❉✐♥â♠✐❝❛ ❝♦♠♣❧❡①❛ ❡ ❢r❛❝t❛✐s
❏✉❧✐❛♥❛ ❈♦♥❝❡✐çã♦ Pr❡❝✐♦s♦
Pr♦❢❡ss♦r❛ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ■❇■▲❈❊✴❯◆❊❙P ✲ ❈❛♠♣✉s ❞❡ ❙ã♦ ❏♦sé ❞♦ ❘✐♦ Pr❡t♦
♣r❡❝✐♦s♦❅✐❜✐❧❝❡✳✉♥❡s♣✳❜r
❲❡❜❡r ❋❧á✈✐♦ P❡r❡✐r❛
Pr♦❢❡ss♦r ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ■❇■▲❈❊✴❯◆❊❙P ✲ ❈❛♠♣✉s ❞❡ ❙ã♦ ❏♦sé ❞♦ ❘✐♦ Pr❡t♦
✇❡❜❡r❢❅✐❜✐❧❝❡✳✉♥❡s♣✳❜r
❋❡r♥❛♥❞♦ ◆❡r❛ ▲❡♥❛r❞✉③③✐
▼❡str❛♥❞♦ ✲❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛
■❇■▲❈❊✴❯◆❊❙P ✲ ❈❛♠♣✉s ❞❡ ❙ã♦ ❏♦sé ❞♦ ❘✐♦ Pr❡t♦ ❢❡r❧❡♥❛r❞✉③③✐❅❣♠❛✐❧✳❝♦♠
❆❜str❛❝t
◆❡st❡ ❛rt✐❣♦✱ tr❛❜❛❧❤❛r❡♠♦s ❝♦♠ ❢✉♥çõ❡s ❝❛ót✐❝❛s ♥♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✱ ❞❡♥tr❡ ❡❧❛s✱ ❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛Qc(z) =z2+c✱ ♦♥❞❡z, c∈C✳ ❉❡s❡♥✈♦❧✈❡r❡♠♦s ♦s r❡s✉❧t❛❞♦s ❜ás✐❝♦s
❡ ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ❚❡♦r✐❛ ❋r❛❝t❛❧ ❡✱ ❡♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛r❡♠♦s ♣❛r❛ ❛❧❣✉♥s t✐♣♦s ❞❡ ❢✉♥çõ❡s ❝♦♠♣❧❡①❛s✱ ♦s ❞♦✐s ♣r✐♥❝✐♣❛✐s t✐♣♦s ❞❡ ❢r❛❝t❛✐s ❞❡ss❛ ❣❡♦♠❡tr✐❛✿ ♦s ❈♦♥❥✉♥t♦s ❞❡ ❏✉❧✐❛ ❡ ❞❡ ▼❛♥❞❡❧❜r♦t✳
P❛❧❛✈r❛s✲❝❤❛✈❡s✿ ❙✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝❛ót✐❝♦s✱ ❢r❛❝t❛✐s✱ ❝♦♥❥✉♥t♦s ❞❡ ❏✉❧✐❛✱ ❝♦♥❥✉♥t♦s ❞❡ ▼❛♥❞❡❧❜r♦t✳
❈♦♠♣❧❡① ❉②♥❛♠✐❝ ❛♥❞ ❋r❛❝t❛❧s
❆❜str❛❝t
■♥ t❤✐s ❛rt✐❝❧❡✱ ✇❡ ✇✐❧❧ ✇♦r❦ ✇✐t❤ ❝❤❛♦t✐❝ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✱ ❛♠♦♥❣ t❤❡♠✱ t❤❡ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥ Qc(z) =z2+c✱ ✇❤❡r❡z, c∈C✳ ❲❡ ✇✐❧❧ ❞❡✈❡❧♦♣ t❤❡ ❜❛s✐❝
❛♥❞ ❢✉♥❞❛♠❡♥t❛❧ r❡s✉❧ts ♦❢ t❤❡ ❋r❛❝t❛❧ ❚❤❡♦r② ❛♥❞ t❤❡♥✱ ✇❡ ✇✐❧❧ ♣r❡s❡♥t t♦ s♦♠❡ t②♣❡s ♦❢ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s✱ t❤❡ t✇♦ ♠❛✐♥ t②♣❡s ♦❢ ❢r❛❝t❛❧s t❤✐s ❣❡♦♠❡tr②✿ t❤❡ ❏✉❧✐❛✬s s❡t ❛♥❞ ▼❛♥❞❡❧❜r♦t✬s s❡t✳
