MESTRADO EM MATEMÁTICA FINANCEIRA TRABALHO FINAL DE MESTRADO DISSERTAÇÃO

▼❆❚❊▼➪❚■❈❆ ❋■◆❆◆❈❊■❘❆

❚❘❆❇❆▲❍❖ ❋■◆❆▲ ❉❊ ▼❊❙❚❘❆❉❖

❉■❙❙❊❘❚❆➬➹❖

❚❊❖❘❊▼❆ ❉❖ ▼❊❘●❯▲❍❖ ❉❊ ❚❆❑❊◆❙✿

❘❊❈❖◆❙❚❘❯➬➹❖ ❉❖ ❊❙P❆➬❖ ❉❊ ❋❆❙❊❙ ❉❊ ❯▼

❙■❙❚❊▼❆ ❉■◆➶▼■❈❖ ❯❙❆◆❉❖ ❙➱❘■❊❙ ❚❊▼P❖❘❆■❙

❙❆❘❆ ❇➪❘❇❆❘❆ ❉❯❚❘❆ ▲❖P❊❙

▼❆❚❊▼➪❚■❈❆ ❋■◆❆◆❈❊■❘❆

❚❘❆❇❆▲❍❖ ❋■◆❆▲ ❉❊ ▼❊❙❚❘❆❉❖

❉■❙❙❊❘❚❆➬➹❖

❚❊❖❘❊▼❆ ❉❖ ▼❊❘●❯▲❍❖ ❉❊ ❚❆❑❊◆❙✿

❘❊❈❖◆❙❚❘❯➬➹❖ ❉❖ ❊❙P❆➬❖ ❉❊ ❋❆❙❊❙ ❉❊ ❯▼

❙■❙❚❊▼❆ ❉■◆➶▼■❈❖ ❯❙❆◆❉❖ ❙➱❘■❊❙ ❚❊▼P❖❘❆■❙

❙❆❘❆ ❇➪❘❇❆❘❆ ❉❯❚❘❆ ▲❖P❊❙

❖❘■❊◆❚❆➬➹❖✿ ❏❖➹❖ ▲❖P❊❙ ❉■❆❙

▼❡s♠♦ ♥ã♦ t❡♥❞♦ t♦❞♦s ♦s ❞❛❞♦s

❞❡✈❡♠♦s ❡ ♣♦❞❡♠♦s t❡♥t❛r r❡❝♦♥str✉✐r ❛ ✈❡r❞❛❞❡✳

❆❣r❛❞❡❝✐♠❡♥t♦s

❆♦ Pr♦❢❡ss♦r ❏♦ã♦ ▲♦♣❡s ❉✐❛s✳ ❆ s✉❛ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ❡ ✐❞❡✐❛s ❞✉r❛♥t❡ ❛ ♣r❡♣❛✲ r❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦ ❢♦r❛♠ ✐♠♣r❡s❝✐♥❞í✈❡✐s✳

➚ ♠✐♥❤❛ ♠ã❡ ✕ ❢♦♥t❡ ❝♦♥tí♥✉❛ ❞❡ ✐♥s♣✐r❛çã♦ ♥❛ ❢❛❝❡ ❞❡ t♦❞❛s ❛s ❛❞✈❡rs✐❞❛❞❡s ✕ ♣❡❧♦ ❛♠♦r✱ ❝❛r✐♥❤♦✱ ❜♦❛ ❞✐s♣♦s✐çã♦ ❡ ❜♦♥s ✈❛❧♦r❡s✳

➚ ♠✐♥❤❛ ✐r♠ã✱ ❝ú♠♣❧✐❝❡ ❡ ❛♠✐❣❛✳

❆♦ ▲✉ís✱ q✉❡ t❛♥t♦ ♠❡ ❛♣♦✐♦✉ ❞✉r❛♥t❡ ❡st❡s ♠❡s❡s ❞❡ tr❛❜❛❧❤♦✳

❆♦s ♠❡✉s Pr♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛✱ q✉❡ ♠❡ s❡r✈✐r❛♠ ❞❡ ✐♥s♣✐r❛çã♦ ❡ q✉❡ t❛♥t♦ ♠❡ ❡♥s✐♥❛r❛♠ ❞❡s❞❡ q✉❡ ❡♥tr❡✐ ♥♦ ■❙❊●✳

❆ t♦❞❛ ❛ ❡q✉✐♣❛ ❞❛ ▲✉s✐t❛♥✐❛✱ ♣❡❧❛ s✉❛ ❝♦♠♣r❡❡♥sã♦ ❡ ✢❡①✐❜✐❧✐❞❛❞❡✳

➚ ❋✉♥❞❛çã♦ ❊❝♦♥ó♠✐❝❛s ✲ ❋✉♥❞❛çã♦ ♣❛r❛ ♦ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛s ❈✐ê♥❝✐❛s ❊❝♦✲ ♥ó♠✐❝❛s✱ ❋✐♥❛♥❝❡✐r❛s ❡ ❊♠♣r❡s❛r✐❛✐s ♣❡❧♦ ❛♣♦✐♦ q✉❡ ♠❡ ❞❡r❛♠ ♥♦ ✷♦_{❛♥♦ ❞❡ ▼❡str❛❞♦}

❡ ❡♠ ❡s♣❡❝✐❛❧ à ❆♥❛ ❘✐t❛ ❙❛♥t♦s ♣❡❧❛ s✉❛ ❝♦♠♣r❡❡♥sã♦✳

❆ t♦❞♦s ♠✉✐t♦ ♦❜r✐❣❛❞❛✱

❙❛r❛ ▲♦♣❡s

❈♦♥t❡ú❞♦

✶ ■♥tr♦❞✉çã♦ ✸

✷ ❚❡♦r❡♠❛ ❞❡ ❚❛❦❡♥s ✹

✷✳✶ ❖ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❞❡ ♠❡❞✐çã♦ é ❛❜❡rt♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷ ❖ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❞❡ ♠❡❞✐çã♦ é ❞❡♥s♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✷✳✶ ▼❡r❣✉❧❤♦ ❞❡ P♦♥t♦s P❡r✐ó❞✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✷✳✷ ■♠❡rsã♦ ❞❡ M ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷✳✸ ▼❡r❣✉❧❤♦ ❞❡ ❙❡❣♠❡♥t♦s ❞❡ Ór❜✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✹ ▼❡r❣✉❧❤♦ ❞❡ M ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸ ❈♦♥❝❧✉sã♦ ❞❛ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❚❛❦❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✹ ❊①t❡♥sã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❚❛❦❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✸ ❊①♣♦❡♥t❡s ❞❡ ▲②❛♣✉♥♦✈ ✸✹

✹ ■♠♣❧❡♠❡♥t❛çã♦ ✸✺

✹✳✶ ▼ét♦❞♦ ❞♦s ❋❛❧s♦s ❱✐③✐♥❤♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✷ ❉❡❝❛✐♠❡♥t♦ ❞❛ ❋✉♥çã♦ ❞❡ ❆✉t♦❝♦rr❡❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✺ ❈♦♥❝❧✉sã♦ ✸✼

✻ ❆♥❡①♦s ✸✽

✻✳✶ ❉❡✜♥✐çõ❡s ❞❡ ❚♦♣♦❧♦❣✐❛ ❉✐❢❡r❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✻✳✷ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✻✳✸ ❘❡s✉❧t❛❞♦s ◆❡❝❡ssár✐♦s à ❉❡♠♦♥str❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✻✳✹ ▼✉❞❛♥ç❛ ❞❡ ❇❛s❡ ❡ ▼❛tr✐③❡s ❙❡♠❡❧❤❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✻✳✺ ❉❡t❡r♠✐♥❛♥t❡ ❞❛ ▼❛tr✐③ ❞❡ ❱❛♥❞❡r♠♦♥❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

❘❡❢❡rê♥❝✐❛s ✺✸

✶ ■♥tr♦❞✉çã♦

◗✉❛♥❞♦ s❡ ♣r❡t❡♥❞❡ ❡st✉❞❛r ❡♠ ❞❡t❛❧❤❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠ s✐st❡♠❛ ❡♠ ♠✉✐t❛s s✐t✉❛çõ❡s ❛♣❡♥❛s s❡ t❡♠ ❛❝❡ss♦ à ❡✈♦❧✉çã♦ ♥♦ t❡♠♣♦ ❞❡ ❝❡rt❛s ♠❡❞✐çõ❡s s♦❜r❡ ❛s ✈❛r✐á✈❡✐s q✉❡ ❝♦♠♣õ❡♠ ❡ss❡ s✐st❡♠❛✳ ❙❡ ❝♦♥s✐❞❡r❛r♠♦s q✉❡ ❛s ♦❜s❡r✈❛çõ❡s ❞❡st❛s ♠❡❞✐çõ❡s ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞❛s ♣♦r ✉♠❛ sér✐❡ t❡♠♣♦r❛❧ ❡ q✉❡ ❡st❛ r❡♣r❡s❡♥t❛ ❛ tr❛❥❡❝tór✐❛ ❞❡ ✉♠ ❞❛❞♦ s✐st❡♠❛ ❞✐♥â♠✐❝♦✱ t♦r♥❛✲s❡ ❡♥tã♦ ✐♠♣♦rt❛♥t❡ ❝♦♠♣r❡❡♥❞❡r q✉❡ ♣r♦♣r✐❡❞❛❞❡s ❞♦ s✐st❡♠❛ ❣❧♦❜❛❧ ❝♦♥s❡❣✉✐♠♦s ♦❜t❡r ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞❛ sér✐❡ t❡♠♣♦r❛❧✳

❯♠❛ ❞❛s té❝♥✐❝❛s ❞❡ ❛♥á❧✐s❡ ❞❡ sér✐❡s t❡♠♣♦r❛✐s q✉❡ t❡♠ ❝♦♠♦ ♦❜❥❡❝t✐✈♦ ♦❜t❡r ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ✈❛r✐á✈❡✐s ♥ã♦ ♦❜s❡r✈á✈❡✐s é ❛ ❝❤❛♠❛❞❛ ❘❡❝♦♥str✉çã♦ ❞♦ ❊s♣❛ç♦ ❞❡ ❋❛s❡s q✉❡ t❡♠ ♣♦r ❜❛s❡ ♣r✐♥❝í♣✐♦s ♣r♦✈❛❞♦s ♣♦r ❚❛❦❡♥s ✲ ❚❡♦r❡♠❛ ❞♦ ▼❡r❣✉❧❤♦ ❞❡ ❚❛❦❡♥s✲✶✾✽✶✳ ❊st❡ t❡♦r❡♠❛ ♣❡r♠✐t❡ r❡❝♦♥str✉✐r ✉♠ ❡s♣❛ç♦ ❞❡ ❢❛s❡s m✲❞✐♠❡♥s✐♦♥❛❧ s✐♠✐❧❛r ❛♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡s ♦r✐❣✐♥❛❧ ❛ ♣❛rt✐r ❞❡ ♠❡❞✐çõ❡s ❞❡ ✉♠❛ ú♥✐❝❛ ✈❛r✐á✈❡❧ ❡ ❡st❡ ❡s♣❛ç♦ r❡❝♦♥str✉í❞♦ ❛♣r❡s❡♥t❛ ✉♠❛ s✉❛✈❡ ✈❛r✐❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡♠ r❡❧❛çã♦ ❛♦ ❡s♣❛ç♦ ♦r✐❣✐♥❛❧ ✭♣r❡s❡r✈❛♥❞♦ ♦s ✐♥✈❛r✐❛♥t❡s ❣❡♦♠étr✐❝♦s ❞♦ s✐st❡♠❛✱ t❛✐s ❝♦♠♦ ♦s ❡①♣♦❡♥t❡s ❞❡ ▲②❛♣✉♥♦✈✮✳

◆❡st❡ tr❛❜❛❧❤♦ ❛♥❛❧✐s❛✲s❡ ❡♠ ❞❡t❛❧❤❡ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ ▼❡r❣✉❧❤♦ ❞❡ ❚❛❦❡♥s ✭❚❛❦❡♥s ❊♠❜❡❞❞✐♥❣ ❚❤❡♦r❡♠✮ ❡ ♣❛r❛ ✐ss♦ é ♥❡❝❡ssár✐♦ t❡r ♣r❡s❡♥t❡s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❞❡ t♦♣♦❧♦❣✐❛ ❞✐❢❡r❡♥❝✐❛❧ ❡ ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s ✐♥❡r❡♥t❡s à ❝♦♠♣r❡❡♥sã♦ ❞♦s ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s✳ ❊ss❛s ❞❡✜♥✐çõ❡s✱ ❜❡♠ ❝♦♠♦ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❛ q✉❡ s❡ r❡❝♦rr❡ ❛♦ ❧♦♥❣♦ ❞❛ ❞❡♠♦♥str❛çã♦ ❡♥❝♦♥tr❛♠✲s❡ ♥♦s ❆♥❡①♦s✳

