡t✐✈❛♠❡♥t❡ ❡ i ∈ N✳ ❆❣♦r❛ ❝♦♠♦ M é ❝♦♠♣❛❝t♦✱ t❡♠♦s q✉❡ t♦❞❛ ❝✉❜❡rt✉r❛ δ (p ), . . . , B δ (p k ) ( ) ε ε 1 ( ) ❛❜❡rt♦ ❛❞♠✐t❡ ✉♠ s✉❜ ❝♦❜❡rt✉r❛ ✜♥✐t♦ B 2 2 ✳

      ■st♦ é✱ [ k ǫ M ⊂ B (p i ) i =1 t (B (p i )) ∩ B (x) 6= ∅ δ ( ) uu ǫ 2 ✭✷✳✺✮ ǫ

      ❆✜r♠❛çã♦ ✷✳✺✳ ❊①✐st❡ t > 0 t❛❧ q✉❡ φ 2 2 ♣❛r❛ i = 1, . . . , k✳ P❛r❛ ♣r♦✈❛r ❡st❛ ❛✜r♠❛çã♦ ♣r❡❝✐s❛♠♦s ♦s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s✳ uu ǫ

      (p i ) ∩ B (x) 6= ∅ ❘❡s✉❧t❛❞♦ ✷✳✸✳ ❊①✐st❡ T > 0 t❛❧ q✉❡ B T uu 2 ♣❛r❛ i = 1, . . . , k✳

      (p i ) i ∈ per(φ t ) ❉❡ ❢❛t♦✱ ♣❡❧❛ ❤✐♣ót❡s❡ ❝♦♠♦ W é ❞❡♥s♦ ❡♠ M ♣❛r❛ t♦❞♦ p ✱ uu uu

      (p i ) (p i ) t❡♠♦s q✉❡ M ⊂ W ✳ ❉❛í✱ ❞❛❞♦ x ∈ M t❡♠✲s❡ q✉❡ x ∈ W ✱ s♦♠❡♥t❡✱ s❡ uu x x (p i ) 6= ∅ ∩ W

      ♣❛r❛ t♦❞♦ ✈✐③✐♥❤❛♥ç❛ ❞❡ x ❞❡♥♦t❛❞♦ ♣♦r V ✱ t❡♠✲s❡ q✉❡ V ✳ ❈♦♠♦ x = B (x) ǫ ✐st♦ ♦❝♦rr❡ ♣❛r❛ t♦❞❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x ✱ t♦♠❡♠♦s V 2 ✳

      ❉❛í q✉❡✱ ǫ uu B 2 (x) ∩ W (p i ) 6= ∅. uu uu i (p i ) ⊂ W (p i ) >

      ❆❣♦r❛✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r r ✱ i = 1, ...k ❝♦♠ B r i t❛❧ q✉❡✱ ǫ uu B 2 i (x) ∩ B (p i ) 6= ∅. r uu uu

      (r i ) (p i ) ⊂ B (p i ) ▲♦❣♦✱ t♦♠❛♥❞♦ T = max t❡♠♦s q✉❡ B r T ♣❛r❛ t♦❞♦ 1≤i≤k i i

      = 1, ..., k ✳ ❉❡ ♦♥❞❡✱ ǫ uu

      B 2 (x) ∩ B (p i ) 6= ∅. T ❆ss✐♠ s❡ ❝♦♥❝❧✉í ❛ ♣r♦✈❛ ❞♦ ❘❡s✉❧t❛❞♦ t (B (p i )) ∩ B (x) 6= ∅ i T uu ǫ

      ❘❡s✉❧t❛❞♦ ✷✳✹✳ ❊①✐st❡ c > 0 t❛❧ q✉❡ φ 2 ♦♥❞❡ |t| < c.t (i = 1, . . . , k)

      ✳

      ✹✵ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      ❡ i = 1, .., k✳ ❧♦❣♦ ❝♦♠♦ φ t

      (i = 1, . . . , k) ✳

      (p i )) ∩ B ǫ 2 (x) 6= ∅ ♦♥❞❡ |t| < c i .t i

      ✳ ■st♦ é✱ ❡①✐st❡♠ c i > t❛❧ q✉❡ φ t (B uu T

      ♦♥❞❡ |t| < c i .t i (i = 1, . . . , k)

      φ t (B uu T (p i )) ∩ B ǫ 2 (x) 6= ∅

      (z i ) ⊂ B ǫ 2 (x) ✱ t❡♠♦s q✉❡

      ✳ ❆❣♦r❛✱ ❝♦♠♦ B δ i

      ✱ s❡❣✉❡ q✉❡ φ t (B uu T (p i )) ∩ B δ i (z i ) 6= ∅

      ❉❛q✉✐✱ ❝♦♠♦ B uu T (φ t (p i )) ⊂ φ t (B uu T (p i ))

      ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛s ❢♦❧❤❛s ✐♥stá✈❡✐s ❢♦rt❡s✱ t❡♠♦s✱ B uu T (φ t (p i )) ∩ B δ i (z i ) 6= ∅.

      (φ t (p i )) ∩ B δ i (p i ) 6= ∅ ✱ ♥♦✈❛♠❡♥t❡

      (p i ) ∈ B uu Tt (p i )) ❚❡♠♦s q✉❡ B uu T

      ♣❛r❛ |t| < c i .t i

      ❋✐❣✉r❛ ✷✳✶✿ ❘❡s✉❧t❛❞♦ ❉❡ ❢❛t♦✱ ❞❡s❞❡ q✉❡ B uu T

      ✱ t❛❧ q✉❡ φ t (p i ) ∈ B δ i (p i )

      ❆❣♦r❛✱ t♦♠❡♠♦s c i >

      > ✳

      (p i ) ♦♥❞❡ δ i

      B δ i (z i ) ⊂ B ǫ 2 (x) ✳ ▲♦❣♦ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛s ❢♦❧❤❛s ✐♥stá✈❡✐s ❢♦rt❡s ❡①✐st❡ B δ i

      é ❛❜❡rt❛ ❡①✐st❡ δ i > t❛❧ q✉❡

      ✳ ❈♦♠♦ B ǫ 2 (x)

      ∈ B uu T (p i ) ∩ B ǫ 2 (x)

      ❡ i = 1, .., k✳ P♦✐s✿ ❙❡❥❛ z i

      ♣❛r❛ |t| < c i .t i

      ❡①✐st❡♠ c i t❛❧ q✉❡ φ t (B uu T (p i )) ∩ B ǫ 2 (x) 6= ∅

      (x) é ❛❜❡rt♦✱

      (p i ) ∩ B ǫ 2 (x) 6= ∅ ♣❡❧♦ ❘❡s✉❧t❛❞♦ ❡ B ǫ 2

      ❆❣♦r❛✱ t♦♠❛♥❞♦ c = min 1≤i≤k (c i ) t❡♠✲s❡ q✉❡

      ✹✶ uu ǫ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s i (i = 1, . . . , k) φ t (B (p i )) ∩ B (x) 6= ∅ T 2

      ♣❛r❛ |t| < c.t ✳ t (B (p i )) ∩ B (x) 6= ∅ i uu ǫ ■st♦ é✱ ❡①✐st❡ c > 0 t❛❧ q✉❡ φ T ♦♥❞❡ |t| < c.t 2

      (i = 1, . . . , k) ✳

      ❉❡ ♦♥❞❡ s❡ ❝♦♥❝❧✉í ❛ ♣r♦✈❛ ❞♦ ❘❡s✉❧t❛❞♦

      > >

      ❋✐❣✉r❛ ✷✳✷✿ ❘❡s✉❧t❛❞♦ ❘❡s✉❧t❛❞♦ ✷✳✺✳ ❙❡ t é s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❡♥tã♦ uu uu ǫ i i ∈ Z B (p i ) ⊂ φ n t (B (p i )) T i i 2

      ♣❛r❛ n q✉❡ ❞❡♣❡♥❞❡ ❞❡ t ❡ n ♦♥❞❡ (i = 1, . . . , k)✳ ❉❡ ❢❛t♦✱ ❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ♠♦str❛r ❡st❡ r❡s✉❧t❛❞♦ ♦❜s❡r✈❡♠♦s q✉❡✿ l j ∈ R l j ∈ Z , t , n

      ❖❜s❡r✈❛çã♦ ✹✳ ❉❛❞♦s t ❡①✐st❡♠ n ❡ c > 0✱ t❛❧ q✉❡ |n l t l − n j t j | < c.t i i = min (t l , t j )

      ♦♥❞❡ t ✳ 1≤l,j≤k i t i | < ct i ❉❛í✱ ❡①✐st❡ t > 0 ❛r❜✐tr❛r✐❛♠❡♥t❡ ❣r❛♥❞❡ t❛❧ q✉❡ |t − n ♣❛r❛ ❛❧❣✉♠ n ∈ Z i q✉❡ ❞❡♣❡♥❞❡ ❞❡ t ❡ (i = 1, . . . , k)✳