❑❡②✇♦r❞s✿ ❈❤❛♦t✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠s✱ ❢r❛❝t❛❧s✱ ❏✉❧✐❛✬s s❡t✱ ▼❛♥❞❡❧❜r♦t✬s s❡t✳
Pr❡❝✐♦s♦✱ ❈✳P✳ P❡r❡✐r❛✱ ❲✳ ❋✳ ▲❡♥❛r❞✉③③✐✱ ❋✳ ◆✳
✲ ✶✲ ❉✐♥â♠✐❝❛ ❝♦♠♣❧❡①❛
✶ ■♥tr♦❞✉çã♦
❆ ❚❡♦r✐❛ ❞♦ ❈❛♦s ♦✉✱ ✐♥❢♦r♠❛❧♠❡♥t❡✱ ❛ ❚❡♦r✐❛ ❞❛ ■♠♣r❡✈✐s✐❜✐❧✐❞❛❞❡✱ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ❜❡❧íss✐♠❛ ❣❡♦♠❡tr✐❛✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ●❡♦♠❡tr✐❛ ❋r❛❝t❛❧✳ ❊ss❛ ❣❡♦♠❡tr✐❛ tr❛t❛✱ ❞❡♥✲ tr❡ ♦✉tr❛s ❝♦✐s❛s✱ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❝♦♠♣♦rt❛♠❡♥t♦s ❞❡ ❞❡t❡r♠✐♥❛❞♦s ♦❜❥❡t♦s✱ ❝❤❛♠❛❞♦s ❢r❛❝t❛✐s✱ ♣❛r❛ ♦s q✉❛✐s ❛ ❣❡♦♠❡tr✐❛ ❡✉❝❧✐❞✐❛♥❛ ♥ã♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛✳ ❚❛✐s ♦❜❥❡t♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❢❛❝✐❧♠❡♥t❡ ♥♦ ♥♦ss♦ ❝♦t✐❞✐❛♥♦✱ ♣♦r ❡①❡♠♣❧♦✱ ✉♠❛ ❢♦❧❤❛ ❞❡ s❛♠❛♠❜❛✐❛✱ ✉♠ ❜ró❝♦❧✐s✱ ♦ ❢♦r♠❛t♦ ❞❛s ♥✉✈❡♥s✱ ❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠❛ ♠♦♥t❛♥❤❛✱ ❡♥tr❡ ♦✉tr♦s✳ ❖ s✉r❣✐♠❡♥t♦ ❞❡ss❛ ❣❡♦♠❡tr✐❛ s❡ ❞❡✉ ♥❛ ❞é❝❛❞❛ ❞❡ ✼✵✱ ❝♦♠ ♦s tr❛❜❛❧❤♦s ❞♦ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ❇❡♥♦ît ▼❛♥❞❡❧❜r♦t✱ q✉❡ ❝✉♥❤♦✉ ❛ ♣❛❧❛✈r❛ ❢r❛❝t❛❧✱ ❞❡r✐✈❛❞❛ ❞♦ ❧❛t✐♠ ❢r❛❝t✉s ✭q✉❡ s✐❣♥✐✜❝❛ q✉❡❜r❛❞♦ ♦✉ ❢r❛t✉r❛❞♦✮✱ ♣❛r❛ ❞❡♥♦t❛r ♦❜❥❡t♦s ❝✉❥❛ ❞✐♠❡♥sã♦ ♥ã♦ ❡r❛ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✱ ❡ ❝♦♥s❡✲ q✉❡♥t❡♠❡♥t❡ ✐♠♣♦ssí✈❡✐s ❞❡ s❡r❡♠ tr❛t❛❞♦s ❛tr❛✈és ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✳ ➱ ❡str❛♥❤♦ ♣❡♥s❛r♠♦s ♥❛ ❡①✐stê♥❝✐❛ ❞❡ss❡s ♦❜❥❡t♦s✱ ♣♦✐s ♥♦ss❛ ✐♥t✉✐çã♦ ❡stá ❛❝♦st✉♠❛❞❛ ❝♦♠ ♦❜❥❡t♦s ❞❡ ❞✐♠❡♥sã♦ ✐♥t❡✐r❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❝✉r✈❛s ✭❞✐♠❡♥sã♦ ✉♠✮✱ ♣❧❛♥♦s✱ ❝✐❧✐♥❞r♦s ✭❞✐♠❡♥sã♦ ❞♦✐s✮ ❡ ♦❜❥❡t♦s ♠❛❝✐ç♦s ✭❞✐♠❡♥sã♦ três✮✳ ▼❛♥❞❡❧❜r♦t✱ ♥❡ss❛ ♠❡s♠❛ é♣♦❝❛✱ ✐❧✉str♦✉ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞♦ ❝♦♠♣✉t❛❞♦r✱ ❛❧❣✉♥s ❞❡ss❡s ♦❜❥❡t♦s✱ ♣♦ré♠✱ ❞❡✈✐❞♦ ❛ ❧✐♠✐t❛çã♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❞❛ é♣♦❝❛✱ ❡ss❛ t❡♦r✐❛ ❝❛♠✐♥❤♦✉ ❛ ♣❛ss♦s ♠♦❞❡r❛❞♦s ❡✱ s♦♠❡♥t❡ ❝♦♠ ♦ ❛✈❛♥ç♦ t❡❝♥♦❧ó❣✐❝♦✱ ❤♦✉✈❡ ✉♠ ♣r♦❣r❡ss♦ s✐❣♥✐✜❝❛t✐✈♦ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛ s❡✉s ❝♦♥❝❡✐t♦s ❡ ♣r♦♣r✐❡❞❛❞❡s ❣❡♦♠étr✐❝❛s✳
❉♦✐s ✐♠♣♦rt❛♥t❡s ❢r❛❝t❛✐s✱ q✉❡ t❡♠ ✉♠ ♣❛♣❡❧ ❞❡ ❞❡st❛q✉❡ ♥❡ss❛ ❣❡♦♠❡tr✐❛✱ sã♦ ♦s ❈♦♥❥✉♥✲ t♦s ❞❡ ▼❛♥❞❡❧❜r♦t ❡ ♦s ❈♦♥❥✉♥t♦s ❞❡ ❏✉❧✐❛ ✭q✉❡ ♥❛ ✈❡r❞❛❞❡ ❞❡✈❡r✐❛♠ ❝❤❛♠❛r✲s❡ ❈♦♥❥✉♥t♦s ❞❡ ❋❛t♦✉✲❏✉❧✐❛✱ ♣♦✐s ❢♦r❛♠ ♦s ❞♦✐s ♠❛t❡♠át✐❝♦s ❢r❛♥❝❡s❡s P✐❡rr❡ ❋❛t♦✉ ❡ ●❛st♦♥ ❏✉❧✐❛ q✉❡ ✐♥tr♦❞✉③✐r❛♠ ♦s ♣r♦❝❡ss♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦s ♠❡s♠♦s✮✳
❖ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t é ✉♠ ❞♦s ❢r❛❝t❛✐s ♠❛✐s ❝♦♥❤❡❝✐❞♦s✱ ❜❛t✐③❛❞♦ ♣❡❧♦ ♣ró♣r✐♦ ▼❛♥❞❡❧❜r♦t✳ ❙❡✉ s✉r❣✐♠❡♥t♦ s❡ ❞❡✉ q✉❛♥❞♦ ❡❧❡ tr❛❜❛❧❤❛✈❛ ❝♦♠ ❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❝♦♠♣❧❡①❛
z = z2 + c ❡ ❡st✉❞❛✈❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s ✐t❡r❛❞❛s ❞❡ss❛ ❢✉♥çã♦✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱
▼❛♥❞❡❧❜r♦t ❡st❛✈❛ ✐♥t❡r❡ss❛❞♦ ❡♠ ♦❜t❡r ♦s ✈❛❧♦r❡s ❝♦♠♣❧❡①♦s ❞❡c✱ ♣❛r❛ ♦s q✉❛✐s ❛ s❡q✉ê♥❝✐❛
❞❛s ✐t❡r❛❞❛s ❞❡ ✵ ♥ã♦ ✏❡s❝❛♣❛✈❛ ♣❛r❛ ♦ ✐♥✜♥✐t♦✧✳ ❖ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r ❡ss❡s ✈❛❧♦r❡s é ♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✳