◆❛ s❡❝çã♦ ✷ ❞❡♠♦♥str❛✲s❡ ♦ ❚❡♦r❡♠❛ ❞♦ ▼❡r❣✉❧❤♦ ❞❡ ❚❛❦❡♥s s❡❣✉✐♥❞♦ ❬✷❪ ❡ ❡①t❡♥❞❡✲s❡ ♦ r❡s✉❧t❛❞♦ ♣r♦✈❛❞♦ ♥✉♠ ❝♦♥t❡①t♦ ♠❛✐s ❣❡r❛❧✳ ◆❛ s❡❝çã♦ ✸ ❡st✉❞❛✲s❡ ✉♠❛ ✐♠♣❧✐❝❛çã♦ ❞♦ t❡♦r❡♠❛ r❡❧❛t✐✈❛♠❡♥t❡ ❛♦s ❡①♣♦❡♥t❡s ❞❡ ▲②❛♣✉♥♦✈ ❞♦ s✐st❡♠❛ r❡❝♦♥str✉í❞♦ ❛tr❛✈és ❞❛s ♠❡❞✐çõ❡s ❞♦ s✐st❡♠❛ ♦r✐❣✐♥❛❧✳ ❋✐♥❛❧♠❡♥t❡✱ ♥❛ s❡❝çã♦ ✹✱ ❛♣r❡s❡♥t❛♠✲s❡ ❛❧❣✉♠❛s ♠❡t♦❞♦❧♦❣✐❛s ✉t✐❧✐③❛❞❛s ♥❛ ♣rát✐❝❛ ♣❛r❛ ♦❜t❡r ✐♥❢♦r♠❛çã♦ út✐❧ s♦❜r❡ ♦ s✐st❡♠❛ ♦r✐❣✐♥❛❧ ❛tr❛✈és ❞❛ r❡❝♦♥str✉çã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡s✳

✷ ❚❡♦r❡♠❛ ❞❡ ❚❛❦❡♥s

❉❡♥♦t❡✲s❡ ♣♦r ❉✐✛k

(M) ♦ ❝♦♥❥✉♥t♦ ❞♦s ❞✐❢❡♦♠♦r✜s♠♦s ❡♠ M q✉❡ sã♦ ❦ ✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡ ♣♦r Ck_{(M,R)♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s C}k _{❞❡M ❡♠ R✳ ❆s r❡st❛♥t❡s}

❞❡✜♥✐çõ❡s ✉t✐❧✐③❛❞❛s ♥♦ ❡♥✉♥❝✐❛❞♦ ❡ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❚❛❦❡♥s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ♥♦s ❆♥❡①♦s✳

❚❡♦r❡♠❛ ✶✳ ✭❚❡♦r❡♠❛ ❞❡ ❚❛❦❡♥s✮ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐❛❧ ❝♦♠♣❛❝t❛ ❞❡ ❞✐♠❡♥sã♦ m✳ P❛r❛ ✉♠ ♣❛r C1_{✲❣❡♥ér✐❝♦ (φ, y) t❛❧ q✉❡ φ ∈❉✐✛}1_{(M)✱ y}

∈ C1_(M,R)✱ ❛ ❛♣❧✐❝❛çã♦

Φ(φ,y):M →R2m+1

Φ(φ,y)(x) = (y(x), y(φ(x)), y(φ2(x)), ..., y(φ2m(x)))

é ✉♠ ♠❡r❣✉❧❤♦✳

❖❜s❡r✈❛çã♦✿ ❯♠❛ ✈❡③ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❉✐✛2

(M)×C2_{(M,R)é ❞❡♥s♦ ❡♠ ❉✐✛}1

(M)×

C1_{(M,R) ✱ ✈❡r ❬✶❪✱ ♥❛ ❞❡♠♦♥str❛çã♦ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ♦ ♣❛r (φ, y) ♣❡rt❡♥❝❡ ❛} ❉✐✛2_(M)

×C2_{(M,R) ♣❛r❛ ♣r♦✈❛r ♦ r❡s✉❧t❛❞♦ ❡♥✉♥❝✐❛❞♦ ♥♦ ❚❡♦r❡♠❛ ✶✳}

❉❡ ✉♠❛ ❢♦r♠❛ s✐♠♣❧✐✜❝❛❞❛ ♦ q✉❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❚❛❦❡♥s ♥♦s ❞✐③ é q✉❡ s❡ t✐✈❡r♠♦s ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ q✉❡ ❞❡♣❡♥❞❡ ❞❡m ✈❛r✐á✈❡✐s✱ ❡ s❡ ❡s❝♦❧❤❡r♠♦s ✉♠❛ ❢✉♥çã♦ ❞❡ ♠❡❞✐çã♦ ♣❛r❛ ❡ss❡ s✐st❡♠❛ ❡ ❝♦♠ ❡ss❛s ♠❡❞✐çõ❡s ❝♦♥str✉✐r♠♦s ♦s ✈❡❝t♦r❡s✿

(y(x), y(φ(x)), y(φ2_{(x)), ..., y(φ}2m_(x))

(y(φ(x)), y(φ2_{(x)), ..., y(φ}2m+1_(x)) ✳✳✳

♣♦❞❡♠♦s r❡♣r♦❞✉③✐r ❛ ❞✐♥â♠✐❝❛ ❞♦ s✐st❡♠❛ ♦r✐❣✐♥❛❧✳

❊st❡ r❡s✉❧t❛❞♦ é s✉r♣r❡❡♥❞❡♥t❡ ✉♠❛ ✈❡③ q✉❡ ♣♦❞❡♠♦s s✐♠♣❧❡s♠❡♥t❡ ♦❜s❡r✈❛r ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦ ✉♠❛ ❞❛s ✈❛r✐á✈❡✐s ❞♦ s✐st❡♠❛ ❡ ❡s❝♦❧❤❡♥❞♦ ❛❞❡q✉❛❞❛♠❡♥t❡ ❛ ❞✐♠❡♥sã♦ ❞♦s ✈❡❝t♦r❡s ❞❡ r❡❝♦♥str✉çã♦✱ ❝♦♥s❡❣✉✐♠♦s ❝♦♠♣r❡❡♥❞❡r ❛ ❡✈♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❞❡ m ✈❛r✐á✈❡✐s✱ ❝♦♠♦ ✐❧✉str❛ ❛ ✜❣✉r❛ s❡❣✉✐♥t❡✿

❋✐❣✉r❛ ✶✿ ❚❡♦r❡♠❛ ❞❡ ❚❛❦❡♥s

❖ ❚❡♦r❡♠❛ ❛✜r♠❛ ❛✐♥❞❛ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣❛r❡s(φ, y)q✉❡ t♦r♥❛♠ ❛ ❛♣❧✐❝❛çã♦

❞❡ ❞❡❧❛② Φ(φ,y) ✉♠ ♠❡r❣✉❧❤♦ é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❣❡♥ér✐❝♦ ❞❡ ❉✐✛1(M)×C1(M,R)✳ ◆❛ ♣rát✐❝❛ ✐st♦ s✐❣♥✐✜❝❛ q✉❡✱ ❡♠❜♦r❛Φ(φ,y) ♥ã♦ s❡❥❛ ✉♠ ♠❡r❣✉❧❤♦ ❞❡ M ♣❛r❛ t♦❞♦s ♦s ♣❛r❡s(φ, y)_∈❉✐✛1(M)_×C1_{(M,R)✱ ❞❛❞♦ ✉♠ ♣❛r(φ, y)♣❛r❛ ♦ q✉❛❧ ❛ ♣r♦♣r✐❡❞❛❞❡} ❢❛❧❤❡ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ✉♠ ♣❛r(φ′_{, y}′_{) ❛r❜✐tr❛r✐❛♠❡♥t❡ ♣ró①✐♠♦ t❛❧ q✉❡ Φ}

(φ′_,y′₎ s❡❥❛ ✉♠ ♠❡r❣✉❧❤♦ ❞❡M ❡♠R2m+1✳ ■st♦ é✱ ❜❛st❛ ✉♠❛ ♣❡q✉❡♥❛ ♣❡rt✉r❜❛çã♦ ❛♦ ♣❛r ✐♥✐❝✐❛❧

♣❛r❛ q✉❡ ❛ ♣r♦♣r✐❡❞❛❞❡ s❡ ✈❡r✐✜q✉❡✳

◆♦t❡✲s❡ ❛✐♥❞❛ q✉❡ ♦ t❡♦r❡♠❛ ❣❛r❛♥t❡ q✉❡ t❡♠♦s ✉♠ ♠❡r❣✉❧❤♦ s❡ ❛ ❞✐♠❡♥sã♦ ❞♦s ✈❡❝t♦r❡s ❞❡ r❡❝♦♥str✉çã♦ ❢♦r ✐❣✉❛❧ ❛2m+ 1✱ ♠❛s ♣♦❞❡♠♦s t❡r ✉♠❛ ❜♦❛ r❡♣r♦❞✉çã♦

❞♦ s✐st❡♠❛ ♦r✐❣✐♥❛❧ s❡ ❡ss❛ ❞✐♠❡♥sã♦ ❢♦r ✐♥❢❡r✐♦r✳ P♦r ❡①❡♠♣❧♦✱ ♥♦ ❝❛s♦ ❞♦ ❛tr❛❝t♦r ❞❡ ▲♦r❡♥③✱ ❝♦♥s❡❣✉✐♠♦s r❡❝♦♥str✉✐r ❛ ❞✐♥â♠✐❝❛ ♦r✐❣✐♥❛❞❛ ♣♦r ✸ ✈❛r✐á✈❡✐s ❛tr❛✈és ❞❡ ✈❡❝t♦r❡s ❞❡ r❡❝♦♥str✉çã♦ ❞❡R3 ❝♦♠ ♦❜s❡r✈❛çõ❡s ❞❡ ❛♣❡♥❛s ✉♠❛ ❞❛s ✈❛r✐á✈❡✐s ❬✾❪✳

❆s ✜❣✉r❛s s❡❣✉✐♥t❡s ❛♣r❡s❡♥t❛♠ ✉♠❛ tr❛❥❡❝tór✐❛ ❞♦ s✐st❡♠❛ ✭à ❡sq✉❡r❞❛✮ ❡ ❛ r❡❝♦♥str✉çã♦ ❞❛ tr❛❥❡❝tór✐❛ ✉t✐❧✐③❛♥❞♦ ❛♣❡♥❛s ❛s ♦❜s❡r✈❛çõ❡s ❞❛ s❡❣✉♥❞❛ ❝♦♦r❞❡♥❛❞❛ ✭à ❞✐r❡✐t❛✮✳

❋✐❣✉r❛ ✷✿ ❆tr❛❝t♦r ❞❡ ▲♦r❡♥③ ❡ ❘❡❝♦♥str✉çã♦✭❝♦♠σ= 10ρ= 28β= 8/3τ= 8δt= 0.01✮

❚❡♦r❡♠❛ ✷✳ ✭❚❡♦r❡♠❛ ❞❡ ❚❛❦❡♥s ❱✷✮ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐❛❧ ❝♦♠♣❛❝t❛ ❞❡ ❞✐♠❡♥sã♦ m ❡ φ :M _→M ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ C2 _{❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿} ✶✳ ❊①✐st❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s ❞❡φ ❝♦♠ ♣❡rí♦❞♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧

❛ 2m✱

✷✳ ❙❡ x é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❝♦♠ ♣❡rí♦❞♦ k _≤ 2m ❡♥tã♦ ♦s ✈❛❧♦r❡s ♣ró♣r✐♦s ❞❛ ❞❡r✐✈❛❞❛ ❞❡ φk _{sã♦ t♦❞♦s ❞✐st✐♥t♦s✳}

❊♥tã♦ ♣❛r❛ y_∈C2_{(M,R) ❣❡♥ér✐❝♦✱ ❛ ❛♣❧✐❝❛çã♦}

Φ(φ,y)(x) = (y(x), y(φ(x)), y(φ2(x)), ..., y(φ2m(x))) é ✉♠ ♠❡r❣✉❧❤♦✳

■r❡♠♦s ♣r✐♠❡✐r♦ ❞❡♠♦♥str❛r ❛ ✈❡rsã♦ ✷ ❞♦ t❡♦r❡♠❛ ❡ ❛♣ós ❡st❡ ❡st❛r ❡st❛❜❡❧❡❝✐❞♦ ❣❡♥❡r❛❧✐③❛♠♦s ♦s r❡s✉❧t❛❞♦s ♣♦r ❢♦r♠❛ ❛ ♦❜t❡r ❛ ✈❡rsã♦ ✶✳