      P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ❞❡st❛ ♦❜s❡r✈❛çã♦ ✈❡r ❆❣♦r❛✱ ❝♦♠❡❝❡♠♦s ❢❛③❡r ❛ ♣r♦✈❛ ❞♦ ❘❡s✉❧t❛❞♦

      ✹✷ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      ) ) i (p i ) ⊂ φ t i (B (p i )) ⊂ φ 2t i (B (p i )) ⊂ uu uu uu ǫ ǫ ǫ

      ❉❡s❞❡ q✉❡ p é ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❡ B 2 2 2 + ... ⊂ ... i ∈ Z

      ❡①✐st❡ m t❛❧ q✉❡✱ uu uu ǫ B T i i (p i ) ⊂ φ m t (B (p i ). + 2 l j ∈ Z l l − , m t

      ❖❜s❡r✈❡♠♦s q✉❡ ♣♦❞❡ ❛❝♦♥t❡❝❡r q✉❡ ❡①✐st❛♠ m t❛✐s q✉❡ |m m j t j | > ct i i = min (t l , t j ) ♦♥❞❡ t ✳ 1≤l,j≤k

    • + P❛r❛ s♦❧✉❝✐♦♥❛r ✐st♦✱ ✉s❡♠♦s ❛ ❖❜s❡r✈❛çã♦ ❖✉ s❡❥❛ ❡①✐st❡ M t❛❧ q✉❡ i ∈ Z m t m t

      |M i i i − M l l l | < ct i i i | < ct i i = M i i t m ▲♦❣♦✱ ❡①✐st❡ t s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ t❛❧ q✉❡ |t − n ♦♥❞❡ n

      ❡ (i = 1, . . . , k)✳

      ) ) uu uu uu ǫ ǫ m t (B (p i )) ⊂ φ n t (B (p i )) i i (p i ) ⊂ < n

      ❆❣♦r❛✱ ❝♦♠♦ φ i i i i ❞❡s❞❡ q✉❡ m ❡ B T uu ǫ 2 2 φ m t (B (p i )) i i 2 ✱ t❡♠♦s q✉❡ uu uu ǫ

      B T i i (p i ) ⊂ φ n t (B (p i )) 2 ❆ss✐♠✱ ❝♦♥❝❧✉✐♥❞♦ ❛ ♣r♦✈❛ ❞♦ ❘❡s✉❧t❛❞♦ ❆❣♦r❛✱ ♣❛r❛ ❝♦♥❝❧✉✐r ❝♦♠ ❛ ♣r♦✈❛ ❞❛ ❆✜r♠❛çã♦ ❜❛st❛ t♦♠❛r t = max (n i t i ) 1≤i≤k

      ❉❛q✉✐✱ uu uu ǫ ǫ φ n i t i (B (p i )) ⊂ φ t (B (p i )) 2 2

      ❛ss✐♠✱ uu uu ǫ B (p ) ⊂ φ (B (p )). T i t i 2

      ✹✸ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      

    <

      ❋✐❣✉r❛ ✷✳✸✿ ❘❡s✉❧t❛❞♦ uu ǫ T (p i ) ∩ B (x) 6= ∅ ▲♦❣♦✱ ❝♦♠♦ ♣❡❧♦ ❘❡s✉❧t❛❞♦ t❡♠♦s q✉❡ B 2

      ❛ss✐♠✱ uu ǫ ǫ φ t (B (p i )) ∩ B (x) 6= ∅. 2 2 ■st♦ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞❛ ❆✜r♠❛çã♦ −t (W )∩B δ ( )(p j ) 6= ∅ ε

      ❆✜r♠❛çã♦ ✷✳✻✳ P❛r❛ ❛❧❣✉♠ j ♦♥❞❡ 1 ≤ j ≤ k t❡♠♦s q✉❡ φ 2 ❉❡ ❢❛t♦✱ t❡♠♦s ♣♦r q✉❡✱ [ k ǫ

      M ⊂ B i =1 δ ( ) (p i ) 2 −t (W ) ⊂ M ❧♦❣♦✱ ❝♦♠♦ φ t❡♠✲s❡ q✉❡ [ k ǫ φ (W ) ⊂ B δ (p i ). −t ( ) [ k i =1 2

      ǫ −t (W ) δ (p i ) B

      ■st♦ é✱ s❡ x ∈ φ ❡♥tã♦ x ∈ ( ) ✳ i =1 δ (p j ) −t (W ) ∩ B δ (p j ) ǫ ǫ 2 ❉❛q✉✐✱ ❡①✐st❡ j ∈ [1, k] t❛❧ q✉❡ x ∈ B ( ) ✱ ❞❡ ♦♥❞❡ x ∈ φ ( ) ✳ 2 2

      ✹✹ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      ❆ss✐♠✱ ♣❛r❛ ❛❧❣✉♠ j ∈ [1, k] t❡♠♦s q✉❡ ǫ φ −t ( ) (W ) ∩ B δ (p j ) 6= ∅. 2

      ❉❡ ♦♥❞❡ s❡ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞❛ ❆✜r♠❛çã♦ uu s ǫ (p j ) ∩ φ −t (B (x)) −t (W ) ∩ W (q)

      ❆✜r♠❛çã♦ ✷✳✼✳ ❊①✐st❡♠ q ∈ W ǫ 2 ❡ y ∈ φ t❛❧ q✉❡ d(q, y) < ✳ 2 ❉❡ ❢❛t♦✱ ♣❡❧❛ ❆✜r♠❛çã♦ t❡♠♦s q✉❡ ❡①✐st❡ t > 0 t❛❧ q✉❡ ♣❛r❛ i = 1, . . . , k t❡♠✲s❡ q✉❡ uu ǫ ǫ

      φ t (B (p i )) ∩ B (x) 6= ∅ 2 2 ❡♥tã♦✱ uu ǫ ǫ

      B uu uu ǫ 2 (p i ) ∩ φ −t (B (x)) 6= ∅. 2 (p i ) ⊂ W (p i )

      ▲♦❣♦✱ ❝♦♠♦ B ♣❛r❛ t♦❞♦ i = 1, ..., k t❡♠♦s q✉❡✱ 2 uu ǫ W (p ) ∩ φ (B (x)) 6= ∅ i −t 2 uu ǫ

      (p j ) ∩ φ (B (x)) 6= ∅ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ j ∈ [1, k] t❡♠♦s q✉❡ W −t ✳ 2

      ❉❡ ♦♥❞❡ t❡♠♦s q✉❡ ❡①✐st❡ uu ǫ q ∈ W (p j ) ∩ φ −t (B (x)). 2 ǫ ✭✷✳✻✮ −t ( ) (W )∩B δ (p j ) 6= ∅

      ❆❣♦r❛✱ ❝♦♠♦ ♣❡❧❛ ❆✜r♠❛çã♦ t❡♠♦s q✉❡ φ uu ǫ 2 ♣❛r❛ ❛❧❣✉♠ j ∈ [1, k] (p ) ∩ φ (B (x)) 6= ∅ j −t ❡ q ∈ W 2 ✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ✈✐③✐♥❤❛♥ç❛ ♣r♦❞✉t♦ s ǫ −t (W ) ∩ W (q)

      ❧♦❝❛❧ t❡♠♦s q✉❡ ❡①✐st❡ y ∈ φ t❛❧ q✉❡ d(q, y) < ✳ 2 ❉❡ ♦♥❞❡ s❡ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞❛ ❆✜r♠❛çã♦ t (y) ∈ B ǫ (x)

      ❆✜r♠❛çã♦ ✷✳✽✳ φ ✳ ǫ ❉❡ ❢❛t♦✱ ❞❡s❞❡ q✉❡ y ❡ q ❡st❡❥❛♠ ♥❛ ♠❡s♠❛ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❡ d(q, y) < ✱ t (q), φ t (y)) < d(q, y) < ǫ 2

      ♣❡❧❛ ❆✜r♠❛çã♦ t❡♠♦s q✉❡✱ d(φ ✳ 2 ■st♦ é✱

      ǫ d . (φ t (q), φ t (y)) <

      ✭✷✳✼✮ uu

      2 ǫ (p j ) ∩ φ (B (x)) −t

      P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ q ∈ W ǫ ǫ 2 ♣♦r s❡❣✉❡ q✉❡ q ∈ φ (B (x)) t (q) ∈ B (x) −t ✱ ❞❡ ♦♥❞❡ φ ✳ 2 2

      ❆ss✐♠✱ ǫ d . (x, φ t (q)) <

      ✭✷✳✽✮

      2

      ✹✺ ✷✳✶✳ Pr♦♣♦s✐çõ❡s ❡ ▲❡♠❛s Pré✈✐♦s

      ❋✐❣✉r❛ ✷✳✹✿ ❆✜r♠❛çã♦ ▲♦❣♦ ✉s❛♥❞♦ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r t❡♠♦s q✉❡✱

      ǫ ǫ d (x, φ t (y)) ≤ d(x, φ t (q)) + d(φ + t (q), φ t (y)) < = ǫ

      2

      2 ❞❡ ♦♥❞❡✱ d (x, φ t (y)) < ǫ.