❖ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛✱ ♦✉tr♦ ❢r❛❝t❛❧ ♠✉✐t♦ ❢❛♠♦s♦✱ ❡stá ✐♥t✐♠❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✱ ♣♦ré♠ s❡✉ s✉r❣✐♠❡♥t♦ s❡ ❞❡✉ ♥♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳ ✭♦ q✉❡ t♦r♥❛✲ ♦ ♠❛✐s ✈❡❧❤♦ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✮✱ ❡♥q✉❛♥t♦ ♦ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ●❛st♦♥ ❏✉❧✐❛ ❡st✉❞❛✈❛ ❛ ✐♥t❡r❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❡ ❢✉♥çõ❡s r❛❝✐♦♥❛✐s✳ ❆ ♣r✐♥❝✐♣❛❧ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✱ é ❛ ♠❛♥❡✐r❛ ♥❛ q✉❛❧ ❛ ❢✉♥çã♦ é ✐t❡r❛❞❛✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t é ♦❜t✐❞♦ ✐t❡r❛♥❞♦✲s❡ z = z2 +c✱ ❝♦♠ z s❡♠♣r❡ ❝♦♠❡ç❛♥❞♦ ❡♠ 0 ❡ ✈❛r✐❛♥❞♦
♦ ✈❛❧♦r ❞❡ c✳ ❏á ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ é ♦❜t✐❞♦ ✐t❡r❛♥❞♦✲s❡ z = z2 +c✱ ♣❛r❛ ✉♠ ✈❛❧♦r ✜①♦
❞❡ c ❡ ✈❛r✐❛♥❞♦ ♦s ✈❛❧♦r❡s ❞❡ z✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t é ♦ ❡s♣❛ç♦
❞❡ ♣❛râ♠❡tr♦s✱ ♦✉ ♦ c✲♣❧❛♥♦✱ ❡♥q✉❛♥t♦ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ é ♦ ❡s♣❛ç♦ ❞✐♥â♠✐❝♦✱ ♦✉ ♦
③✲♣❧❛♥♦✳
◆❡ss❡ tr❛❜❛❧❤♦✱ ✐♥tr♦❞✉③✐r❡♠♦s ❢♦r♠❛❧♠❡♥t❡ ❡ss❡s ❞♦✐s ❝♦♥❥✉♥t♦s✱ ❞❡s❝r❡✈❡r❡♠♦s s❡✉s ❝♦♠♣♦rt❛♠❡♥t♦s ❡ ❞❛r❡♠♦s ❡①❡♠♣❧♦s ❞♦s ♠❡s♠♦s✳
✷ ❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s
◆❡st❛ s❡çã♦✱ ✐♥tr♦❞✉③✐r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s✱ ♦s q✉❛✐s s❡r✈✐rã♦ ❞❡ ❛❧✐❝❡r❝❡ ♣❛r❛ ♥♦ss♦ tr❛❜❛❧❤♦✳
❉❡✜♥✐çã♦ ✶✳ ❆ ór❜✐t❛ ❞❡ ✉♠ ♣♦♥t♦x0♣❡❧❛ ❢✉♥çã♦F é ❛ s❡q✉ê♥❝✐❛x0, F(x0),· · ·, Fn(x0),· · ·✳
❉❡✜♥✐çã♦ ✷✳ ❯♠ ♣♦♥t♦ ✜①♦ ❞❡ ✉♠❛ ❢✉♥çã♦ F é ✉♠ ♣♦♥t♦ x0 s❛t✐s❢❛③❡♥❞♦ F(x0) = x0 ❡✱
♠❛✐s ❣❡r❛❧♠❡♥t❡✱ Fn(x
0) =x0, ♣♦✐s
Fn(x
0) = Fn−1(F(x0)) =Fn−1(x0) = · · ·=F(x0) = x0.
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ór❜✐t❛ ❞❡ ✉♠ ♣♦♥t♦ ✜①♦ x0 é ❛ s❡q✉ê♥❝✐❛ ❝♦♥st❛♥t❡ x0, x0,· · · .