✷✳✶ ❖ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❞❡ ♠❡❞✐çã♦ é ❛❜❡rt♦

Pr❡t❡♥❞❡♠♦s ♠♦str❛r q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❞❡ ♠❡❞✐çã♦ y q✉❡ t♦r♥❛♠ ❛ ❛♣❧✐❝❛çã♦ Φ(φ,y) ✉♠ ♠❡r❣✉❧❤♦ ❞❡ M✱ é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✳✶ P❛r❛ ✐ss♦ ❝♦♠❡ç❛♠♦s ♣♦r ❞❡♠♦♥str❛r q✉❡ ❛ ❛♣❧✐❝❛çã♦F✱ ❞❡✜♥✐❞❛ ♣♦r✿

F :C2(M,R_)→C2_(M,R2m+1₎

F(y) = Φ(φ,y)(x) = (y(x), y(φ(x)), y(φ2(x)), ..., y(φ2m(x))) é ❝♦♥tí♥✉❛✳

✶_{♥❛ t♦♣♦❧♦❣✐❛C}1

Pr♦♣♦s✐çã♦ ✸✳ ❆ ❛♣❧✐❝❛çã♦F(y) = Φ(φ,y)= (y, y◦φ, y◦φ2, ..., y◦φ2m)é ❝♦♥tí♥✉❛✳

❉❡♠♦♥str❛çã♦✳ ❉✐✈✐❞✐♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❡♠ três ❧❡♠❛s✿ ♣r✐♠❡✐r♦ ♠♦str❛♠♦s q✉❡ ❛ ❛♣❧✐❝❛❝ã♦y7→y◦φ é ❝♦♥tí♥✉❛✱ ❞❡ s❡❣✉✐❞❛ ♠♦str❛✲s❡ ♣♦r ✐♥❞✉çã♦ q✉❡ ❛ ❛♣❧✐❝❛çã♦ y7→y◦φn _{é ❝♦♥tí♥✉❛ ❡ ✜♥❛❧♠❡♥t❡ ❝♦♥❝❧✉í♠♦s q✉❡ F é ❝♦♥tí♥✉❛✳}

▲❡♠❛ ✶✿ ❆ ❛♣❧✐❝❛çã♦F1 :C2(M,R)→C2(M,R)❞❡✜♥✐❞❛ ♣♦ry7→y◦φ é ❝♦♥tí♥✉❛✳ ❙❡❥❛{(Ui, hi), i∈Λ} ✉♠ ❜♦♠ ❛t❧❛s ✜♥✐t♦ ❞❡M ❡ s❡❥❛ Wi =h−i 1B(1)✳ Pr❡t❡♥❞❡✲

♠♦s ♠♦str❛r q✉❡ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ N(ǫ) =T_iN1_(y,(U

i, hi),(R, id), Wi, ǫ) ❞❡

y t❛❧ q✉❡ s❡y_{b∈ N}(ǫ)❡♥tã♦ F1(_by)_{∈ N} =T_iN1_(y◦φ,(U

i, hi),(R, id), Wi, ǫ′)✳

❖s ❝♦♥❥✉♥t♦s{Wi, i∈Λ}❢♦r♠❛♠ ✉♠❛ ❝♦❜❡rt✉r❛ ❞❡ M ❡ ❝♦♠♦φ é ✉♠ ❞✐❢❡♦♠♦r✲

✜s♠♦✱ t❛♠❜é♠ ♦s ❝♦♥❥✉♥t♦s{φ−1_W

i, i∈Λ}❡{φ−1Wi∩Wj, i, j ∈Λ}sã♦ ❝♦❜❡rt✉r❛s

❞❡M✳

❈♦♠♦ ❛s ❢✉♥çõ❡s Dhiφh−j1 :φ−1Wi∩Wj → Rm×m sã♦ ❝♦♥tí♥✉❛s✱ tê♠ ❞♦♠í♥✐♦

❝♦♠♣❛❝t♦ ❡ sã♦ ❡♠ ♥ú♠❡r♦ ✜♥✐t♦✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ A t❛❧ q✉❡ _kDhiφh−j1k < A

♣❛r❛ t♦❞♦ ♦ i, j _∈Λ

❙❡❥❛ x _∈ Wj✱ ❧♦❣♦ x ∈ φ−1Wi ∩Wj ♣❛r❛ ❛❧❣✉♠ i ∈ Λ ❡ s❡❥❛ x′ = φ(x) ∈ Wi✱ ❡

❡s❝♦❧❤❛✲s❡ ǫ <min{ǫ′_, ǫ′

A}✳ ❊♥tã♦✿

|yb_◦φh−_j1(hjx)−y◦φh−j1(hjx)|=|by(x′)−y(x′)|=

=_|y_b◦φh−_i 1(hix′)−y◦φh−i1(hix′)|< ǫ < ǫ′

❡

kD_by◦φh−j1(hjx)−Dy◦φh−j1(hjx)k=kDyhb i−1hiφh−j1(hjx)−Dyhi−1hiφh−j1(hjx)k=

=_kDyh_b −1

i (hix′)Dhiφhj−1(hjx)−Dyhi−1(hix′)Dhiφh−j1(hjx)k

<_kDyh_b −_i 1(hix′)−Dyh−i 1(hix′)kkDhiφhj−1(hjx)k< ǫA < ǫ′

▲♦❣♦ yb◦φ(x)∈ N ❡ ♣♦rt❛♥t♦ F1 é ❝♦♥tí♥✉❛✳

▲❡♠❛ ✷✿ ❆ ❛♣❧✐❝❛çã♦ Fn : C2(M,R) → C2(M,R) ❞❡✜♥✐❞❛ ♣♦r (y, φ) 7→ y◦φn : é

❝♦♥tí♥✉❛✳

❙❛❜❡♠♦s q✉❡ é ✈❡r❞❛❞❡ ♣❛r❛ n = 1 ❡ ♣r❡t❡♥❞❡♠♦s ♠♦str❛r q✉❡ s❡ y _◦φn−1 _é ❝♦♥tí♥✉❛ ❡♥tã♦ y_◦φn _{é ❝♦♥tí♥✉❛✳ ❚❡♠♦s y}

◦φn _{= (y}

◦φn−1_{)◦φ q✉❡ é✱ ♣♦r ❤✐♣ót❡s❡✱}

✉♠❛ ❝♦♠♣♦s✐çã♦ ❞❡ ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s ❡ ♣♦r ✐ss♦ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛✳ ▲❡♠❛ ✸✿ ❆ ❛♣❧✐❝❛çã♦ F(y) = Φ(φ,y) = (y, y◦φ, y◦φ2, ..., y◦φ2m) é ❝♦♥tí♥✉❛✳

❊s❝r❡✈❡♥❞♦ F =T ◦F ♦♥❞❡F = (F0, F1, ..., F2m)❡

T : [C2_(M,R)]2m+1 _{→ C}2_(M,R2m+1_{) é ❛ ❛♣❧✐❝❛çã♦ q✉❡ tr❛♥s❢♦r♠❛ ♦ ❝♦♥❥✉♥t♦ ❞❡}

2m+ 1❢✉♥çõ❡s r❡❛✐s ♥♦ ✈❡❝t♦r ❞❡ ❞✐♠❡♥sã♦2m+ 1❝✉❥❛s ❝♦♠♣♦♥❡♥t❡s sã♦ ❛s ❢✉♥çõ❡s

r❡❛✐s✳

P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r s❛❜❡♠♦s q✉❡ F é ❝♦♥tí♥✉❛❀ ❧♦❣♦✱ ❛♣❡♥❛s t❡♠♦s ❞❡ ♠♦str❛r q✉❡ T é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛✱ ♦ q✉❡ ❢❛r❡♠♦s ❞❡♠♦♥str❛♥❞♦ q✉❡ ♣❛r❛ ❝❛❞❛ ❝♦♥✲ ❥✉♥t♦ ❞❡ ❢✉♥çõ❡s (f0, f1, ..., f2m) ❡♠ [C2(M,R)]

2m+1_{✱ ♣❛r❛ t♦❞❛ ❛ ✈✐③✐♥❤❛♥ç❛}

N

❞❡ T(f0, f1, ..., f2m) = f ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ N(ǫ) ❞❡ (f0, f1, ..., f2m) t❛❧ q✉❡

TN(ǫ)⊂ N✳

❙❡❥❛ {(Ui, hi), i∈ Λ} ✉♠ ❜♦♠ ❛t❧❛s ✜♥✐t♦ ❞❡ M✱Wi =h−i 1B(1)✳ ❉❛❞❛ q✉❛❧q✉❡r

✈✐③✐♥❤❛♥ç❛ N ❡♠ C2_(M,R2m+1_{) ❞❡f ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❛ ❢♦r♠❛} \

i

N1(f,(Ui, hi),(R2m+1, id), Wi, ǫ′) ❝♦♥t✐❞❛ ❡♠ N✳

❚♦♠❡✲s❡ ǫ < ǫ′

2m+1 ❡ ❞❡✜♥❛✲s❡ N(ǫ) = 2m

O

j=1 \

i

N1(fj,(Ui, hi),(R, id), Wi, ǫ) ❡ s❡❥❛

(f0,b f1, ...,b f2bm)∈ N(ǫ)✱ fb=T(f0,b f1, ...,b f2bm) ❡ x∈Wi✳

❊♥tã♦✿

|f hb −_i 1(hix)−f h−i 1(hix))| ≤

2m

X

j=0

|fbjh−i 1(hi(x))−fjh−i 1(hi(x))|<

2m

X

j=0

ǫ= (2m+1)ǫ < ǫ′ ❡ ❛✐♥❞❛

kDf hb −_i 1(hi(x))−Df h−i 1(hi(x))k ≤

2m

X

j=0

kDfbjh−i 1(hi(x))−Dfjh−i 1(hi(x))k<

2m

X

j=0 ǫ < ǫ′

▲♦❣♦ fb∈ N ❡T é ❝♦♥tí♥✉❛✳

❈♦♥s✐❞❡r❡✲s❡ ❛❣♦r❛ ♦ ❝♦♥❥✉♥t♦ S ❞❛s ❛♣❧✐❝❛çõ❡s C2_{✱ f : M → R}2m+1 _{q✉❡ sã♦} ♠❡r❣✉❧❤♦s ❞❡ M✳ ❊st❡ ❝♦♥❥✉♥t♦ é ❛❜❡rt♦ ❡♠ C2_(M,R2m+1_{) ❬✸❪✱ ❧♦❣♦ ❝♦♠♦ F é} ❝♦♥tí♥✉❛ ♦❜t❡♠♦s ♦ r❡s✉❧t❛❞♦ ♣r❡t❡♥❞✐❞♦✿ F−1_{(S) = {y ∈ C}2_{(M,R) : F(y) ∈S} é} ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✳

✷✳✷ ❖ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❞❡ ♠❡❞✐çã♦ é ❞❡♥s♦

P❛r❛ ♠♦str❛r q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❢✉♥çõ❡s ❞❡ ♠❡❞✐çã♦ y q✉❡ t♦r♥❛♠ Φ(φ,y) ✉♠ ♠❡r❣✉❧❤♦ é ❞❡♥s♦ t❡♠♦s ❞❡ ♠♦str❛r q✉❡ ♣❛r❛ ❝❛❞❛y∈C2_{(M,R)q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛} ❞❡y ❝♦♥té♠ ✉♠❛ ❢✉♥çã♦ y′ ♣❛r❛ ❛ q✉❛❧ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ❞❡❧❛② é ✉♠ ♠❡r❣✉❧❤♦✳ P❛r❛ ✐ss♦ é s✉✜❝✐❡♥t❡ ❡♥❝♦♥tr❛r ✉♠ ❜♦♠ ❛t❧❛s ❞❡ M ❡ ♠♦str❛r q✉❡ ♣❛r❛ q✉❛❧q✉❡r ǫ s❡ t❡♠ y′ _∈ T

iN1(y,(Ui, hi),(R, id), Wi, ǫ) ❝♦♠ y′ ∈ C2(M,R) ❡ t❛❧ q✉❡ Φ(φ,y′₎ é ✉♠ ♠❡r❣✉❧❤♦ ❞❡M ✳

■r❡♠♦s ❝♦♥str✉✐r ❡①♣❧✐❝✐t❛♠❡♥t❡ ✉♠ y′ _{❛❞❡q✉❛❞♦ s♦♠❛♥❞♦ ❢✉♥çõ❡s ❛ y✿}

y′ =y+

N

X

j=1

ajψj ✭✷✳✶✮

♦♥❞❡ N é ✜♥✐t♦✱ aj ∈R❡ ψj :M →R ❢✉♥çõ❡sC∞✳ ❆❥✉st❛r❡♠♦s y′ ✈ár✐❛s ✈❡③❡s ♣♦r

❢♦r♠❛ ❛ ❞♦t❛r✱ ❛ ❝❛❞❛ ❛❥✉st❛♠❡♥t♦✱y′ _{❝♦♠ ❛❧❣✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡s❡❥❛❞❛✳ ❚❡♠♦s ♥❛✲}

t✉r❛❧♠❡♥t❡ ❞❡ ❣❛r❛♥t✐r q✉❡ ❝❛❞❛ ✉♠ ❞♦s ❛❥✉st❛♠❡♥t♦s é ♣♦ssí✈❡❧ ❡ q✉❡ à ♠❡❞✐❞❛ q✉❡ ❞♦t❛♠♦sy′ _{❞❡ ♥♦✈❛s ♣r♦♣r✐❡❞❛❞❡s ❝♦♥s❡❣✉✐♠♦s ♣r❡s❡r✈❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❛♥t❡r✐♦r❡s✳}