      ❋✐♥❛❧♠❡♥t❡ ❞❛q✉✐✱ s❡❣✉❡ q✉❡ t (y) ∈ W φ t (y) ∈ B ǫ (x).

      ❊ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ φ ♣❡❧❛ ❆✜r♠❛çã♦ ❚❡♠♦s q✉❡

      φ t (y) ∈ B ǫ (x) ∩ W ❞❡ ♦♥❞❡✱ B ǫ (x) ∩ W 6= ∅.

      ❆ss✐♠✱ x ∈ W . ■st♦ é✱ ❞❛❞♦ x ∈ M ♠♦str❛♠♦s q✉❡ x ∈ W ✳ ❖ q✉❡ ♦ ♠❡s♠♦ ❞✐③❡r q✉❡ M ⊂ W ❡ ❝♦♠♦ ♣♦r ♦✉tr♦ ❧❛❞♦ t❡♠♦s q✉❡ W ⊂ M✱

      ✹✻ ✷✳✷✳ ❋♦❧❤❡❛çõ❡s ❈♦♥❥✉♥t❛♠❡♥t❡ ■♥t❡❣rá✈❡✐s s❡❣✉❡ q✉❡

      W uu uu = M.

      (x) ▲♦❣♦✱ ❝♦♠♦ W ∈ F é ❛r❜✐trár✐♦✱ t♦♠❡♠♦s W = W ✱ ❛ss✐♠ uu

      W (x) = M.

      ❋✐❣✉r❛ ✷✳✺✿ ❆✜r♠❛çã♦

      ✷✳✷ ❋♦❧❤❡❛çõ❡s ❈♦♥❥✉♥t❛♠❡♥t❡ ■♥t❡❣rá✈❡✐s δ (x)

      ❙❡❥❛ N ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ♣r♦❞✉t♦ ❞❡ x ❝♦♠♦ ♥♦ ❚❡♦r❡♠❛ ❙❡ y ❡ z δ (x) > ❡stã♦ ♥❛ ♠❡s♠❛ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧ ❢♦rt❡ ❡♠ N ✱ ❡♥tã♦ ❡①✐st❡ ✉♠ δ t❛❧ y,z : B ′ (y) −→ B (z) s s q✉❡ ❛ ❛♣❧✐❝❛çã♦ F δ δ ❞❛❞❛ ♣❡❧❛ ♣r♦❥❡çã♦ ❛♦ ❧♦♥❣♦ ❞❛ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧ ❢♦rt❡ é ❜❡♠ ❞❡✜♥✐❞❛✳ uu ss ❉❡✜♥✐çã♦ ✷✳✶✳ ❆s ❢♦❧❤❡❛çõ❡s F ❡ F sã♦ ❝❤❛♠❛❞❛s ❝♦♥❥✉♥t❛♠❡♥t❡ δ

      (x) ✐♥t❡❣rá✈❡❧ ❡♠ N ✱ s❡ ♣❛r❛ y ❡ z q✉❡ ❡stã♦ ♥❛ ♠❡s♠❛ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧

      > ❢♦rt❡ ❡ δ t❡♠♦s q✉❡✿ ss s ss s s

      F ′ onde u ′ y,z (W (u) ∩ B (y)) ⊂ W (F y,z (u)) ∩ B (z), ∈ B (y). δ δ δ

      ✹✼ ✷✳✷✳ ❋♦❧❤❡❛çõ❡s ❈♦♥❥✉♥t❛♠❡♥t❡ ■♥t❡❣rá✈❡✐s uu ss F

      ❡ F sã♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s❡ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ M δ (x) u s ❡stã♦ ❝♦♥t✐❞❛ ❡♠ t❛❧ ✈✐③✐♥❤❛♥ç❛ N ✳ ❚❛♠❜é♠ ❞✐r❡♠♦s q✉❡ E ❡ E sã♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳

      ❋✐❣✉r❛ ✷✳✻✿ ❋♦❧❤❡❛çõ❡s ❈♦♥❥✉♥t❛♠❡♥t❡ ■♥t❡❣rá✈❡✐s uu ss ❆❣♦r❛✱ ♣❛ss❛♠♦s ❛ ❡♥✉♥❝✐❛r ✉♠ r❡s✉❧t❛❞♦ q✉❡ é ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❢♦❧❤❡❛çõ❡s F

      ❡ F ❝♦♥❥✉♥t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳ uu ss Pr♦♣♦s✐çã♦ ✷✳✷✳ F ❡ F sã♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ✐♥t❡❣rá✈❡✐s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ u s E

      ⊕ E é ✐♥t❡❣rá✈❡❧✳ u s uu ss

      ⊕ E ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛♠♦s q✉❡ E é ✐♥t❡❣rá✈❡❧ ❡♥tã♦ F ❡ F sã♦

      ❝♦♥❥✉♥t❛♠❡♥t❡ ✐♥t❡❣rá✈❡✐s✳ u s ss ⊕ E y,z (W (u)) ss P♦✐s✱ ❝❛s♦ ❝♦♥tr❛r✐♦ E é t❛♥❣❡♥t❡ ❛ ❞✉❛s s✉❜ ✈❛r✐❡❞❛❞❡s F ❡ u s

      W ⊕ E

      (F y,z (u)) ✳ ❖ q✉❛❧ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ❥á q✉❡ E é t❛♥❣❡♥t❡ ❛ ✉♠❛ ú♥✐❝❛

      ❢♦❧❤❡❛çã♦ ✭✈❡r ✜❣✉r❛ ❞❡ ♦♥❞❡ ss s ss s s F ′ ′ y,z (W (u) ∩ B (y)) ⊂ W (F y,z (u)) ∩ B (z) e u ∈ B (y). δ δ δ uu ss

      ❆❣♦r❛✱ s✉♣♦♥❤❛♠♦s q✉❡ F ❡ F sã♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ✐♥t❡❣rá✈❡✐s ❡♥tã♦✱ u s ⊕ E

      ♠♦str❛r❡♠♦s q✉❡ E é ✐♥t❡❣rá✈❡❧✳ ❉❡ ❢❛t♦✿ uu ss 1

      ❙❡ F ❡ F sã♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ✐♥t❡❣rá✈❡✐s ❡♥tã♦ ❤á ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ C u s ⊕ E

      ú♥✐❝❛ ❛tr❛✈és ❞❡ ❝❛❞❛ ♣♦♥t♦ x ❞❡ M✱ ❝✉❥♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ é E x x ✳ ❚❡♠♦s u s ⊕ E

      ❛ss✐♠ ✉♠❛ ❢♦❧❤❡❛çã♦ F ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠ ❝♦♠ ✜❜r❛❞♦ t❛♥❣❡♥t❡ E ✳ 1 ❆✜r♠❛çã♦ ✷✳✾✳ ❆ ❢♦❧❤❡❛çã♦ F é ❞❡ ❝❧❛ss❡ C ✳

      ✹✽ ✷✳✷✳ ❋♦❧❤❡❛çõ❡s ❈♦♥❥✉♥t❛♠❡♥t❡ ■♥t❡❣rá✈❡✐s ❋✐❣✉r❛ ✷✳✼✿

      ❉❡ ❢❛t♦✱ ♣❛r❛ ❝❛❞❛ p ∈ M✱ s❡❥❛♠ L(p) ✉♠❛ ❢♦❧❤❛ ❞❡ F q✉❡ ❝♦♥té♠ p✱ B ⊂ L(p) ✉♠❛ ❜♦❧❛ ❛❜❡rt❛ ❝♦♥t❡♥❞♦ p✱ ❡ n −1 1

      η : B −→ R t ✉♠ ♠❡r❣✉❧❤♦ C ✳ t : M −→ M ❆❣♦r❛✱ ❝♦♠♦ φ ❧❡✈❛ ❢♦❧❤❛s ❞❡ F ❡♠ ❢♦❧❤❛s ❞❡ F ❞❡s❞❡ q✉❡ φ é

      ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ♣❛r❛ ❝❛❞❛ t ✜①♦✳ ❉❡✜♥❛♠♦s✱ [

      U φ = t (B) |t|<δ

      ♦♥❞❡ δ > 0 é s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ t❛❧ q✉❡ ❛ ❛♣❧✐❝❛çã♦ h : B × (−δ, δ) −→ U

      (x, t) 7−→ h(x, t) = φ t (x) s❡❥❛ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✳ ❆❣♦r❛✱ ❞❡✜♥❛ n n −1

      ψ :U −→ R (= R × R) φ t (x) 7−→ ψ(φ t (x)) = (η(x), t).