P♦♥t♦s ✜①♦s sã♦ ♦❜t✐❞♦s r❡s♦❧✈❡♥❞♦✲s❡ ❛ ❡q✉❛çã♦F(x) = x.
❉❡✜♥✐çã♦ ✸✳ ❯♠❛ ór❜✐t❛ é ❝❤❛♠❛❞❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ ♦✉ ❝í❝❧♦✱ ❞❡ ♣❡rí♦❞♦n✱ s❡ ♦ ♣♦♥t♦ x0 é
✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❞❡ ♣❡rí♦❞♦n✱ ✐st♦ é✱ t❡♠♦sFn(x
0) =x0✱ ♣❛r❛ ❛❧❣✉♠n >0❡Fk(x0)6=x0
♣❛r❛ k < n.
❙❡x0 ❢♦r ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❞❡ ♣❡rí♦❞♦k✱ ❡♥tã♦x0 s❡rá ✉♠ ♣♦♥t♦ ✜①♦ ♣❛r❛Fk✱ ♦✉ s❡❥❛✱
Fk(x
0) =x0. ▲♦❣♦✱ ♦s ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s ❞❡ ♣❡rí♦❞♦ k, ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s r❡s♦❧✈❡♥❞♦✲s❡
❛ ❡q✉❛çã♦Fk(x) = x.
❉❡✜♥✐çã♦ ✹✳ ❯♠ ♣♦♥t♦x0é ❝❤❛♠❛❞♦ ♣♦♥t♦ ❡✈❡♥t✉❛❧♠❡♥t❡ ✜①♦✱ ♦✉ ❡✈❡♥t✉❛❧♠❡♥t❡ ♣❡r✐ó❞✐❝♦✱
s❡x0 ♥ã♦ é ✉♠ ♣♦♥t♦ ✜①♦ ♦✉ ♣❡r✐ó❞✐❝♦✱ ♠❛s ❛❧❣✉♠ ♣♦♥t♦ ❞❛ ór❜✐t❛ ❞❡x0 é ✜①♦ ♦✉ ♣❡r✐ó❞✐❝♦✳
❚❡♦r❡♠❛ ✺✳ ✭❞♦ P♦♥t♦ ❋✐①♦✮✿ ❙✉♣♦♥❤❛ q✉❡ F : [a, b] → [a, b] é ❝♦♥tí♥✉❛✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ♣♦♥t♦ ✜①♦ ♣❛r❛F ❡♠ [a, b]✳
❉❡✜♥✐çã♦ ✻✳ ❙✉♣♦♥❤❛ q✉❡ x0 é ✉♠ ♣♦♥t♦ ✜①♦ ❞❡ F✳ ❊♥tã♦✱ x0 é ✉♠ ♣♦♥t♦ ✜①♦ ❛tr❛t♦r✱
s❡ |F′(x
0)| < 1✳ ❖ ♣♦♥t♦ x0 é ✉♠ ♣♦♥t♦ ✜①♦ r❡♣✉❧s♦r✱ s❡ |F′(x0)| > 1✳ ❋✐♥❛❧♠❡♥t❡✱ s❡
|F′(x
0)|= 1✱ ❡♥tã♦ ♦ ♣♦♥t♦ ✜①♦ é ❝❤❛♠❛❞♦ ❞❡ ♥❡✉tr♦ ♦✉ ✐♥❞✐❢❡r❡♥t❡✳
❚❡♦r❡♠❛ ✼✳ ✭❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❘❡♣✉❧s♦r✮✿ ❙✉♣♦♥❤❛ q✉❡ x0 é ✉♠ ♣♦♥t♦ ✜①♦
r❡♣✉❧s♦r ❞❡F✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ✐♥t❡r✈❛❧♦ I q✉❡ ❝♦♥té♠x0 ❡♠ s❡✉ ✐♥t❡r✐♦r ❡ ❝♦♠ ❛s s❡❣✉✐♥t❡s
❝♦♥❞✐çõ❡s s❛t✐s❢❡✐t❛s✿ s❡ x∈I ❡ x6=x0✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ n >0 t❛❧ q✉❡ Fn(x)6∈I✳
Pr❡❝✐s❛r❡♠♦s t❛♠❜é♠ ❞❡ ❛❧❣✉♥s tó♣✐❝♦s ❞❡ ❛r✐t✐♠ét✐❝❛ ❝♦♠♣❧❡①❛✳ ▲❡♠❜r❡✲s❡ q✉❡i2 =−1✱
❛ss✐♠ ❞❡✜♥✐♠♦s✿
+ := (a+bi) + (c+di) = (a+c) + (b+d)i · := (a+bi)·(c+di) = (ac−bd) + (ad+cb)i
❉❛❞♦z =a+bi✱ ♦ ♠ó❞✉❧♦ ♦✉ ♥♦r♠❛ ❞❡ z✱ ❝✉❥❛ ♥♦t❛çã♦ é |z|✱ é
❞❛❞❛ ♣♦r✿
r=|z|=√a2+b2.