▲❡♠❛ ✹✳ ❙❡❥❛y_∈C2_{(M,R)❡ ψ}

j :M →R, j = 1, ..., N ❢✉♥çõ❡s C∞✱ ❝♦♠ N ✜♥✐t♦✳

❙❡❥❛ a = (a1, ...aN)T ∈ RN✳ P❛r❛ ❝❛❞❛ ✈✐③✐♥❤❛♥ç❛ N ❞❡ y ❡①✐st❡ ❛❧❣✉♠ δ > 0 t❛❧

q✉❡ s❡ kak< δ ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r

y′ =y+

N

X

j=1 ajψj

♣❡rt❡♥❝❡ ❛ N✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛{(Ui, hi), i∈Λ} ✉♠ ❜♦♠ ❛t❧❛s ✜♥✐t♦ ❞❡ M ❡ s❡❥❛

T

iN1(y,(Ui, hi),(R, id), Wi, ǫ) ⊂ N✳ P❛r❛ ❝❛❞❛ j✱ 1 ≤ j ≤ N ❡ i ∈ Λ ❛ ❢✉♥çã♦

ψj :Wi → R é ❝♦♥tí♥✉❛ ❝♦♠ ❞♦♠í♥✐♦ ❝♦♠♣❛❝t♦ ❡ ♣♦r ✐ss♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ B

t❛❧ q✉❡ |ψj|< B✳

❚❡♠♦s ❛ss✐♠ ✿

|y′h−_i 1(hix)−yh−i 1(hix)|=| N

X

j=1

ajψjh−i 1(hix)| ≤ N

X

j=1

|aj||ψj(x)| ≤B N

X

j=1

|aj|

❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❝♦♠♦Dψjh−i 1 sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ♣♦❞❡♠♦s ✉t✐❧✐③❛r ♦s ♠❡s✲

♠♦s ❛r❣✉♠❡♥t♦s ❡ ♦❜t❡r✿

kDy′h−i 1(hix)−Dyh−i 1(hix)k ≤B′ N

X

j=1

|aj|

❊ ❛ss✐♠ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ s❡ka_k< δ ❡♥tã♦ y′ _{∈ N✳}

Pr❡t❡♥❞❡♠♦s ❛❣♦r❛ ❝♦♥str✉✐r ❢✉♥çõ❡s ❞❡ ♠❡❞✐çã♦y′ _{❞❛ ❢♦r♠❛ ✭✷✳✶✮ q✉❡ ♦r✐❣✐♥❛♠}

❛♣❧✐❝❛çõ❡s ❞❡ ❞❡❧❛② q✉❡ sã♦ ♠❡r❣✉❧❤♦s ❞❡M ❢❛③❡♥❞♦ s✉❝❡ss✐✈❛s ❛❧t❡r❛çõ❡s ❛ y′ _q✉❡

❛ ❞♦t❛♠ ❞❡ ♣r♦♣r✐❡❞❛❞❡s q✉❡ s❡ ✈❡r✐✜❝❛♠ ❡♠ ♣❛rt❡s ❞❡M s✉❝❡ss✐✈❛♠❡♥t❡ ♠❛✐♦r❡s✳ ✷✳✷✳✶ ▼❡r❣✉❧❤♦ ❞❡ P♦♥t♦s P❡r✐ó❞✐❝♦s

❙❡❥❛ P2m ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s ❞❡ φ ❝♦♠ ♣❡rí♦❞♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛

2m✳ ◗✉❛♥❞♦ t❡♥t❛♠♦s ♠❡r❣✉❧❤❛r M ❡♠ R2m+1 _{❛tr❛✈és ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ❞❡❧❛②} ♦s ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s ❝♦♠ ♣❡rí♦❞♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ 2m ❝r✐❛♠ ♣r♦❜❧❡♠❛s ♣♦✐s ♣❛r❛ ❡st❡s ♣♦♥t♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❞❛s s✉❛s ✐♠❛❣❡♥s ♣♦rΦ(φ,y) s❡rã♦ ✐❣✉❛✐s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ ♣♦♥t♦s ✜①♦s ❞❡ φ t♦❞❛s ❛s ❝♦♦r❞❡♥❛❞❛s sã♦ ✐❣✉❛✐s ❡ ♣♦rt❛♥t♦ Φ ♥ã♦ s❡rá ✉♠

♠❡r❣✉❧❤♦✳ P♦r ❡①❡♠♣❧♦✱ s❡ φ ❢♦r ❛ ✐❞❡♥t✐❞❛❞❡✱ ✐st♦ é✱ φ(x) = x ❡♥tã♦ Φ(φ,y)(x) =

(y(x), y(x), ..., y(x))❡ ♣♦rt❛♥t♦Φ(φ,y)♥ã♦ s❡rá ✉♠ ♠❡r❣✉❧❤♦ ❞❡M q✉❛❧q✉❡r q✉❡ s❡❥❛ ❛ ❢✉♥çã♦ ❞❡ ♠❡❞✐çã♦ ❡s❝♦❧❤✐❞❛✳

■♠♣õ❡✲s❡ ♥♦ ❚❡♦r❡♠❛ ✷ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s ❞❡ φ ❝♦♠ ♣❡rí♦❞♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛2ms❡❥❛ ✜♥✐t♦✱ ✐✳❡✱ q✉❡ P2m é ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❡ ❛ss✐♠ t❡♠♦s q✉❡

♣❛r❛ q✉❛❧q✉❡r ♣♦♥t♦ xi ∈ P2m ❡①✐st❡ ✉♠❛ s✉❛ ✈✐③✐♥❤❛♥ç❛ q✉❡ ♥ã♦ ❝♦♥té♠ ♥❡♥❤✉♠

♦✉tr♦ ♣♦♥t♦ ❞❡ P2m✳

P♦r ❢♦r♠❛ ❛ ❣❛r❛♥t✐r q✉❡ ♥❡♥❤✉♠ ❞❡st❡s ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s t❡♠ ❛ ♠❡s♠❛ ✐♠❛❣❡♠✱ ✐st♦ é✱ q✉❡Φ s❡❥❛ ✉♠❛ ❢✉♥çã♦ ✐♥❥❡❝t✐✈❛ ❡♠ P2m✱ t❡♠♦s ❞❡ ❛❥✉st❛r y ❞❡ ❢♦r♠❛ ❛ q✉❡

t♦♠❡ ✈❛❧♦r❡s ❞✐❢❡r❡♥t❡s ♣❛r❛ ❞✐❢❡r❡♥t❡s ♣♦♥t♦s ❡ ❢❛③❡♠♦s ❡ss❡ ❛❥✉st❛♠❡♥t♦ ✉t✐❧✐③❛♥❞♦ ❢✉♥çõ❡s ❜✉♠♣ ✿

❙❡❥❛♠x1, x2 _{∈ P}2m ♣❛r❛ ♦s q✉❛✐syt♦♠❛ ♦ ♠❡s♠♦ ✈❛❧♦r ❡ s❡❥❛♠(Ui, hi)❛s ❝❛rt❛s

q✉❡ ❝♦♥tê♠x1 ❝♦♠ h1x1 ♦ ❝❡♥tr♦ ❞❡ B(3) ❡ ❞❡✜♥❛✲s❡ ❛ ❢✉♥çã♦ ψ :M →R ♣♦r

ψ(x) =     

λ(h1x) ♣❛r❛ x ∈ h−11B(3)

0 ❝✳❝

♦♥❞❡λ :Rm_→R é ✉♠❛ ❢✉♥çã♦ ❜✉♠♣ ❝♦♠ s✉♣♦rt❡ ❡♠ _B(3) ❡ ✐❣✉❛❧ ❛ ✶ ❡♠ _B(1)✳

▲♦❣♦ s❡ ❞❡✜♥✐r♠♦s y′ _{=y+aψ ❝♦♠ a ∈ R t❡♠♦s q✉❡ y}′_{(x1) 6= y}′_{(x2) ✭❞✐❢❡r❡♠}

♣♦ra✮✳

❯t✐❧✐③❛♥❞♦ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❛r❣✉♠❡♥t♦s ♣♦❞❡♠♦s tr❛t❛r ♦ ❝❛s♦ ❡♠ q✉❡ ♠❛✐s ❞♦ q✉❡ ❞♦✐sx′

istê♠ ❛ ♠❡s♠❛ ✐♠❛❣❡♠ ✭♥❡st❛ s✐t✉❛çã♦ t❡rí❛♠♦s ❞❡ ❢❛③❡r ♠❛✐s ❞♦ q✉❡ ✉♠❛

❛❧t❡r❛çã♦✮✱ ❡ ❞❡st❛ ❢♦r♠❛ ❝♦♥s❡❣✉✐♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❢✉♥çã♦ ❞❡ ♠❡❞✐çã♦y ✐♥❥❡❝t✐✈❛ ❡♠ P2m ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ♣❛r❛ y ❣❡♥ér✐❝♦✱ Φ(φ,y) é ✐♥❥❡❝t✐✈❛ q✉❛♥❞♦ r❡str✐♥❣✐❞❛ ❛ P2m✳

Pr❡t❡♥❞❡♠♦s ❛❣♦r❛ ♠♦str❛r q✉❡ Φ(φ,y) é ✉♠❛ ✐♠❡rsã♦ ❡♠ P2m ❡ ♣❛r❛ ✐ss♦ t❡♠♦s

❞❡ ❣❛r❛♥t✐r q✉❡DΦ(φ,y)h−i 1(hixi)t❡♠ ❢✉❧❧ r❛♥❦ ❡♠ t♦❞♦ ♦ xi ∈P2m✳

❈♦♥s✐❞❡r❡♠✲s❡ ♣r✐♠❡✐r♦ ♦s ♣♦♥t♦s ✜①♦s ❞❡φ✿

■♠❡rsã♦ ❞♦s P♦♥t♦s ❋✐①♦s✿ ❙❡❥❛x1 ✉♠ ♣♦♥t♦ ✜①♦ ❞❡ φ ❡♥tã♦ ❛ ❦✲és✐♠❛ ❝♦❧✉♥❛ ❞❛ ♠❛tr✐③ DΦ(φ,y)h−11(h1x1) é✿

Dyφk−1_h−1

1 (h1x1) =Dyh−11(h1x)Dh1φk−1h−11(h1x1) ❡ s❡ ❡s❝r❡✈❡r♠♦s v =Dyh−1

1 (h1x)❡ J =Dh1φh−11(h1x1) ♦❜t❡♠♦s✿ Dyφk−1h−₁1(h1x1) = vJk−1

Pr❡t❡♥❞❡♠♦s ❡♥tã♦ ♠♦str❛r q✉❡ ♦ ❝♦♥❥✉♥t♦ {v, vJ, vJ2_{, ..., vJ}2m

}❝♦♥té♠ m ✈❡❝✲ t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ❝♦♠♦ ♣♦r ❤✐♣ót❡s❡ J t❡♠ m ✈❛❧♦r❡s ♣ró♣r✐♦s ❞✐st✐♥t♦s λj ❝♦♠ j = 1, ..., m ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ m ✈❡❝t♦r❡s ♣ró♣r✐♦s ❧✐♥❡❛r♠❡♥t❡

✐♥❞❡♣❡♥❞❡♥t❡sej ❝♦♠j = 1, ..., m✳ ❊①♣r❡ss❛♥❞♦ ♦ ✈❡❝t♦rv ❝♦♠♦ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐✲

♥❡❛r ❞♦s ✈❡❝t♦r❡s ♣ró♣r✐♦s ❞❡J✱ ✐✳❡✱v =P_jαjej ❧♦❣♦ ♣❛r❛ ❝❛❞❛j t❡♠♦s✿ ejJ =λjej

❧♦❣♦ s❡v =P_jαjej ❡♥tã♦ ✿ vJ =PjαjejJ =Pjαjλjej✳

◆❛ ❜❛s❡ B = _{e1, e2, ..., em} ♦s ✈❡❝t♦r❡s {v, vJ, vJ2, ..., vJ2m} ♣♦❞❡♠ s❡r r❡♣r❡✲

s❡♥t❛❞♦s ♣❡❧❛ ♠❛tr✐③ s❡❣✉✐♥t❡✿

        