      ✹✾ ✷✳✷✳ ❋♦❧❤❡❛çõ❡s ❈♦♥❥✉♥t❛♠❡♥t❡ ■♥t❡❣rá✈❡✐s 1 ❈❧❛r❛♠❡♥t❡ ψ é C ❡ ❛ ❝♦❧❡çã♦ ❞❡ t♦❞❛s ❛s ❝❛rt❛s ❞❡✜♥✐❞❛s ❞❡st❡ ♠♦❞♦ 1

      ❞❡t❡r♠✐♥❛♠ ❛ s♦❧✉çã♦ ❞❡ ❢♦❧❤❡❛çã♦ F✱ ✐st♦ é✱ F é C ✳ ❋✐❣✉r❛ ✷✳✽✿ ❆✜r♠❛çã♦ u s

      ⊕ E ❖❜s❡r✈❡♠♦s q✉❡ ❞❛ Pr♦♣♦s✐çã♦ s❡ ❝♦♥❝❧✉✐ q✉❡✱ s❡ E ♥ã♦ é ✐♥t❡❣rá✈❡❧ uu ss

      ❡♥tã♦ F ❡ F ♥ã♦ sã♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ✐♥t❡❣rá✈❡✐s✳ ❆ ❝♦♥tr❛ ♣♦s✐t✐✈❛ ❞❛ s❡❣✉✐♥t❡ uu ss ♣r♦♣♦s✐çã♦ ❛✜r♠❛ q✉❡✱ s❡ F ❡ F ♥ã♦ sã♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ✐♥t❡❣rá✈❡✐s ❡♥tã♦ uu ss u s

      W (x) (x) ⊕ E

      ❡ W sã♦ ❞❡♥s♦s ❡♠ M ♣❛r❛ ❝❛❞❛ x ∈ M✳ ■st♦ é✱ s❡ E ♥ã♦ é uu ss (x) (x)

      ✐♥t❡❣rá✈❡❧ ❡♥tã♦ W ❡ W sã♦ ❞❡♥s♦s ❡♠ M ♣❛r❛ ❝❛❞❛ x ∈ M✳ Pr♦♣♦s✐çã♦ ✷✳✸✳ ❙❡ ❛❧❣✉♠ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❢♦rt❡ ♦✉ ✐♥stá✈❡❧ ❢♦rt❡ ♥ã♦ é ❞❡♥s♦ uu ss ❡♠ M ❡♥tã♦ F ❡ F sã♦ ❥✉♥t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳

      ❉❡♠♦♥str❛çã♦✿ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ ❞❛❞♦ ✉♠ x ∈ M ❛r❜✐trár✐♦ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ δ (x)

      > ♣r♦❞✉t♦ N ❞❡ x ♦♥❞❡ δ > 0 t❛❧ q✉❡ ❞❛❞♦ y, z ❡ δ ❝♦♠♦ ♥❛ ❉❡✜♥✐çã♦ t❡♠✲s❡ q✉❡✿ ss s ss s s ′ (y) F y,z (W (u) ∩ B (y)) ⊂ W (F y,z (u)) ∩ B (z) δ δ

      ♦♥❞❡ u ∈ B δ ✳ ❉❡ ❢❛t♦✿ δ (x)

      ❖ t❡♦r❡♠❛ ❞❡ ✈✐③✐♥❤❛♥ç❛ ♣r♦❞✉t♦ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ N ✳ ❆❣♦r❛ ♠♦str❡♠♦s q✉❡✱

      ✺✵ ✷✳✷✳ ❋♦❧❤❡❛çõ❡s ❈♦♥❥✉♥t❛♠❡♥t❡ ■♥t❡❣rá✈❡✐s ss s ss s F ′ y,z (W (u) ∩ B (y)) ⊂ W (F y,z (u)) ∩ B (z). δ δ

      P♦r ❤✐♣ót❡s❡✱ ❝♦♠♦ ❛❧❣✉♠❛ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❢♦rt❡ ♦✉ ✐♥stá✈❡❧ ❢♦rt❡ ♥ã♦ é ′ ′ ′ uu ss ∈ M (x ) (x )

      ❞❡♥s♦ ❡♠ M ❡♥tã♦ ❡①✐st❡ x t❛❧ q✉❡ W ♦✉ W ♥ã♦ é ❞❡♥s❛ ❡♠ uu M

      (x ) ✳ ❙✉♣♦♥❤❛♠♦s q✉❡ W ♥ã♦ é ❞❡♥s❛ ❡♠ M✱ ❡♥tã♦ ♣❡❧♦ ▲❡♠❛ ❡①✐st❡ ✉♠ t (p) uu

      ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ p ❞❡ φ ❞❡ ♣❡rí♦❞♦ r t❛❧ q✉❡ W ♥ã♦ é ❞❡♥s❛ ❡♠ M✱ ❧♦❣♦ ♣❡❧❛ Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡✱ 1 uu

      (p) ✐✮ M é ✉♠ ✜❜r❛❞♦ s♦❜r❡ S ❝♦♠ ✜❜r❛ W ✱ ❡ t r | uu

      ✐✐✮ φ é ❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ φ W ✱ ♦♥❞❡ r é ♦ ♣❡rí♦❞♦ ❞❡ p✳ (p) P♦r ♦✉tr♦ ❧❛❞♦ t❡♠♦s q✉❡✿ uu uu uu uu

      é F ✲s❛t✉r❛❞♦ ❞❡s❞❡ q✉❡ W é F ✲s❛t✉r❛❞♦✳ ss

    • W (p) (p) uu
    • W (p)

      é F ✲s❛t✉r❛❞♦✳ uu uu ss (p) (p)

      P♦✐s✱ ❝❛s♦ ❝♦♥tr❛r✐♦ s❡ W ♥ã♦ é F ✲s❛t✉r❛❞♦✱ ❡①✐st❡ u ∈ W t❛❧ ss ss uu uu (u) 6⊂ W (p) (u) (p) q✉❡ W ✳ ❊♥tã♦✱ ❡①✐st❡ v ∈ W t❛❧ q✉❡ v 6∈ W ✳ uu t (p) W

      ♣❛r❛ ❛❧❣✉♠ t ∈ (0, r)✳ ❉❛q✉✐✱ s❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ q✉❡ v ∈ φ ss

      (u) ❆❣♦r❛✱ ❝♦♠♦ v ∈ W t❡♠♦s q✉❡✱ d lim (φ nr (v), φ nr (u)) = 0. n →∞ t W (p) nr (v) ∈ uu ✭✷✳✾✮

      P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ v ∈ φ s❡❣✉❡ q✉❡✱ φ uu uu uu φ φ W φ φ W φ W nr t (p) = t nr (p) = t (p) uu uu

      ✱ ♣♦✐s φ nr W (p) = W (p)

      ❞❡s❞❡ q✉❡ r é ♦ ♣❡rí♦❞♦ ❞❡ p✳ ❆ss✐♠✱ uu φ W . nr (v) ∈ φ t (p) uu uu uu ✭✷✳✶✵✮ W

      (p) nr (u) ∈ φ nr (p) = W (p) ❊✱ ❝♦♠♦ u ∈ W ❡♥tã♦ φ ✳ ❆ss✐♠✱ uu

      φ nr (u) ∈ W (p) uu uu ✭✷✳✶✶✮ φ W , W t (p) (p) = 0

      ▲♦❣♦✱ ♣♦r s❡❣✉❡ q✉❡✱ d ✳ uu uu t (p) (p) W ❉❛q✉✐✱ ❝♦♠♦ φ ❡ W sã♦ ❝♦♠♣❛❝t♦s ❞❡s❞❡ q✉❡ M é ❝♦♠♣❛❝t♦ t W (p) ∩ W (p) 6= ∅ uu uu t❡♠♦s q✉❡✱ φ ✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦ ❛♦ ▲❡♠❛

       uu ss

      W

    • φ t (p) ∈ (0, r)

      é F ✲s❛t✉r❛❞♦ ♦♥❞❡ t ✳

      ✺✶ ✷✳✷✳ ❋♦❧❤❡❛çõ❡s ❈♦♥❥✉♥t❛♠❡♥t❡ ■♥t❡❣rá✈❡✐s uu uu 1 = φ t (x ) ∈ φ t W (p) ∈ W (p) 1 t❛❧ q✉❡ x 1 P♦✐s✱ s❡❥❛ y uu ss ss ss uu ✱ ❧♦❣♦ ❝♦♠♦ W (p) (x ) ⊂ W (p) t (W (x)) ⊂ 1 uu é F ✲s❛t✉r❛❞♦ t❡♠♦s q✉❡ W ❡♥tã♦ φ ss ss ss

      φ W t (p) t (W (x )) = W (φ t (x )) = W (y ) ✱ ❧♦❣♦ ❝♦♠♦ φ 1 1 1 ss uu t❡♠♦s

      W (y 1 ) ⊂ φ t (p) q✉❡✱ W ✳ uu uu uu uu

      W

    • φ t (p) ∈ (0, r) (p)

      é F ✲s❛t✉r❛❞♦ ♦♥❞❡ t ✱ ❞❡s❞❡ q✉❡ W é F ✲ s❛t✉r❛❞♦✳ t W (p) (p) uu uu

      ◆♦t❡ q✉❡✱ φ é ✉♠ ✜❜r❛❞♦✱ ❞❡s❞❡ q✉❡ W é ✉♠ ✜❜r❛❞♦ ❡ uu φ r | W ❞❡✐①❛ ❝❛❞❛ ✜❜r❛❞♦ ✐♥✈❛r✐❛♥t❡✳ ❆❧é♠ ❞✐ss♦ ♣❡❧♦ ▲❡♠❛ ❛s ✜❜r❛s sã♦ (p) ❞✐s❥✉♥t❛s✳ uu u = φ t (p) W