α1 α2 · · · αm

α1λ1 α2λ2 · · · αmλm

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ α1λ2m

1 α2λ22m · · · αmλ2mm

         ✭✷✳✷✮

❙❡ ❝♦♥s✐❞❡r❛r♠♦s ❛♣❡♥❛s ❛s ♣r✐♠❡✐r❛sm❧✐♥❤❛s ❞❛ ♠❛tr✐③ ❛♥t❡r✐♦r t❡♠♦s q✉❡ ❛ ❝❛r❛❝✲ t❡ríst✐❝❛ ❞❡ss❛ ♠❛tr✐③ s❡rá ✐❣✉❛❧m s❡ ❡ só s❡ ♦ s❡✉ ❞❡t❡r♠✐♥❛♥t❡ ❢♦r ♥ã♦ ♥✉❧♦✳ ❯♠❛ ✈❡③ q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ♠❛tr✐③ ❡♠R✐st♦ é✿ _{det :}Rm_×Rm _→R❡♥tã♦✱ ♦✉ ❛ ❢✉♥çã♦ é ✐❞❡♥t✐❝❛♠❡♥t❡ ✐❣✉❛❧

❛ ③❡r♦ ♦✉ ♦s s❡✉s ③❡r♦s ❢♦r♠❛♠ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦✳

P♦r ❢♦r♠❛ ❛ ♠♦str❛r q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ♥ã♦ é ✐❞❡♥t✐❝❛♠❡♥t❡ ✐❣✉❛❧ ❛ ③❡r♦ ♣♦❞❡✲ ♠♦s ❡s❝♦❧❤❡r(α1, α2, ..., αm) = (1,1, ...,1)❡ ♦❜t❡r ❛ss✐♠ ✉♠❛ ♠❛tr✐③ ❞❡ ❱❛♥❞❡r♠♦♥❞❡

✭q✉❡ t❡♠ ❢✉❧❧ r❛♥❦✮✳ _        

1 1 _{· · ·} 1

λ1 λ2 · · · λm

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ λ2m

1 λ22m · · · λ2mm

        

◆♦t❡✲s❡ q✉❡ ❛ ♠❛tr✐③ q✉❛❞r❛❞❛ q✉❡ s❡ ♦❜tê♠ ❛♦ ❡s❝♦❧❤❡r ❛s ♣r✐♠❡✐r❛sm❧✐♥❤❛s ❞❛ ♠❛tr✐③ ✭✷✳✷✮ ❡ ❛ q✉❡ é ♦❜t✐❞❛ ❡s❝♦❧❤❡♥❞♦ ❛sm ♣r✐♠❡✐r❛s ❧✐♥❤❛s ❞❡ DΦ(φ,y)h−11(h1x1) sã♦ ♠❛tr✐③❡s s❡♠❡❧❤❛♥t❡s✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ tê♠ ❛ ♠❡s♠❛ ❝❛r❛❝t❡ríst✐❝❛✳

❉❡st❛ ❢♦r♠❛ ♠♦str♦✉✲s❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❡❝t♦r❡sv ♣❛r❛ ♦s q✉❛✐s ❛ ❞❡r✐✈❛❞❛ ❞❡

Φ(φ,y) t❡♠ ❢✉❧❧ r❛♥❦ é ❛❜❡rt♦ ❡ ❞❡♥s♦ ❡♠Rm ❡ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r v′ t❛❧ q✉❡ ❛ ❞✐❢❡✲ r❡♥ç❛a=v′_{−v t❡♠ ♥♦r♠❛ ❛r❜✐tr❛r✐❛♠❡♥t❡ ♣❡q✉❡♥❛ ❡ ♦ ❝♦♥❥✉♥t♦{v}′_{, v}′_J2_{, ..., v}′_J2m_}

❢♦r♠❛ ✉♠❛ ❜❛s❡ ❞❡Rn✳

❙✉♣♦♥❤❛✲s❡ ❛❣♦r❛ q✉❡yé ✉♠❛ ❢✉♥çã♦ ❞❡ ♠❡❞✐çã♦ ♣❛r❛ ❛ q✉❛❧Φ(φ,y)♥ã♦ é ✐♠❡rs✐✈❛

❡♠ x1✳ ❙❡ ❞❡✜♥✐r♠♦s ❛s ❢✉♥çõ❡s C∞ _ψ

j :M →R 1≤j ≤m ❝♦♠♦

ψi(x) =

    

µi(x)λ(h1x) ♣❛r❛ x ∈ h−11B1

0 ❝✳❝

♦♥❞❡µi :U1 →R é ❛ i✲és✐♠❛ ❢✉♥çã♦ ❝♦♦r❞❡♥❛❞❛ ❞❡h1 ❡

y′_{(x) = y(x) +}

m

X

i=1

aiψi(x)

❡♥tã♦ t❡♠♦sDy′_h−1

1 (h1x) = Dyh−11(h1x) +a✉♠❛ ✈❡③ q✉❡

∂ψi

∂uk(u) =δi,k ❝♦♠ u=h1x

❡uk é ❛ ❦✲és✐♠❛ ❝♦♦r❞❡♥❛❞❛ ❞❡ u✳

❊s❝♦❧❤❡♥❞♦ a❝♦♠ ♥♦r♠❛ ❛r❜✐tr❛r✐❛♠❡♥t❡ ♣❡q✉❡♥❛ ❡♥tã♦ ❝♦♥s❡❣✉✐♠♦s ❡♥❝♦♥tr❛r y′ _{♥✉♠❛ ✈✐③✐♥❤❛ç❛ ❞❡ y t❛❧ q✉❡ {Dy}′_h−1

1 (h1x)Jk, k = 1, ...,2m} ❢♦r♠❛ ✉♠❛ ❜❛s❡ ❞❡

Rn ❡ t❛❧ q✉❡_Φ(φ,y′₎ é ✉♠❛ ✐♠❡rsã♦ ❞❡ x1✳ ❯♠❛ ✈❡③ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ✜①♦s ❞❡ φ é ✜♥✐t♦✱ ❡♥tã♦ ♣r❡❝✐s❛♠♦s ❛♣❡♥❛s ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❛❥✉st❛♠❡♥t♦s ❛ y ♣♦r ❢♦r♠❛ ❛ ❣❛r❛♥t✐r q✉❡ t❡♠♦s ✉♠❛ ✐♠❡rsã♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s ✜①♦s✳

■♠❡rsã♦ ❞♦s P♦♥t♦s P❡r✐ó❞✐❝♦s✿ ❈♦♥s✐❞❡r❡♠✲s❡ ❛❣♦r❛x1 ❡x2 ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s t❛✐s q✉❡φ(x1) =x2 ❡φ(x2) = x1✳ ❈♦♠♦x1 6=x2 ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❝♦♥❥✉♥t♦s ❛❜❡r✲ t♦s ❞✐s❥✉♥t♦s q✉❡ ❝♦♥té♠x1 ❡ x2 ❤♦♠❡♦♠ór✜❝♦s ❛ ❜♦❧❛s ❛❜❡rt❛s B1 ❡ B2 ❝❡♥tr❛❞❛s ❡♠ h1(x1) ❡h2(x2)✳

❆♥❛❧✐s❡✲s❡ ❛ q✉❡stã♦ ❞❡ ✐♠❡rs✐✈✐❞❛❞❡ ❡♠ x1✱ ✐st♦ é✱ ♦ r❛♥❦ ❞❡DΦ(φ,y)h−11(h1x1)✱ ❝✉❥❛2i+ 1 ✲ és✐♠❛ ❝♦❧✉♥❛ é ✐❣✉❛❧ ❛ ✿

Dyφ2i_h−1

1 (h1x1) = Dyh−11h1φ2ih−11(h1x1) = Dyh−11(h1x1)Dh1φ2ih−11(h1x1) = vJi ❡ ❛ 2i✲és✐♠❛ ❝♦❧✉♥❛ é✿

Dyφ2i−1_h−1

1 (h1x1) = Dyφ2iφ−1h−11(h1x1) = Dyφ−1h−11(h1x1)Dh1φ2ih−11(h1x1) =wJi ♣♦✐s x1 = φ2i_{(x1) ❡ ♦♥❞❡ s❡ ❞❡✜♥❡ J}i _{= Dh1φ}2i_h−1

1 (h1x1)✱ v = Dyh−11(h1x1) ❡ w=Dyφ−1_h−1

1 (h1x1)✳

◗✉❡r❡♠♦s ❛❣♦r❛ ❛✈❡r✐❣✉❛r s❡ {v, wJ, vJ, wJ2_{, ..., wJ}m_{, vJ}m

} ❝♦♥té♠ m ✈❡❝t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ t❛❧ ❝♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡ s❡ ❡s❝r❡✈❡r♠♦sv❡w❡♠ t❡r♠♦s

❞♦s ✈❡❝t♦r❡s ♣ró♣r✐♦s ❞❡J t❡♠♦s✿

v = P_jαjej ❡ w = P_jβjej ❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r vJk = P_jαjλj2kej ❡ wJk =

P

jβjλ2 k j ej

◆❡st❛ ❜❛s❡DΦ(φ,y)h−₁1(h1x)t♦♠❛ ❛ ❢♦r♠❛✿

               

α1 α2 _{· · ·} αm

β1λ1 β2λ2 _{· · ·} βmλm

α1λ1 α2λ2 · · · αmλm

✳✳✳ ✳✳✳

β1λ1 β2λ2 · · · βmλm

α1λm1 α2λm2 · · · αmλmm

               

❡ s❡ r❡❛rr❛♥❥❛r♠♦s ❛s ❝♦❧✉♥❛s ♣♦❞❡♠♦s t❡r✿                   

α1 α2 · · · αm

α1λ1 α2λ2 · · · αmλm

✳✳✳ ✳✳✳

α1λm1 α2λm2 · · · αmλmm

β1λ1 β2λ2 _{· · ·} βmλm

✳✳✳ ✳✳✳

β1λm

1 β2λm2 · · · βmλmm

                  

❙❡ ❛❣♦r❛ ❡❧✐♠✐♥❛r♠♦s ❛s ú❧t✐♠❛s ❧✐♥❤❛s ♦❜t❡♥❞♦ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛✱ s❡ ❛s ♣r✐♠❡✐r❛sm❧✐♥❤❛s ❢♦r❡♠ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡♥tã♦ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ m_×m s❡rá ❞✐❢❡r❡♥t❡ ❞❡ 0✳

❙❡ ❡s❝♦❧❤❡r♠♦s ✿ (α1, α2, ..., αm, β1, β2, ..., βm) = (1,1, ...,1, β₁∗, β₂∗, ..., βm∗)✱

❝♦♠ β∗

1, β2∗, ..., βm∗ ✜①♦s ♦❜t❡♠♦s ✉♠❛ ♠❛tr✐③ ❞❡ ❱❛♥❞❡r♠♦♥❞❡ q✉❡ ❥á s❛❜❡♠♦s ♥ã♦

t❡r ❞❡t❡r♠✐♥❛♥t❡ ✐❣✉❛❧ ❛ ③❡r♦✳

❆ss✐♠ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s(v, w)♣❛r❛ ♦s q✉❛✐s ❛ ❞❡r✐✈❛❞❛ t❡♠ ❢✉❧❧ r❛♥❦ é ❛❜❡rt♦

❡ ❞❡♥s♦ ❡♠ Rm_×Rm ❡ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r _v′ ❡ _w′ t❛✐s q✉❡ _{k(v, w) −(v}′_{, w}′_)k é

❛r❜✐tr❛r✐❛♠❡♥t❡ ♣❡q✉❡♥❛ ❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ {v′_{, w}′_{, v}′_J2_{, w}′_J2_{, ..., v}′_Jm_{, w}′_Jm

} ❢♦r♠❛

✉♠❛ ❜❛s❡ ❞❡Rm✳

❙✉♣♦♥❤❛✲s❡ ❛❣♦r❛ q✉❡yé ✉♠❛ ❢✉♥çã♦ ❞❡ ♠❡❞✐çã♦ ♣❛r❛ ❛ q✉❛❧Φ(φ,y)♥ã♦ é ✐♠❡rs✐✈❛ ❡♠ x1✳ P♦❞❡♠♦s ❞❡✜♥✐r Ψi :M →R ❡χi :M →R1≤i≤m ❝♦♠♦ ✿

Ψi(x) =

    

µ1,i(x)λ(h1x) ♣❛r❛ x ∈ h−₁1B1

0 ❝✳❝

❡ χi(x) =

    

µ2,i(x)λ(h2x) ♣❛r❛ x ∈ h−21B2

0 ❝✳❝

♦♥❞❡ µk,i é ❛ i✲és✐♠❛ ❢✉♥çã♦ ❝♦♦r❞❡♥❛❞❛ ❞❡hk✱ k = 1,2 ❡

y′(x) =y(x) +

m

X

i=1

aiΨi(x) + m

X

i=1

biχi(x)

Pr❡t❡♥❞❡♠♦s ❛❣♦r❛ ♠♦str❛r q✉❡ ❡①✐st❡♠ ✈❡❝t♦r❡s a ❡ b ❝♦♠ ♥♦r♠❛ ❛r❜✐tr❛r✐❛✲ ♠❡♥t❡ ♣❡q✉❡♥❛ t❛❧ q✉❡Φ(φ,y′₎ é ✐♠❡rs✐✈❛ ❡♠ x1✳✷

❚❡♠♦s Dy′_h−1

1 (h1x1) =Dyh−11(h1x1) +a=v+a ❡♥tã♦ ❡s❝♦❧❤❡♠♦sa=v−v′ ❡ Dy′_φ−1_h−1

1 (h1x1) =Dy′h2−1(h2x2)Dh2φ−1h1−1(h1x1) = (Dyh−21(h2x2)+b)A=w+bA ♦♥❞❡ A=Dh2φ−1_h−1

1 (h1x1) é ✉♠❛ ♠❛tr✐③ ✐♥✈❡rtí✈❡❧ ✭♣♦✐s φ é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✮ ❡ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r✿ b= (w′_−w)A−1_✳

❊st❛s ❡s❝♦❧❤❛s ♣❛r❛ a ❡ b t♦r♥❛♠ Φ(φ,y′₎ ✉♠❛ ✐♠❡rsã♦ ❡♠ x1 ❡ ✉♠❛ ✈❡③ q✉❡ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ❛ ♥♦r♠❛ ❞❡(v₋v′_{, w−w}′_{)❛r❜✐tr❛r✐❛♠❡♥t❡ ♣❡q✉❡♥❛✱ t❛♠❜é♠ ❛s}

♥♦r♠❛s ❞❡ a ❡b s❡rã♦ ❛r❜✐tr❛r✐❛♠❡♥t❡ ♣❡q✉❡♥❛s✳

❊st❡s ❛r❣✉♠❡♥t♦s ♣♦❞❡♠ s❡r ❡st❡♥❞✐❞♦s ♣♦r ❢♦r♠❛ ❛ ❝♦❜r✐r t♦❞♦s ♦s ♣♦♥t♦s ♣❡✲ r✐ó❞✐❝♦s ❞❡φ ❝♦♠ ♣❡rí♦❞♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛2m✱ ✐✳❡✱ ♣❛r❛ ❛♦ ❢❛③❡r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡

❛❥✉st❛♠❡♥t♦s ❛ y ♣♦❞❡♠♦s ❝♦♥str✉✐r ✉♠❛ ❢✉♥çã♦ ❞❡ ♠❡❞✐çã♦ y′ _{t❛❧ q✉❡ Φ}

(φ,y′₎ s❡❥❛ ✉♠❛ ✐♠❡rsã♦ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❡♠P2m✳

✷_{◆♦t❡✲s❡ q✉❡ ❝♦♠♦ B}

1 ❡ B2 sã♦ ❞✐s❥✉♥t♦s ❡♥tã♦ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ♣♦♥t♦ x t❛❧ q✉❡ Ψi(x) ❡

χj(x)s❡❥❛♠ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♥ã♦ ♥✉❧♦s✳

■♠❡rsã♦ ♥✉♠❛ ❱✐③✐♥❤❛♥ç❛ ❈♦♠♣❛❝t❛ ❞❡ P2m ❯♠❛ ✈❡③ q✉❡ Φ(φ,y′₎ é ✉♠❛ ✐♠❡rsã♦ ❞❡ P2m ♣❡❧♦ ❚❡♦r❡♠❛ ✷✷✱ ❝♦♥s❡❣✉✐♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❜♦❧❛ ❞❡ r❛✐♦ ri ❡

❝❡♥tr♦ ❡♠xi bi(ri, xi)❞❡ ❝❛❞❛ ♣♦♥t♦ x∈P2m t❛❧ q✉❡Φ(φ,y′₎ s❡❥❛ ✉♠ ♠❡r❣✉❧❤♦ ❞❡ss❛ ✈✐③✐♥❤❛♥ç❛ ❞❡xi✳

Φ(φ,y) é ✉♠❛ ✐♠❡rsã♦ ♥❛ ✉♥✐ã♦ ❞❡st❛s ❜♦❧❛s ❡ é ✐♥❥❡❝t✐✈❛ ❡♠ ❝❛❞❛ bi(ri, xi) ♠❛s

♣♦❞❡♠♦s t❡r Φ(φ,y)(bi(ri, xi))TΦ(φ,y)(bj(rj, xj)) 6= ∅ ❝♦♠ i 6= j✳ ◆♦ ❡♥t❛♥t♦✱ ❝♦♠♦

Φ(φ,y) é ❝♦♥tí♥✉❛✱ s❡ t♦♠❛r♠♦s r❛✐♦s ♠❛✐s ♣❡q✉❡♥♦s✱ ri✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❜♦❧❛s

♠❛✐s ♣❡q✉❡♥❛s ❝✉❥❛s ✐♠❛❣❡♥s ♥ã♦ s❡ ✐♥t❡rs❡❝t❡♠ ❡ ❛ss✐♠ Φ(φ,y) s❡rá ✉♠❛ ✐♠❡rsã♦ ✐♥❥❡❝t✐✈❛ ♥❛ ✉♥✐ã♦ ❞❡st❛s ❜♦❧❛s ♠❛✐s ♣❡q✉❡♥❛s✳

❈♦♥s✐❞❡r❡♠✲s❡ ❛❣♦r❛ ❜♦❧❛s ❢❡❝❤❛❞❛s bi✱ ❞❡ ❝❡♥tr♦ ❡♠ x∈P2m r❛✐♦ r₂i ✭t❛❧ q✉❡ bi

é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❛s ❜♦❧❛s ❛❜❡rt❛s ♠❛✐s ♣❡q✉❡♥❛s r❡❢❡r✐❞❛s ♥♦ ú❧t✐♠♦ ♣❛rá❣r❛❢♦✮ ❡ s❡❥❛Vy =S_ibi q✉❡ é ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ ❡ ♣♦r ✐ss♦ é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❝♦♠♣❛❝t❛ ❞❡

P2m ❡♥tã♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r ❛ss✐♠ q✉❡Φ(φ,y′₎é ✉♠❛ ✐♠❡rsã♦ ✐♥❥❡❝t✐✈❛ ❞❡st❡ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦✳

✷✳✷✳✷ ■♠❡rsã♦ ❞❡ M

❖ ♣ró①✐♠♦ ♣❛ss♦ ❞❛ ❞❡♠♦♥str❛çã♦ ❝♦♥s✐st❡ ❡♠ ♠♦str❛r q✉❡ ❡♠ q✉❛❧q✉❡r ✈✐③✐✲ ♥❤❛♥ç❛ ❞❡ y✱ ❞❡♥♦t❡✲s❡ ♣♦r _Uy✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ♦✉tr❛ ❢✉♥çã♦ ❞❡ ♠❡❞✐çã♦

q✉❡ ❞á ♦r✐❣❡♠ ❛ ✉♠❛ ✐♠❡rsã♦ ❞❡ t♦❞❛ ❛ ✈❛r✐❡❞❛❞❡M✳

❆ ❡str❛té❣✐❛ é ❛ s❡❣✉✐♥t❡✿ ❝♦♠❡ç❛♠♦s ♣♦r ❝♦❜r✐r M ❝♦♠ ❝♦♥❥✉♥t♦s ❝♦♠♣❛❝t♦s ❡ ♠♦str❛r q✉❡ ❢❛③❡♥❞♦ ❛❧t❡r❛çõ❡s ❛r❜✐tr❛r✐❛♠❡♥t❡ ♣❡q✉❡♥❛s ♥❛ ❢✉♥çã♦ ❞❡ ♠❡❞✐çã♦✱ ♣♦❞❡♠♦s ♣r♦❞✉③✐r ✉♠ ❞❡❧❛② ♠❛♣ q✉❡ é ✉♠❛ ✐♠❡rsã♦ ♥✉♠ ❞❡st❡s ❝♦♥❥✉♥t♦s✳ ❋❛③❡♥❞♦ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣❡rt✉r❜❛çõ❡s ❝♦♥s❡❣✉✐♠♦s ✐♠❡r❣✐r t♦❞♦s ♦s ♦✉tr♦s ❝♦♥❥✉♥t♦s ❝♦♠♣❛❝t♦s ✭q✉❡ sã♦ ❡♠ ♥ú♠❡r♦ ✜♥✐t♦ ♣♦✐sM é ❝♦♠♣❛❝t❛✮✳

❈♦♥str✉çã♦ ❞❡ ✉♠ ❛t❧❛s ❛❞❡q✉❛❞♦ ❚♦♠❡✲s❡ ✉♠ ❛t❧❛s ❛r❜✐trár✐♦ ❞❡ M✳ ❯♠❛ ✈❡③ q✉❡ t♦❞♦s ♦s ♣♦♥t♦s ❡♠P2m s❡ ❡♥❝♦♥tr❛♠ ♥✉♠ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❝❛rt❛ ♣♦❞❡♠♦s

❡s❝♦❧❤❡r ♦✉tr♦ ❞♦♠í♥✐♦U′

i t❛❧ q✉❡✿ Ui′ =Ui∩bi ❡ ❛❥✉st❛♥❞♦ ♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛✲

❞❛s ✭❡ t♦♠❛♥❞♦ s❡ ♥❡❝❡ssár✐♦ ✉♠ ❞♦♠í♥✐♦ ♠❛✐♦r✮ ❝♦♥s❡❣✉✐♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❝❛rt❛

(Ui, hi) ❝♦♠ Ui ⊂ bi ❡ Ui = h−i 1B(3)✳ P♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❞❡st❡s ♣❛r❛ ❝❛❞❛

xi ∈ P2m ❡ ❝✉❥♦s ❞♦♠í♥✐♦s ❞❛s ❝❛rt❛s sã♦ ❞✐s❥✉♥t♦s✳ ❙❡ t♦♠❛r♠♦s Wi = h−i 1B(i)

❡♥tã♦ {Wi :xi ∈ P2m} é ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ ❞❡ P2m✳

❈♦♥str✉çã♦ ❞❛ ❈♦❜❡rt✉r❛ ❈♦♠♣❛❝t❛ ❞❡M ❈♦♥s✐❞❡r❡✲s❡ ❛❣♦r❛ ♦ ❝♦♠♣❧❡♠❡♥✲ t❛r ❞❡ P2m✱ P2Cm q✉❡ é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✳ P❛r❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡st❡ ❝♦♥❥✉♥t♦✱ ♦s

♣♦♥t♦s {x, φ(x), ...φ2m_(x)

} sã♦ t♦❞♦s ❞✐st✐♥t♦s✳ ▲♦❣♦ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ❝♦♥✲

❥✉♥t♦ ❛❜❡rt♦ Ux ⊂ P2Cm q✉❡ ❝♦♥té♠ x t❛❧ q✉❡ ♦s ❝♦♥❥✉♥t♦s {Ux, φUx, ...φ2mUx} sã♦

❞✐s❥✉♥t♦s✳ P♦❞❡♠♦s t♦♠❛rUx❝♦♠♦ s❡♥❞♦ ✉♠ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❝❛rt❛ ❡ ❡♥❝♦♥tr❛r ✉♠

s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s (Ux, hx) ❝♦♠ Ux =h−x1B(3)✳

❋✐♥❛❧♠❡♥t❡ t❡♠♦s ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ ❞❡ M✿ {Wx = hx−1B(1) : x ∈ P2Cm} ∪

{Wi : xi ∈ P2m} ❡ ❞❡st❛ ❡①tr❛í♠♦s ✉♠❛ s✉❜❝♦❜❡rt✉r❛ ✜♥✐t❛ ✭q✉❡ ❝♦♥té♠ t♦❞♦s ♦s

❝♦♥❥✉♥t♦s ❞❡{Wi :xi ∈ P2m}✮✳

❙❡❥❛♠Wi✱1≤i≤k♦s ❝♦♥❥✉♥t♦s q✉❡ ❝♦♥tê♠ ♦s ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s ❡Wi✱k < i ≤l

♦s ❝♦♥❥✉♥t♦s ❝♦♥t✐❞♦s ❡♠ PC

2m✳ P♦r ❝♦♥str✉çã♦✱ Φ(φ,y) é ✉♠ ♠❡r❣✉❧❤♦ ❞♦ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦Sk

i=1Wi✳ ❋❛❧t❛✲♥♦s ♣♦rt❛♥t♦ ❛❥✉st❛r ❛ ❢✉♥çã♦ ❞❡ ♠❡❞✐çã♦ ❞❡ ❢♦r♠❛ ❛ q✉❡ s❡❥❛ ✉♠❛ ✐♠❡rsã♦ ♥♦s r❡st❛♥t❡sWi✬s✳