      ❉❡♥♦t❡♠♦s K ❛ ✜❜r❛ q✉❡ ❝♦♥t❡♠ ♦ ♣♦♥t♦ u ♣❛r❛ ❛❧❣✉♠ < r 0 < t

      ✳ ss ss (y) (y) ⊂ K u

      ❆✜r♠❛çã♦ ✷✳✶✵✳ ❙❡ u ∈ W ❡♥tã♦ W ✳ u u (u) ⊂ K u ss ss ❉❡ ❢❛t♦✱ ❝♦♠♦ u ∈ K ❡ K é F ✲s❛t✉r❛❞♦ t❡♠♦s q✉❡ W ✳ ss ss ss

      (y) (u) = W (y) ❉❛í ❝♦♠♦ u ∈ W t❡♠♦s q✉❡ W ✳ ❆ss✐♠✱ ss W (y) ⊂ K u . uu uu

      (y) (y) ⊂ K u ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ s❡ ♣r♦✈❛ q✉❡✱ s❡ u ∈ W ❡♥tã♦ W ✳ ss s ss s

      (y) ∩ B ′ (y) y,z (W (y) ∩ B ′ (y)) ❆✜r♠❛çã♦ ✷✳✶✶✳ ❙❡ u ∈ W δ ❡♥tã♦ F δ ❡st❛ ♥❛ ss s

      (y) ∩ B (y) ♠❡s♠❛ ✜❜r❛ q✉❡ W δ ✳ ss s ss

      (y) ∩ B ′ (y) W (y) ∩ s ❉❡ ❢❛t♦✱ ❈♦♠♦ u ∈ W δ ❡♥tã♦ ♣❡❧❛ ❆✜r♠❛çã♦ B ′ δ ✳ (y) ⊂ K u y,z (W (y) ∩ B ′ (y)) ⊂ K u ss s δ

      ❆❣♦r❛✱ ♠♦str❡♠♦s q✉❡ F ✳ ′ ′ ss s ∈ F (W (y) ∩ B ′ (y)) ∈ K y,z u

      ■st♦ é✱ s❡ q δ ❡♥tã♦ q ✳ ss s ss s ′ ′ ∈ F y,z (W (y) ∩ B (y)) ∈ W (y) ∩ B (y) ′ ′ ❈♦♠♦ q δ ❡♥tã♦ ❡①✐st❡ u δ t❛❧ q✉❡ q = F y,z (u )

      ✳ ′ uu uu ′ ′ ′ ∈ K u u (u ) ⊂ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ u ❡ K é F ✲s❛t✉r❛❞♦ t❡♠♦s q✉❡✱ W ′ ′ ′

      K u = F y,z (u ) ′ uu ′ ✳ ▲♦❣♦✱ ❝♦♠♦ u ❡ q ❡♥tã♦ ♥❛ ♠❡s♠❛ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧ t❡♠♦s q✉❡ q ∈ W (u )

      ✳ ❉❡ ♦♥❞❡ s❡❣✉❡ q✉❡✱ q .

      ∈ K u ′ ′ ss s ✭✷✳✶✷✮ ∈ W (y) ∩ B ′ (y) ⊂ K u ∈ K u

      ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ u δ s❡❣✉❡ q✉❡✱ u ✳

      ✺✷ ✷✳✷✳ ❋♦❧❤❡❛çõ❡s ❈♦♥❥✉♥t❛♠❡♥t❡ ■♥t❡❣rá✈❡✐s ′ ′ ∈ K u ∩ K u u ∩ K u 6= ∅

      ❉❛q✉✐✱ u ✐st♦ é✱ K ❡♥tã♦✱ ♣❡❧♦ ▲❡♠❛ t❡♠♦s q✉❡✱ K . u = K u ❆❣♦r❛✱ ♣♦r s❡❣✉❡ q✉❡✱ q . y,z (W (y) ∩ B (y)) (y) ∩ B (y) ss s ss s ∈ K u ❆ss✐♠✱ F δ ❡st❛ ♥❛ ♠❡s♠❛ ✜❜r❛ q✉❡ W δ ✳ y,z (W (y) ∩ B ′ (y)) ⊂ W (F y,z (y)) ∩ B (z) ss s ss s

      ❆✜r♠❛çã♦ ✷✳✶✷✳ F δ δ ✳ y,z (W (y) ∩ B (y)) 6⊆ W (F y,z (y)) ∩ ss s ss s ss s ss s P♦r ❝♦♥tr❛❞✐çã♦✱ s✉♣♦♥❤❛♠♦s q✉❡ F δ B δ (z) y,z (u) ∈ F y,z (W (y) ∩ B ′ (y)) (y) ∩ B ′ (y) δ δ

      ❡♥tã♦✱ ❡①✐st❡ q = F ❝♦♠ u ∈ W ss s (F (y)) ∩ B (z) y,z t❛❧ q✉❡ q 6∈ W δ ✳ y,z : B (y) −→ B (z) (y) y,z (u) ∈ s s s ′ ′ s s ▲♦❣♦✱ ❝♦♠♦ F δ δ ❡ u ∈ B δ t❡♠♦s q✉❡✱ q = F [ ss

      B W δ δ ✳ (z) ⊂ W (z) = (φ t (z)) t ∈(−δ,δ) ❉❛q✉✐✱ s❡❣✉❡ q✉❡ ss q ∈ W (φ t (z))

      < δ < r 6= 0 = 0

      ♣❛r❛ ❛❧❣✉♠ −δ < t ✳ ◆♦t❡ q✉❡ t ✱ ❝❛s♦ ❝♦♥tr❛r✐♦ s❡ t ❡♥tã♦✱ ss ss ss q ∈ W (z) (F (y)) = W (z) y,z

      ♦ q✉❡ s❡r✐❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s q 6∈ W ✳ z = φ t (p) 1 W ∈ (0, r) uu 1 ❆❣♦r❛✱ s❡❥❛ K ♣❛r❛ ❛❧❣✉♠ t ✳ ❊♥tã♦✱ uu

      φ −t 1 (K z ) = W (p).

      ✭✷✳✶✸✮ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦❜s❡r✈❡♠♦s q✉❡ uu uu

      φ t +t 1 W (p) = φ t φ t 1 W (p) = φ t (K z ) = K φ t0 (z) ❡♥tã♦✱ uu φ W . t +t (z) 1 (p) = K φ t0 ✭✷✳✶✹✮ ❆❧é♠ ❞✐ss♦✱ ♦❜s❡r✈❡♠♦s q✉❡ φ = K q

      ✐✮ K t0 (z) ✳ ss (φ t (z))

      P♦✐s✱ ❝♦♠♦ q ∈ W ❡♥tã♦✱ ♣❡❧❛ ❆✜r♠❛çã♦ t❡♠♦s q✉❡✱ ss W

      (φ t (z)) ⊂ K q t (z) ∈ K q φ = ✱ ❞❛q✉✐✱ φ ❛ss✐♠✱ ♣❡❧❛ ❆✜r♠❛çã♦ K t0 (z)

      K q q = K u ✳ ✐✐✮ K ✳

      ✺✸ ✷✳✷✳ ❋♦❧❤❡❛çõ❡s ❈♦♥❥✉♥t❛♠❡♥t❡ ■♥t❡❣rá✈❡✐s P♦✐s✱ ❝♦♠♦ u ❡ q = F y,z (u) ❡♥tã♦ ♥❛ ♠❡s♠❛ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧ t❡♠♦s q✉❡ q

      K φ t0 (z) = K q = K u = K y = K z ❞❡ ♦♥❞❡✱

      (y) ∩ B s δ ′ (y)) ⊂ W ss (F y,z (y)) ∩ B s δ (z) ✳

      ❖ q✉❡ é ✉♠ ❝♦♥tr❛❞✐çã♦ ❛♦ ▲❡♠❛ ❆ss✐♠✱ F y,z (W ss

      ▲♦❣♦✱ W uu (p) ∩ φ t W uu (p) = W uu (p) 6= ∅.

      ❚❡♠♦s q✉❡✱ W uu (p) = φ −t W uu (p) ⇒ φ t W uu (p) = W uu (p).

      (p) = W uu (p).

      (K φ t0 (z) ) = φ −(t +t 1 ) φ t +t 1 W uu

      ❉❛í✱ ♣♦r ♦✉tr♦ ❧❛❞♦ ❝♦♠♦ φ −(t +t 1 )

      W uu (p) .