■♠❡rsã♦ ❞♦s ❈♦♥❥✉♥t♦s ❈♦♠♣❛❝t♦s ❙❡❥❛ i ♦ ♠❡♥♦r í♥❞✐❝❡ ♠❛✐♦r q✉❡ k ♣❛r❛ ♦ q✉❛❧ Φ(φ,y) ❢❛❧❤❛ ❡♠ s❡r ✉♠❛ ✐♠❡rsã♦ ❞❡ Wi✳ ❙❡❥❛ x ✉♠ ❡❧❡♠❡♥t♦ ❞❡ Ui ❡ s❡❥❛♠

µj : Uj → R, j = 1, ..., m ❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ❞❡ hi✱ ✐✳❡✱ hix = (µ1x, ..., µmx) ❡

s❡❥❛u=h1x ❡ uj =µjx✳

❊♥tã♦ ❛ ♠❛tr✐③ ❏❛❝♦❜✐♥❛♥❛ ❞❡ Φ(φ,y) ❡♠ hix é✿

              

∂yh−i1(u)

∂u1 · · ·

∂yh−i 1(u)

∂um

∂yφh−_i 1(u)

∂u1 · · ·

∂yφh−_i 1(u)

∂um

✳✳✳ ✳✳✳

∂yφ2m_h−1

i (u)

∂u1 · · ·

∂yφ2m_h−1

i (u)

P❛r❛ ❛❧❣✉♠u_∈hiWi(=B(1)) ❡st❛ ♠❛tr✐③ ♥ã♦ t❡♠ ❢✉❧❧ r❛♥❦✿ t❡♠♦s ❞❡ ❛ t♦r♥❛r ❢✉❧❧

r❛♥❦ ❛tr❛✈és ❞❡ ♣❡rt✉r❜❛çõ❡s ❡♠ y✳ ❊♠ ❝❛❞❛ ✉♠❛ ❞❛s ❛❧t❡r❛çõ❡s ♠♦❞✐✜❝❛♠♦s ✉♠❛ ❡ ❛♣❡♥❛s ✉♠❛ ❝♦❧✉♥❛ ❞❛ ♠❛tr✐③ t♦r♥❛♥❞♦✲❛ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛s ❝♦❧✉♥❛s à s✉❛ ❡sq✉❡r❞❛✳ ❆♦ ✜♠ ❞❡ ✉♠ ♠á①✐♠♦ ❞❡m ❛❧t❡r❛çõ❡s ♦❜t❡♠♦s ✉♠❛ ♠❛tr✐③ ❝♦♠ ❢✉❧❧ r❛♥❦✳

❙✉♣♦♥❤❛✲s❡ q✉❡ ❛s ♣r✐♠❡✐r❛ss❝♦❧✉♥❛s ❞❡DΦ(φ,y)sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ♣❛r❛ t♦❞♦ ♦u_∈B(1)✳ ❉❡✜♥❛✲s❡ ❝♦♠♦ ❤❛❜✐t✉❛❧♠❡♥t❡λ :Rm_{→R✉♠❛ ❢✉♥çã♦ ❜✉♠♣}

✐❣✉❛❧ ❛ ✶ ❡♠B(1) ❝♦♠ s✉♣♦rt❡ ❡♠B(2)✳ ❉❡✜♥❛✲s❡ t❛♠❜é♠ψ :M _→R ❝♦♠♦

ψ(x) =     

µs+1(x)λ(hi(x)) s❡ x ∈ Ui

0 ❝✳❝

❡♥tã♦ ψ(x) = µs+1(x) s❡ x ∈ Wi✱ ψ t❡♠ s✉♣♦rt❡ ❡♠ Ui ❡ ψ ◦φ−j t❡♠ s✉♣♦rt❡ ❡♠

φj_U i✳

❉❡✜♥❛✲s❡ ❛❣♦r❛ ψj :M →R ❝♦♠♦ ψj = ψ◦φ−j ♣❛r❛ 0≤j ≤ 2m✳ ❖s ψj′s tê♠

s✉♣♦rt❡s ❞✐s❥✉♥t♦s✳

❈♦♥str✉✐♠♦s ❛ ❢✉♥çã♦ ❞❡ ♠❡❞✐çã♦ y′ _=y+P2m

j=0aj+1ψj

❉❡st❛ ❢♦r♠❛✱ ♣❛r❛ u_∈hiWi, φkhi−1(u)∈φkUi t❡♠♦s q✉❡✱

y′_φk_h−1

i (u) =yφkh

−1

i (u) +ak+1ψ(φ−kh−i 1(u)) =yφkh

−1

i (u) +ak+1µs+1(h−i 1(u)) =

=yφk_h−1

i (u) +ak+1us+1 ❡ ♣♦rt❛♥t♦ ∂y

′_φk_h−1

i

∂us+1

(u) = ∂yφ

k_h−1

i

∂us+1

(u) +ak+1✳

❯♠❛ ✈❡③ q✉❡ s❛❜❡♠♦s ❝♦♠♦ ❛❥✉st❛r ❛ s+ 1✲és✐♠❛ ❝♦❧✉♥❛ ❞❛ ♠❛tr✐③ ❏❛❝♦❜✐❛♥❛

♣r❡t❡♥❞❡♠♦s s❛❜❡r ❝♦♠♦ é q✉❡ ❛tr❛✈és ❞❡ ✉♠❛ ♣❡rt✉r❜❛çã♦ ❛r❜✐tr❛r✐❛♠❡♥t❡ ♣❡q✉❡♥❛ ❛ t♦r♥❛♠♦s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ss ❝♦❧✉♥❛s à s✉❛ ❡sq✉❡r❞❛✳

▲❡♠❛ ✺✳ ❙❡❥❛ Mn×m ♦ ❡s♣❛ç♦ ❞❛s ♠❛tr✐③❡s ❝♦♠ n ❧✐♥❤❛s ❡ m ❝♦❧✉♥❛s✳ ❖ ❝♦♥❥✉♥t♦

❞❛s ♠❛tr✐③❡s (m_×n) ❝♦♠ ❢✉❧❧ r❛♥❦ é ❛❜❡rt♦ ❡♠ Mn×m✳

❙❡❥❛ Js(x) ❛ ♠❛tr✐③ ❢♦r♠❛❞❛ ♣❡❧❛s ♣r✐♠❡✐r❛s s ❝♦❧✉♥❛s ❞❡ DΦ(φ,y) ❡♠ x ∈ Ui✳

❊♥tã♦✱ ♣♦r ❤✐♣ót❡s❡✱ ♣❛r❛x_∈Wi ✱Js(x)t❡♠ ❢✉❧❧ r❛♥❦✳ ◆♦t❡✲s❡ q✉❡Js é ✉♠❛ ❢✉♥çã♦

❝♦♥tí♥✉❛ ❞❡Ui ♣❛r❛ ♦ ❡s♣❛ç♦ ❞❛s ♠❛tr✐③❡s ❞❡ ❞✐♠❡♥sã♦(2m+1)×s❡♥tã♦ ♣❡❧♦ ▲❡♠❛

✺ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦X ⊂Ui✱ ❝♦♠Wi ⊂X t❛❧ q✉❡ ♣❛r❛ t♦❞♦s ♦s ♣♦♥t♦s ❡♠

X ❛s ♣r✐♠❡✐r❛s s ❝♦❧✉♥❛s ❞❛ ♠❛tr✐③ DΦ(φ,y) sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❉❡♥♦t❡✲s❡ ❛ ❝♦❧✉♥❛ s ❞❛ ♠❛tr✐③ ❏❛❝♦❜✐❛♥❛ ❞❡ Φ(φ,y) ♣♦r

DΦ(φ,y) s

❡ ❞❡✜♥❛✲s❡ ❛❣♦r❛ ❛ ❢✉♥çã♦S :Rs_{×X →}R2m+1

(λ1, λ2, ..., λs, x)→ s

X

j=1 λj

DΦ(φ,y) j

−DΦ(φ,y) s+1

❝♦♠ s≤m ❞❡ ❢♦r♠❛ ❛ q✉❡ Rs_×X t❡♥❤❛ ❞✐♠❡♥sã♦ ✐♥❢❡r✐♦r ❛ R2m+1✳

❆ ❢✉♥çã♦ S é C1 _{✉♠❛ ✈❡③ q✉❡ y ❡ φ sã♦ C}2 _{❡ s❡❣✉❡ ❡♥tã♦ ❞♦ ▲❡♠❛ ✶✾ q✉❡ ♦} ❝♦♠♣❧❡♠❡♥t❛r ❞❡S(Rs_×X)é ❞❡♥s♦ ❡♠R2m+1 ❡ ♣♦❞❡♠♦s ❡♥tã♦ ❡s❝♦❧❤❡r ✉♠ ✈❡❝t♦r

a_∈R2m+1 ❝♦♠ ♥♦r♠❛ ❛r❜✐tr❛r✐❛♠❡♥t❡ ♣❡q✉❡♥❛ t❛❧ q✉❡_{a 6∈S(}Rs_×X)✳

◆♦t❡✲s❡ q✉❡ S(Rs_{×X) é ♦ ❝♦♥❥✉♥t♦ ❞♦s ✈❡❝t♦r❡s✱ a∈R}2m+1 _{q✉❡ ✈❡r✐✜❝❛♠ ✿} a= s X j=1 λj

DΦ(φ,y) j

−DΦ(φ,y) s+1

⇔a+DΦ(φ,y) s+1 = s X j=1 λj

DΦ(φ,y) j

❧♦❣♦ ❡s❝♦❧❤❡♥❞♦ a /∈ S(Rs_×X) ❝♦♥s❡❣✉✐♠♦s ♦❜t❡r ✉♠❛ ♠❛tr✐③ ❝♦♠ ❝❛r❛❝t❡ríst✐❝❛

s+ 1 ❡ s❡❣✉✐♥❞♦ ❡st❡ ❡sq✉❡♠❛ ❛s ✈❡③❡s ♥❡❝❡ssár✐❛s ♦❜t❡♠♦s ✉♠❛ ❢✉♥çã♦ ❞❡ ♠❡❞✐çã♦

y′ _{q✉❡ ❞á ♦r✐❣❡♠ ❛ ✉♠ ❞❡❧❛② ♠❛♣ q✉❡ é ✉♠❛ ✐♠❡rsã♦ ❞❡W}

i ✭❡ t❛♠❜é♠ ❞❡Sk_j=1Wj ✮

❡ ♣❡❧♦ ❧❡♠❛ ✹ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❢✉♥çã♦ ❞❡ ♠❡❞✐çã♦ ❡♠N q✉❡ é ✉♠❛ ✐♠❡rsã♦

❞❡ q✉❛❧q✉❡r Wj ❝♦♠ j < i✳ ❘❡♣❡t✐♥❞♦ ♦ ❛r❣✉♠❡♥t♦ ♣❛r❛ Wi+1✱ ❛♦ ✜♠ ❞❡ ❛❧❣✉♠❛s

✐t❡r❛çõ❡s t❡r❡♠♦s ❡♥❝♦♥tr❛❞♦ ✉♠❛ ✐♠❡rsã♦ ❞❡ t♦❞❛ ❛ ✈❛r✐❡❞❛❞❡M✳

❯♠❛ ✈❡③ q✉❡ Φ(φ,y′₎ é ✉♠❛ ✐♠❡rsã♦ ❞❡ M ❡♥tã♦ ♣❛r❛ ❝❛❞❛ x ∈ M ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x✱ Nx✱ t❛❧ q✉❡ Φ(φ,y′₎ é ✉♠ ♠❡r❣✉❧❤♦ ❞❡ Nx✱ ♣❡❧♦ t❡♦r❡♠❛ ✷✷✱ ❡ s❡

♥♦t❛r♠♦s q✉❡ M é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❜♦❧❛ ❢❡❝❤❛❞❛ βx

❝❡♥tr❛❞❛ ❡♠ x t❛❧ q✉❡ β_{x ⊂ N}x✳ ❖s ✐♥t❡r✐♦r❡s ❞❡st❛s ❜♦❧❛s ❢♦r♠❛♠ ✉♠❛ ❝♦❜❡rt✉r❛

❛❜❡rt❛ ❞❡ M ✭✉♠❛ ✈❡③ q✉❡ ♣❛r❛ ❝❛❞❛ x ❤á ✉♠❛ ❜♦❧❛ βx q✉❡ ❧❤❡ ❝♦rr❡s♣♦♥❞❡✮ ❡

❛ ♣❛rt✐r ❞❡st❛ ❝♦❜❡rt✉r❛ ♣♦❞❡♠♦s ❡①tr❛✐r ✉♠❛ s✉❜❝♦❜❡rt✉r❛ ✜♥✐t❛✳ ❆ ❝♦❧❡❝çã♦ ❞❡ ❜♦❧❛s ❢❡❝❤❛❞❛s ❝♦rr❡s♣♦♥❞❡st❡s{βi : 1 ≤i ≤ n′} ❢♦r♠❛ ✉♠❛ ❝♦❜❡rt✉r❛ ❝♦♠♣❛❝t❛ ❡

❝❛❞❛ ✉♠ ❞♦s s❡✉s ❡❧❡♠❡♥t♦s é ♠❡r❣✉❧❤❛❞♦ ♣♦rΦ(φ,y′)✳

❙❡ s❡❧❡❝❝✐♦♥❛r♠♦s ✉♠❛ ❞❡st❛s ❜♦❧❛s ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❞❡ ♠❡❞✐çã♦ q✉❡ ❞ã♦

♦r✐❣❡♠ ❛ ✉♠ ♠❡r❣✉❧❤♦ é ❛❜❡rt♦ ❧♦❣♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❢✉♥çõ❡s ❞❡ ♠❡❞✐çã♦ q✉❡ ❞ã♦ ♦r✐❣❡♠ ❛ ♠❡r❣✉❧❤♦s ❞❡ t♦❞❛s ❛s ❜♦❧❛s✱ ❞❡♥♦t❡✲s❡ ♣♦rU′

y✱ é ✉♠ ❛❜❡rt♦ ✉♠❛ ✈❡③ q✉❡ é

❛ ✐♥t❡r❡s❡❝çã♦ ✜♥✐t❛ ❞❡ ❝♦♥❥✉♥t♦s ❛❜❡rt♦s✳ U′

y é ✉♠❛ ✈✐③✐♥❤❛ç❛ ❞❡ y′ ❡ ❝♦♠♦y′ ∈ Uy

♣♦❞❡♠♦s t♦♠❛rU′

y ⊂ Uy✳

◆♦t❡✲s❡ q✉❡ ❛s ❢✉♥çõ❡s ❡♠ U′

y ❞ã♦ ♦r✐❣❡♠ ❛ ♠❡r❣✉❧❤♦s ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❜♦❧❛s

✐♥❞✐✈✐❞✉❛❧♠❡♥t❡ ❡ ♥ã♦ sã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛s ❢✉♥çõ❡s ❞❡ ♠❡❞✐çã♦ q✉❡ ❞ã♦ ♦r✐❣❡♠ ❛ ♠❡r❣✉❧❤♦s ❞❡

n

[

i=1 βi✳

❆❣♦r❛✱ ♣❡❧♦ ▲❡♠❛ ❞❡ ▲❡❜❡s❣✉❡ s❛❜❡♠♦s q✉❡ ❡①✐st❡ ǫ >0 t❛❧ q✉❡ ❛ ❜♦❧❛ ❢❡❝❤❛❞❛

❞❡ r❛✐♦ǫ ❝❡♥tr❛❞❛ ❡♠ q✉❛❧q✉❡r ♣♦♥t♦ ❞❡ M ❡stá ❝♦♥t✐❞❛ ♥♦ ✐♥t❡r✐♦r ❞❡βi ♣❛r❛ ♣❡❧♦

♠❡♥♦s ✉♠i✳ ❙❡❣✉❡✲s❡ q✉❡ ❝❛❞❛ǫ✲❜♦❧❛ é ♠❡r❣✉❧❤❛❞❛ ♣♦rΦ(φ,y′₎ ❡ t❛♠❜é♠ ♣♦r t♦❞❛s ❛s ❛♣❧✐❝❛çõ❡s Φ(φ,yb) ❝♦♠ by∈ Uy′✳

❙❡ ❞❡♥♦t❛r♠♦s ❛ ♠étr✐❝❛ ❡♠ M ♣♦rρ ♣♦❞❡♠♦s r❡s✉♠✐r ❡st❡s r❡s✉❧t❛❞♦s ❛tr❛✈és ❞♦ ❧❡♠❛ s❡❣✉✐♥t❡✿

▲❡♠❛ ✻✳ ❙❡ yb _{∈ U}′

y ❡♥tã♦ Φ(φ,by) é ✉♠❛ ✐♠❡rsã♦ ❞❡ M✱ ✉♠ ♠❡r❣✉❧❤♦ ❞❡ Vy ❡

Φ(φ,yb)(x)6= Φ(φ,by)(x′) s❡♠♣r❡ q✉❡ x6=x′ ❡ ρ(x, x′)< ǫ✳ ✷✳✷✳✸ ▼❡r❣✉❧❤♦ ❞❡ ❙❡❣♠❡♥t♦s ❞❡ Ór❜✐t❛

P❛r❛ ❝❛❞❛ x _∈ M s❡❥❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s _{x, φx, ..., φ2m_{x} é ♦ s❡❣♠❡♥t♦ ❞❡}

ór❜✐t❛ ❞❡ x✳ ❈♦♠♦ ✈✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡ ór❜✐t❛s ♣❡r✐ó❞✐❝❛s ♣♦❞❡♠ ❝r✐❛r ♣r♦❜❧❡♠❛s q✉❛♥❞♦ t❡♥t❛♠♦s ❡♥❝♦♥tr❛r ✉♠ ♠❡r❣✉❧❤♦ ❡ ♣♦r r❛③õ❡s s❡♠❡❧❤❛♥t❡s ♦s ♣❛r❡s (x, x′₎

♦♥❞❡ x′ _{♣❡rt❡♥❝❡ ❛♦ s❡❣♠❡♥t❡ ❞❡ ór❜✐t❛ ❞❡ x ♣♦❞❡♠ s❡r t❛♠❜é♠ ♣r♦❜❧❡♠át✐❝♦s}

✉♠❛ ✈❡③ q✉❡ ♥ã♦ ♣♦❞❡♠♦s ❛❧t❡r❛r ❛s ❝♦♦r❞❡♥❞❛s ❞❡ Φ(φ,y)(x) s❡♠ ❛❧t❡r❛r♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡Φ(φ,y)(x′)✳ ❙ã♦ ❡s♣❡❝✐❛❧♠❡♥t❡ ♣r♦❜❧❡♠át✐❝♦s ♦s ♣♦♥t♦s q✉❡ ♣❡rt❡♥❝❡♠ ❛ ✉♠❛ ór❜✐t❛ ❝♦♠ ♣❡rí♦❞♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ 4m ♣♦✐s ♣♦❞❡♠♦s t❡r x = φj_x′ _♣❛r❛

❛❧❣✉♠0_≤j _≤2m ❡ x′ _=φi_{x ♣❛r❛ ❛❧❣✉♠0≤i≤2m✳}

❚❡♥❞♦ ❡♠ ❝♦♥t❛ ❡st❛s ♦❜s❡r✈❛çõ❡s tr❛t❛r❡♠♦s ♥❡st❛ s❡❝çã♦ ♦ ❝❛s♦ ❞❛ ✐♥❥❡❝t✐✈✐❞❛❞❡ ❛♣❡♥❛s ♥♦s s❡❣♠❡♥t♦s ❞❡ ór❜✐t❛ ❡ ♥❛ s❡❝çã♦ s❡❣✉✐♥t❡ ❡①t❡♥❞❡♠♦s ❛ ✐♥❥❡❝t✐✈✐❞❛❞❡ ♣❛r❛ t♦❞❛ ❛ ✈❛r✐❡❞❛❞❡M✳

▲❡♠❛ ✼✳ ❙❡❥❛y′ _{t❛❧ q✉❡Φ}

(φ,y′₎é ✉♠❛ ✐♠❡rsã♦ ✐♥❥❡❝t✐✈❛ ❡♠V_y✳ ❊♠ t♦❞❛ ❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ y′ _{❡♠ C}2_{(M,R) ❡①✐st❡ ✉♠❛ ❢✉♥çã♦✱ y}′′ _{t❛❧ q✉❡ ♣❛r❛ t♦❞♦ ♦ x ∈ M ❡ j t❛❧ q✉❡}

1≤j ≤2m✱ Φ(φ,y′′₎(x)= Φ6 ₍φ,y′′₎(φjx) ❛ ♥ã♦ s❡r q✉❡ x=φjx✳

❉❡♠♦♥str❛çã♦✳ ■r❡♠♦s ❛❥✉st❛r ❛ ❢✉♥çã♦ y′ _{♣❛r❛ ❝❛❞❛ j ♣❛r❛ ♦ q✉❛❧ ♦ ❧❡♠❛ ♥ã♦ s❡}

✈❡r✐✜q✉❡✳ ❚♦♠❡✲s❡j ♥❡ss❛s ❝♦♥❞✐çõ❡s ❡ ❞❡✜♥❛✲s❡ ♦ ❝♦♥❥✉♥t♦S =T2_i=0m φ−i_V y✸✳

P❛r❛ ❝❛❞❛ x _∈ S ❡ 0 _≤ k _≤ 2m t❡♠♦s φk_{x ∈ V}

y ❡ ❝♦♠♦ ❥á ❡st❛❜❡❧❡❝❡♠♦s ❛

✐♥❥❡❝t✐✈✐❞❛❞❡ ❡♠Vy ♦ ❧❡♠❛ ✈❡r✐✜❝❛✲s❡ ❡♠S✳

❙❡❥❛ T ♦ ❢❡❝❤♦ ❞♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ S✱ ✉♠❛ ✈❡③ q✉❡ S é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡

P2m ❡♥tã♦ s❡ x ∈ T t❡♠♦s q✉❡ x 6∈ P2m ❡ {x, φx, ..., φ2mx} sã♦ t♦❞♦s ❞✐❢❡r❡♥t❡s ❡

♣♦❞❡♠♦s ♣♦r ✐ss♦ ❡♥❝♦♥tr❛r ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✱ Ux✱ t❛❧ q✉❡ Ux, φUx, ..., φ2mUx sã♦

t♦❞♦s ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s✳

❚❡♠♦s ❛❣♦r❛ ❞❡ ❝♦♥s✐❞❡r❛r ❞♦✐s ❝❛s♦s✿

❈❛s♦ ✶ ❙❡ x ♥ã♦ é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❝♦♠ ♣❡rí♦❞♦ ❡♥tr❡2m+ 1 ❡4m ❡ ♥❡st❡ ❝❛s♦ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r Ux t❛❧ q✉❡ Ux, φUx, φ2Ux, ..., φ4mUx sã♦ t♦❞♦s ❞✐s❥✉♥t♦s✱

❡ ❝♦♠♦ ❤❛❜✐t✉❛❧♠❡♥t❡✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ Ux é ♦ ❞♦♠í♥✐♦ ❞❡ ✉♠ ❜♦♠ ❛t❧❛s✿

hxUx =B(3) ❡Wx =h−x1B(1)

❈❛s♦ ✷ ❙❡ x é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❝♦♠ ♣❡rí♦❞♦ k✱ ❝♦♠ 2m+ 1≤k ≤4m✱ ♥❡st❡

❝❛s♦ ❡♥❝♦♥tr❛♠♦sUxt❛❧ q✉❡Ux, φUx, φ2Ux, ..., φk−1Ux sã♦ t♦❞♦s ❞✐s❥✉♥t♦s ❡ t♦♠❛♠♦s

Ux ❝♦♠♦ s❡♥❞♦ ♦ ❞♦♠í♥✐♦ ❞❡ ✉♠ ❜♦♠ ❛t❧❛s✳ ❉❡✜♥✐♠♦s t❛♠❜é♠Xx =WxTφ−kWx

✭❡st❡ ❝♦♥❥✉♥t♦ é ❛❜❡rt♦ ❡ ♥ã♦ ✈❛③✐♦ ✉♠❛ ✈❡③ q✉❡ x_∈Wx✮

P♦r ❢♦r♠❛ ❛ s✐♠♣❧✐✜❝❛r ❛ ♥♦t❛çã♦ ♣❛r❛ ♦ ❈❛s♦ ✶ ❡s❝r❡✈❡♠♦s Xx =Wx✳

❆ss✐♠ t❡♠♦s✿

• ◆♦ ❈❛s♦ ✶ ❡ ❈❛s♦ ✷ q✉❛♥❞♦k > 2m+j♥❡♥❤✉♠ ❞♦s ❝♦♥❥✉♥t♦sφ2m+1_X

x, ..., φ2m+jXx

✐♥t❡rs❡❝t❛ 2m

[

l=0 φlUx

• ◆♦ ❝❛s♦ ✷ s❡k _≤2m+j ♥❡♥❤✉♠ ❞♦s ❝♦♥❥✉♥t♦sφ2m+1_X

x, ..., φk−1Xx ✐♥t❡rs❡❝t❛

2m

[

l=0

φlUx ❡ φkXx ⊂Wx, φk+1Xx ⊂φWx, ..., φ2m+jXx ⊂φ2m+j−kXx

✸_{■♥t❡rs❡❝çã♦ ❞♦s ♣♦♥t♦s x∈ M t❛✐s q✉❡ φ}i_x

∈ Vy✱ ♣❛r❛ i= 0, ...,2m q✉❡ é ❛ ✈✐③✐♥❤❛♥ç❛ ❞♦s

♣♦♥t♦s ♣❡r✐ó❞✐❝♦s ❝♦♠ ♣❡rí♦❞♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛2m