      = φ −t (φ −t 1 (K z )) = φ −t

      φ −(t +t 1 ) (K φ t0 (z) ) = φ −(t +t 1 ) (K z )

      K φ t0 (z) = K z . ▲♦❣♦✱

      ♣❡❧❛ ❆✜r♠❛çã♦ K y = K z ✳ ❆ss✐♠✱

      ∈ W uu (u) ❡♥tã♦✱ ♣❡❧❛ ❆✜r♠❛çã♦ t❡♠♦s q✉❡✱ W uu

      (y) ⊂ K z ✱ ❞❛q✉✐✱ y ∈ K z ❛ss✐♠✱

      (y) ❡♥tã♦✱ ♣❡❧❛ ❆✜r♠❛çã♦ t❡♠♦s q✉❡✱ W uu

      ✐✈✮ K y = K z ✳ P♦✐s✱ ❝♦♠♦ y ❡ z ❡♥tã♦ ♥❛ ♠❡s♠❛ ✈❛r✐❡❞❛❞❡ ✐♥stá✈❡❧ t❡♠♦s q✉❡ z ∈ W uu

      K y = K u

      (y) ⊂ K u ✱ ❞❛q✉✐✱ y ∈ K u ❛ss✐♠✱ ♣❡❧❛ ❆✜r♠❛çã♦

       t❡♠♦s q✉❡✱ W ss

      ❧♦❣♦✱ ♣❡❧❛ ❆✜r♠❛çã♦

      ❡♥tã♦ u ∈ W ss (y)

      P♦✐s✱ ❝♦♠♦ u ∈ W ss (y) ∩ B s δ ′ (y)

      ❛ss✐♠✱ ♣❡❧❛ ❆✜r♠❛çã♦ K u = K q ✳ ✐✐✐✮ K u = K y

      (u) ⊂ K q ✱ ❞❛q✉✐✱ u ∈ K q

      ■st♦ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞❛ ❆✜r♠❛çã♦

      ✺✹ ✷✳✸✳ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧

      ❋✐❣✉r❛ ✷✳✾✿ ❆✜r♠❛çã♦ s δ (y) ❉❛q✉✐✱ ❞❡ ♠❛♥❡✐r❛ s✐♠✐❧❛r ♣❛r❛ u ∈ B t❡♠♦s q✉❡✱ ss s ss s F (W (u) ∩ B ′ (y)) ⊂ W (F (u)) ∩ B (z). y,z y,z δ δ

      ■st♦✱ ♣r♦✈❛ ❛ Pr♦♣♦s✐çã♦✳

      ✷✳✸ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧

      ◆❡st❛ s❡çã♦✱ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧✱ ❝✉❥♦ ❡♥✉♥❝✐❛❞♦ é ❞❛❞♦ ❛ s❡❣✉✐r✳ ❚❡♦r❡♠❛ ✷✳✶✳ ❙❡❥❛♠ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛✱ s✉❛✈❡ r

      (r ≥ 1) t ) = M ❡ φ : M × R −→ M ✉♠ ✢✉①♦ ❆♥♦s♦✈ ❞❡ ❝❧❛ss❡ C t❛❧ q✉❡ Ω(φ ✳ ❊♥tã♦ ❡①✐st❡♠ ❡①❛t❛♠❡♥t❡ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿

      ✐✮ ❈❛❞❛ ✈❛r✐❡❞❛❞❡ ❡stá✈❡❧ ❢♦rt❡ ❡ ✐♥stá✈❡❧ ❢♦rt❡ é ❞❡♥s♦ ❡♠ M✱ ♦✉ t ✐✐✮ φ é ❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❆♥♦s♦✈ ❞❡ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ 1

      ❝♦♠♣❛❝t❛ C ❞❡ ❝♦❞✐♠❡♥sã♦ 1 ❡♠ M✳ ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛♠♦s q✉❡ ♥ã♦ ❛❝♦♥t❡❝❡ ♦ ✐t❡♠ (i)✱ ❡♥tã♦ ♠♦str❛r❡♠♦s q✉❡ ❛❝♦♥t❡❝❡ t

      ♦ ✐t❡♠ (ii)✱ ✐st♦ é✱ q✉❡ φ ❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❆♥♦s♦✈ ❞❡ ✉♠❛ 1 s✉❜✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ C ❞❡ ❝♦❞✐♠❡♥sã♦ ✉♠ ❡♠ M✳

      ✺✺ ✷✳✸✳ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧

      ❉❡ ❢❛t♦✿ uu (x) ss ❈♦♠♦ ♥ã♦ ❛❝♦♥t❡❝❡ ♦ ✐t❡♠ (i)✱ ❡①✐st❡ ✉♠ ♣♦♥t♦ x ∈ M t❛❧ q✉❡ W ♦✉ uu

      W (x) (x)

      ♥ã♦ é ❞❡♥s❛ ❡♠ M✳ ❙✉♣♦♥❤❛♠♦s q✉❡ W ♥ã♦ é ❞❡♥s❛ ❡♠ M✱ ❡♥tã♦ t (p) uu ♣❡❧♦ ▲❡♠❛ ❡①✐st❡ ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ p ❞❡ φ t❛❧ q✉❡ W ♥ã♦ é ❞❡♥s❛ ❡♠ M t

      ✳ ❉❛í s❡❣✉❡✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ q✉❡ φ é ❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ φ | uu r W ✱ ♦♥❞❡ r é ♦ ♣❡rí♦❞♦ ❞❡ p✳ (p) uu

      (p) P❛r❛ ❝♦♥❝❧✉✐r✱ ❜❛st❛ ♠♦str❛r q✉❡ W s❡❥❛ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ M✳ P♦✐s t

      ❞❛í t❡rí❛♠♦s q✉❡ φ é ❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❆♥♦s♦✈ ❞❡ ✉♠❛ uu (p) ⊂ M s✉❜✈❛r✐❡❞❛❞❡✳ ❆❧é♠ ❞✐ss♦ ❝♦♠♣❛❝t❛✱ ❞❡s❞❡ q✉❡ M é ❝♦♠♣❛❝t❛ ❡ W t r | uu

      ❢❡❝❤❛❞❛✳ ❊ ❞❡ ❝♦❞✐♠❡♥sã♦ 1 ❡♠ M ❞❡s❞❡ q✉❡ φ é ❛ s✉s♣❡♥sã♦ ❞❡ φ ✳ uu ss W (p) (x) (x)

      ❈♦♠ ❡ss❡ ♣r♦♣ós✐t♦ ♦❜s❡r✈❡♠♦s q✉❡✱ ❝♦♠♦ W ♦✉ W ♥ã♦ é ❞❡♥s❛ ❡♠ uu ss M

      ✱ t❡♠♦s ♣❡❧❛ Pr♦♣♦s✐çã♦ q✉❡ F ❡ F sã♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✱ ❞❛í u s ⊕ E s❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ q✉❡ E é ✐♥t❡❣rá✈❡❧✳ ❉❛q✉✐ ❡①✐st❡ ✉♠❛ ❢♦❧❤❡❛çã♦ F 1 u s

      ⊕ E ❞❡ ❝❧❛ss❡ C t❛❧ q✉❡ ♦ s✉❜❡s♣❛ç♦ E é t❛♥❣❡♥t❡ ❛ F✳ uu

      (p) x = L ❆✜r♠❛çã♦ ✷✳✶✸✳ ❙❡ x ∈ W ❡ L é ✉♠❛ ❢♦❧❤❛ ❞❡ F ❝♦♥t❡♥❞♦ x ❡♥tã♦ uu L ⊂ W

      (p) ✳ u s

      ⊕ E ❉❡ ❢❛t♦✱ ❝♦♠♦ L é ✉♠❛ ❢♦❧❤❛ ❞❡ F ❡♥tã♦ L é t❛♥❣❡♥t❡ ❛ E ❡ t❡♠ ❛ [ ss uu uu

      W (y)

      ❢♦r♠❛ ❞❡ ❞❡s❞❡ q✉❡ L é F ✲s❛t✉r❛❞♦ ❡ F ✲s❛t✉r❛❞♦✳ y ∈W (x) uu [ ss ss W (y) (y)

      ▲♦❣♦✱ ❞❛❞♦ z ∈ L = t❡♠♦s q✉❡✱ z ∈ W ♣❛r❛ ❛❧❣✉♠ uu y ∈W (x) uu y ∈ W (x)

      ✳ uu uu uu uu uu (p) (p) (p)

      ▲♦❣♦✱ ❝♦♠♦ x ∈ W ❡ W é F ✲s❛t✉r❛❞♦✱ ❞❡s❞❡ q✉❡ W é F ✲ uu uu (x) ⊂ W (p) s❛t✉r❛❞♦✱ t❡♠♦s q✉❡ W ✳ ❉❛q✉✐ s❡❣✉❡ q✉❡✱ uu y uu ss ∈ W (p).

      (p) ss ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ W é F ✲s❛t✉r❛❞♦ ✭✈❡r Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡ uu W

      (y) ⊂ W (p) ✳ ❉❡ ♦♥❞❡✱ uu z ∈ W (p).

      ❆ss✐♠✱ uu L ⊂ W (p).

      ■st♦ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞❛ ❆✜r♠❛çã♦ uu (p)

      ❆❣♦r❛✱ s❡ ♠♦str❛♠♦s q✉❡ L = W ❝♦♥❝❧✉✐r✐❛♠♦s ❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛✱ uu 1 (p)

      ♣♦✐s t❡rí❛♠♦s q✉❡ W é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ ❝❧❛ss❡ C ✱ ❥á q✉❡ L ∈ F é ✉♠❛ 1 s✉❜✈❛r✐❡❞❛❞❡ ❞❡ ❝❧❛ss❡ C ❞❡ M✳ uu uu (p) (p))

      P❛r❛ ✐st♦✱ ❜❛st❛ ♠♦str❛r q✉❡ L é ❞❡♥s♦ ❡♠ W ✭✐st♦ é✱ L = W ❡ q✉❡ uu L

      (p) é ❢❡❝❤❛❞♦ ✭✐st♦ é✱ L = L)✱ ♣♦✐s ❝❧❛r❛♠❡♥t❡ ❞❛q✉✐ t❡rí❛♠♦s q✉❡ L = W ✳

      ✺✻ ✷✳✸✳ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧

      L ⊂ W uu (p)

      ✱ ❡♠ ♣❛rt✐❝✉❧❛r ♣❛r❛ V x = [ −ǫ<t<ǫ φ t (L)

      ✱ q✉❡ ❝❧❛r❛♠❡♥t❡ ❝♦♥t❡♠ x ❞❡s❞❡ q✉❡ L = L x ✱ ♦♥❞❡ ǫ < r✳ ❊♥tã♦✱ [ −ǫ<t<ǫ

      φ t (L) ∩ W uu (p) 6= ∅. ▲♦❣♦✱ ❡①✐st❡ t ∈ (−ǫ, ǫ) t❛❧ q✉❡ φ t (L) ∩ W uu

      (p) 6= ∅ ✳ ❆❣♦r❛ ♥♦t❡ q✉❡ ♣♦r t 6= 0

      ✳ ❆ss✐♠✱ ❡①✐st❡ |t| < ǫ ❡ t 6= 0 t❛❧ q✉❡✱

      W uu (p) ∩ φ t (L) 6= ∅.

      ✭✷✳✶✻✮ ▲♦❣♦✱ ❝♦♠♦ ♣❡❧❛ ❆✜r♠❛çã♦

      ❡ W uu (p) ⊂ W uu (p) t❡♠♦s q✉❡✱ W uu

      ❡♥tã♦✱ ♣❛r❛ t♦❞❛ V x ✭✈✐③✐♥❤❛♥ç❛ ❞❡ x✮✱ t❡♠♦s q✉❡

      (p) ∩ φ t (L) ⊂ W uu (p) ∩ φ t (W uu (p)) ✳ ❉❛q✉✐✱ ♣♦r s❡❣✉❡ q✉❡✱

      W uu (p) ∩ φ t (W uu (p)) 6= ∅

      ✳ ■st♦ é✱ ❡①✐st❡ |t| < ǫ ❡ t 6= 0 t❛❧ q✉❡ W uu

      (p) ∩ φ t (W uu (p)) 6= ∅ ✳ ❖ q✉❡ é ✉♠❛

      ❝♦♥tr❛❞✐çã♦ ❛♦ ▲❡♠❛ ❉❛q✉✐ ❝♦♥❝❧✉í♠♦s q✉❡ W uu

      (p) ⊂ L ✳

      ❆ss✐♠✱ W uu (p) ⊂ L

      ✳ Pr♦✈❡♠♦s ❛ s❡❣✉✐♥t❡ ❛✜r♠❛çã♦✳

      V x ∩ W uu (p) 6= ∅

      ▼❛s✱ x ∈ W uu (p)

      ❆✜r♠❛çã♦ ✷✳✶✹✳ L = W uu (p)

      ✳ ❙✉♣♦♥❤❛♠♦s ♣♦r ❛❜s✉r❞♦ q✉❡✱ W uu

      ✐✮ ❈❧❛r❛♠❡♥t❡✱ L ⊂ W uu (p)

      ✳ P♦✐s✱ ♣❡❧❛ ❆✜r♠❛çã♦ t❡♠♦s q✉❡✱ L ⊂ W uu

      (p) ❡♥tã♦ L ⊂ W uu

      (p) = W uu (p)

      ✳ ■st♦ é✱ L ⊂ W uu (p)

      ✳ ✐✐✮ ❆❣♦r❛✱ ♣r♦✈❡♠♦s q✉❡ W uu

      (p) ⊂ L ✳

      P❛r❛ ✐st♦✱ ❜❛st❛ ♠♦str❛r q✉❡ W uu (p) ⊂ L

      (p) 6⊂ L ✳

      (p) ∩ L = ∅ ✭✷✳✶✺✮

      ❊♥tã♦✱ ❡①✐st❡ q 1 ∈ W uu

      (p) t❛❧ q✉❡ q 1 6∈ L

      ✳ ▼❛✐s ❛✐♥❞❛✱ ♦❜s❡r✈❡♠♦s q✉❡✱ ♣❛r❛ t♦❞♦ q ∈ W uu

      (p) t❡♠♦s q✉❡✱ q 6∈ L✳ P♦✐s✱ ❝♦♠♦ q ∈ W uu (p)

      ❡♥tã♦ W uu

      (q) = W uu (p) ❧♦❣♦✱ W uu

      (q) 6⊂ L ✳ ❉❛q✉✐✱ ❝♦♠♦ L é F uu ✲s❛t✉r❛❞♦ t❡♠♦s q✉❡✱ q 6∈ L✳

      ❆ss✐♠✱ W uu

      ❆✜r♠❛çã♦ ✷✳✶✺✳ L é ❢❡❝❤❛❞♦✳

      ✺✼ ✷✳✸✳ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧

      ✭◆♦t❡ q✉❡✱ s❡ x n ∈ L z ♥ã♦ t❡♠♦s ♥❛❞❛ q✉❡ ♣r♦✈❛r✮✳ ❉❛q✉✐✱ ❝♦♠♦ L z ⊂ W uu

      ❖❜s❡r✈❛çã♦ ✺✳ ❊♠ 1992 ❈✳ ❇♦♥❛tt✐ ❡ ❘✳ ▲❛♥❣❡✈✐♥✱ ❝♦♥str✉✐r❛♠ ✉♠ ✢✉①♦ ❞❡ ❆♥♦s♦✈ tr❛♥s✐t✐✈♦ ♥✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛ ❞❡ ❞✐♠❡♥sã♦ 3 q✉❡ ♥ã♦ é ✉♠❛ s✉s♣❡♥sã♦ ❞❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❆♥♦s♦✈✱ ♣❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ✈❡r

      ❝♦♥tr❛❞✐çã♦ ❛♦ ▲❡♠❛ ❉❡ ♦♥❞❡✱ L é ❢❡❝❤❛❞♦✳

      (p) ∩ W uu (p) 6= ∅ ♣❛r❛ 0 < |t| < ǫ < r✱ ♦ q✉❡ é ✉♠❛

      ■st♦ é✱ φ t W uu

      (p) ∩ W uu (p) ✳

      ✭✷✳✶✽✮ ❆ss✐♠✱ ❞❡ t❡♠♦s q✉❡✱ x N ∈ φ t W uu

      (p) t❡♠♦s q✉❡✱ x N ∈ W uu (p).

      ✭✷✳✶✼✮ ❆❧é♠ ❞✐ss♦✱ ❞❡s❞❡ q✉❡ x N ∈ L ❡ L ⊂ W uu

      ✳ ❉❡ ♦♥❞❡✱ ♣❛r❛ ❛❧❣✉♠ 0 < |t| < ǫ x N ∈ φ t (W uu (p)).

      (p) ❡♥tã♦✱ x N ∈ [ 0<|t|<ǫ φ t (W uu (p))

      (p) t❡♠♦s q✉❡✱ x N ∈ V z ∩ [ 0<|t|<ǫ φ t W uu

      ∈ N t❛❧ q✉❡ ♣❛r❛ n > n t❡♠✲s❡ q✉❡ x n ∈ V z ✳ ❆❧é♠ ❞✐ss♦ s❡ x n 6∈ L z ❡♥tã♦ ❡①✐st❡ x N ∈ L ♣❛r❛ ❛❧❣✉♠ N > n t❛❧ q✉❡✱ x N ∈ V z ∩ [ 0<|t|<ǫ φ t (L z ).

      ❙✉♣♦♥❤❛♠♦s ♣♦r ❝♦♥tr❛❞✐çã♦ q✉❡ L ♥ã♦ é ❢❡❝❤❛❞♦✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ {x n } n ∈N ✱ x n ∈ L t❛❧ q✉❡ lim n →∞ x n = z

      ❆❣♦r❛✱ ❝♦♠♦ ❡①✐st❡ {x n } n ∈N ✱ x n ∈ L t❛❧ q✉❡ lim n →∞ x n = z ✱ ❡♥tã♦ ❡①✐st❡ n

      2 ✳

      φ t (L z ) ❝♦♠ 0 < ǫ < r

      ✭L z é ❛ ❢♦❧❤❛ ❞❡ F q✉❡ ❝♦♥t❡♠ ③✮✳ ❙❡❥❛ V z ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ z ❡ ❝♦♥s✐❞❡r❡ [ −ǫ<t<ǫ

      ✳ ❉❛q✉✐ s❡❣✉❡ ❞❛ ❆✜r♠❛çã♦ q✉❡ L z ⊂ W uu (p)

      é ❢❡❝❤❛❞♦ ❡♥tã♦ z ∈ W uu (p)

      ❝♦♠♦ W uu (p)

      ∈ W uu (p) ✳ ▲♦❣♦✱

      (p) ❡♥tã♦ x n

      ∈ L ❡ L ⊂ W uu

      ❡ z 6∈ L✳ ❖❜s❡r✈❡♠♦s q✉❡✱ ❝♦♠♦ x n

      P♦r ♦✉tr♦ ❧❛❞♦ ❊✳ ●❤②s ♣r♦✈♦✉ ❡♠ q✉❡ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ♥♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ✉♥✐tár✐♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛ ❞❡ ❝✉r✈❛t✉r❛ ♥❡❣❛t✐✈❛ é ✉♠ ✢✉①♦ ❞❡ ❆♥♦s♦✈ tr❛♥s✐t✐✈♦ ❛❧é♠ ❞✐ss♦ ♥ã♦ é ✉♠❛ s✉s♣❡♥sã♦✳

      ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

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

      ❬✷❪ ❇❛r❜♦t✱ ❚✳ ●❡♦♠❡tr✐❡ tr❛♥s✈❡rs❡ ❞❡s ✢♦ts ❞✬❆♥♦s♦✈✳ ❚❤❡s❡✳ ▲②♦♥ ❞❡❝❡♠❜r❡ ✶✾✾✷✳

      ❬✸❪ ❇♦♥❛tt✐✱ ❈✳❀ ▲❛♥❣❡✈✐♥✱ ❘✳❀ ❯♥ ❡①❡♠♣❧❡ ❞❡ ✢♦t ❞✬❆♥♦s♦✈ tr❛♥s✐t✐❢ tr❛♥s✈❡rs❡ ❛ ✉♥ t♦r❡ ❡t ♥♦t ❝♦♥❥✉❣✉❡ ❛ ✉♥❡ s✉s♣❡♥s✐♦♥✳ ❊r❣♦❞✐❝ ❚❤❡♦r② ❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s✱ ✶✾✾✹✳

      ❬✹❪ ❇r✐♥✱ ▼✳❀ ❙t♦❝❦✱ ●✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✷✵✵✷✳

      ❬✺❪ ❈❛♠❛❝❤♦✱ ❈✳❀ ▲✐♥s✱ ◆✳ ●❡♦♠❡tr✐❝ ❚❤❡♦r② ♦❢ ❋♦❧✐❛✐t✐♦♥s✳ ❇✐r❦❤❛✉s❡r✱ ❇♦st♦♥✱ ✶st ❡❞✱ ✶✾✽✺✳

      ❬✻❪ ❉♦❝❛r♠♦✱ ▼✳ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ Pr♦❥❡t♦ ❊✉❝❧✐❞❡s✱ ■▼P❆✱ ✶✾✼✾✳

      ❬✼❪ ❋r❛♥❦s✱ ❏✳❀ ❲✐❧❧✐❛♠s✱ ❇✳ ❆♥♦♠❛❧♦✉s ❆♥♦s♦✈ ✢♦✇s✳ ●❧♦❜❛❧ ❚❤❡♦r② ♦❢ ❉✐♥❛♠②❝❛❧ ❙②st❡♠s✳ ❙♣r✐♥❣❡r ▲❡❝t✉r❡ ◆♦t❡s ❡♥ ▼❛t❤❡♠❛t✐❝s ✽✶✾✳ ❙♣r✐♥❣❡r✿ ❇❡r❧✐♥✳ ✶✾✽✵✱ ♣♣✳✶✺✽✲✶✼✹✳

      ❬✽❪ ●❤②s✱ ❊✳❀ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ ✈❡rt♦r ✜❡❧❞s ✐♥ ❞✐♠❡♥s✐♦♥ ✸✳ ◆♦t❡s ❜② ▼❛tt❤✐❛s ❛♥❞ ❙✐❞❞❤❛rt❤❛ ❇❤❛tt❛❝❤❛r②❛✳ ✷✵✶✸✳

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

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

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

      ❬✶✷❪ ▲✐♠❛✱ ❊✳ ▲✳ ❱❛r✐❡❞❛❞❡s ❉✐❢❡r❡♥❝✐➹➼✈❡✐s✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ Pr♦❥❡t♦ ❊✉❝❧✐❞❡s✱ ■▼P❆✱ ✶✾✼✽✳

      ❬✶✸❪ ▼❛ts✉♠♦t♦✱ ❙✳❀ ❈♦❞✐♠❡♥s✐♦♥ ♦♥❡ ❆♥♦s♦✈ ❋❧♦✇✳ ◆♦t❡s ♦❢ t❤❡ ❙❡r✐❡s ♦❢ ▲❡❝t✉r❡s ❤❡❧❞ ❛t t❤❡ ❙❡♦✉❧ ◆❛t✐♦♥❛❧ ❯♥✐✈❡rs✐t②✱ ✶✾✾✺✳

      ✺✾ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

      ❬✶✹❪ ▼♦r❛❧❡s✱ ❈✳ ❆✳ ▲❡❝t✉r❡s ♦♥ s❡❝t✐♦♥❛❧✲❆♥♦s♦✈ ✢♦✇s✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✳ ▼♦♥❛ts❤✳ ▼❛t❤✱ ✶✺✾✭✵✸✮✱ ✷✵✶✵✳

      ❬✶✺❪ ◆✐✈❡♥✱ ■✳ ■rr❛t✐♦♥❛❧ ♥✉♠❜❡rs✱ ❈❛r✉s ▼❛t❤✳ ▼♦♥♦❣r❛♣❤ ♥♦❀ ✶✶✱ ✶✾✺✻✳ ❬✶✻❪ P❛❧✐s✱ ❏✳❀ ▼❡❧♦✱ ❲✳ ■♥tr♦❞✉❝❛♦ ❛♦s ❙✐st❡♠❛s ❉✐♥❛♠✐❝♦s✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿

      Pr♦❥❡t♦ ❊✉❝❧✐❞❡s✱ ■▼P❆✱ ✶✾✼✽✳ ❬✶✼❪ P❧❛♥t❡✱ ❏✳ ❋✳❀ ❆♥♦s♦✈ ✢♦✇s✳ ❆♠❡r✐❝❛♥ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❱✳ ✾✹✱

      ◆♦✳ ✸ ✭❏✉❧✳✱ ✶✾✼✷✮✱ ♣♣✳ ✼✷✾✲✼✺✹✳ ❬✶✽❪ P❧❛♥t❡✱ ❏✳ ❋✳❀ ❙♦❧✈❛❜❧❡ ❣r♦✉♣s ❛❝t✐♥❣ ♦♥ t❤❡ ❧✐♥❡✳ ❚r❛s❛❝t✐♦♥s ♦❢ t❤❡

      ❆♠❡r✐❝❛♠ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✷✼✽✱ ✹✵✶✲✹✶✹✱ ✶✾✽✸✳ ❬✶✾❪ P❡r❦♦✱ ▲✳ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ❛♥❞ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s✱ ✸r❞ ❡❞✳

      ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ✷✵✶✶✳ ❬✷✵❪ P✐❧②✉❣✐♥✱ ❙✳ ❙❤❛❞♦✇✐♥❣ ✐♥ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✲

      ◆❡✇ ❨♦r❦✱ ✶✾✾✾✳ ❬✷✶❪ ❙♠❛❧❡✱ ❙✳ ❉✐✛❡r❡♥t✐❛❜❧❡ ❞②♥❛♠✐❝❛❧ s②st❡♠s✳ ❇✉❧❧❡t✐♥ ♦❢ t❤❡ ❆♠❡r✐❝❛♥

      ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✈✳ ✼✸ ✭✶✾✻✼✮✱ ♣♣✳ ✼✹✼✲✽✶✼✳ ❬✷✷❪ ❱❡r❥♦✈s❦②✱ ❆✳ ❙✐st❡♠❛s ❞❡ ❆♥♦s♦✈✳ ❯♥✐✈❡rs✐❞❛❞ ◆❛❝✐♦♥❛❧ ❞❡ ■♥❣❡♥✐❡r✐❛✱

      ❳■■✲❊▲❆▼✱ ✶✾✾✾✳ ❬✷✸❪ ❩❡❤♥❞❡r✱ ❊✳ ▲❡❝t✉r❡s ♦♥ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s✳ ❊✉r♦♣❡❛♥ ▼❛t❤❡♠❛t✐❝❛❧

      ❙♦❝✐❡t②✱ ✷✵✶✵✳ ❬✷✹❪ ❩❤✉③❤♦♠❛✱ ❱✳❀ ❆r❛♥s♦♥✱ ❙✳❀ ❇❡r❧✐ts❦② ❘✳ ◗✉❛❧✐t❛t✐✈❡ ❚❤❡♦r② ♦❢

      ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ♦♥ ❙✉r❢❛❝❡s✳ ❙♣r✐♥❣❡r Pr❡ss✱ ✶✾✾✾✳

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