❙✉❛ r❡♣r❡s❡♥t❛çã♦ ♣♦❧❛r é ❞❛❞❛ ♣♦r
z =r(cosθ+isenθ).
▼❛✐s ❛✐♥❞❛✱ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ❊✉❧❡r✱ q✉❛♥❞♦ ♠ó❞✉❧♦ ❞❡ z é ✐❣✉❛❧ ❛ 1✿
eiθ =cosθ+isenθ.
❆ss✐♠✱ ❣❡♥❡r❛❧✐③❛♥❞♦
z =reiθ.
❉❡s✐❣✉❛❧❞❛❞❡ ❚r✐❛♥❣✉❧❛r✿ ❙❡ z ❡w sã♦ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❡♥tã♦ • |z+w| ≤ |z|+|w|;
• |z−w| ≥ |z| − |w|.
❉❡✜♥✐çã♦ ✽✳ ❉❛❞♦ z=a+bi✱ ♦ ❝♦♥❥✉❣❛❞♦ ❞❡ z✱ r❡♣r❡s❡♥t❛❞♦ ♣♦rz¯✱ é
¯
z =a−bi.
❉❡✜♥✐çã♦ ✾✳ ✭❘❛✐③ ❈♦♠♣❧❡①❛ ❡ P♦t❡♥❝✐❛çã♦✮✿
zk =rk(cos(kθ) +isen(kθ))✱ ♦♥❞❡k
∈R✳
✸ ❋✉♥çõ❡s ❈♦♠♣❧❡①❛s ▲✐♥❡❛r❡s
❈♦♥s✐❞❡r❡♠♦s
Lα(z) = αz, α∈C
❡ ❛♥❛❧✐s❡♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ s✉❛s ór❜✐t❛s✳
◆♦t❡ q✉❡ z = 0 é ♣♦♥t♦ ✜①♦ ❞❡ Lα✳ ❚♦♠❡ α =ρeiψ ❡ s❡❥❛ z0 =reiθ✳ ❆ss✐♠✱
z1 = Lα(z0) = ρeiψreiθ =ρrei(ψ+θ)
z2 = Lα(z1) = ρeiψρrei(ψ+θ) =ρ2rei(2ψ+θ)
✳✳✳
zn = Lα(zn−1) =Lαn(z0) = ρnrei(nψ+θ)
❊①✐st❡♠ três ♣♦ss✐❜✐❧✐❞❛❞❡s✿ ✶✳ ❙❡ ρ <1✱ ❡♥tã♦ ρn
→ 0 q✉❛♥❞♦ n → +∞✳ ❯♠❛ ✈❡③ q✉❡ |ei(nψ+θ)| = 1,∀n✱ s❡❣✉❡ q✉❡
|zn| →0,q✉❛♥❞♦ n →+∞✳ ■st♦ é✱ s❡ ρ <1✱ ❡♥tã♦ t♦❞❛s ❛s ór❜✐t❛s ✈ã♦ ❛ 0✳
✷✳ ❙❡ ρ > 1✱ ❡♥tã♦ ♦ ❝♦♥trár✐♦ ❛❝♦♥t❡❝❡✳ ❈♦♠♦ ρn → ∞, q✉❛♥❞♦ n → ∞✱ s❡❣✉❡ q✉❡ ♦s
♣♦♥t♦s ♥ã♦ ♥✉❧♦s t❡♥❞❡♠ ❛♦ ✐♥✜♥✐t♦✳ ❊♠ ❛♥❛❧♦❣✐❛ ❝♦♠ ♦ ❝❛s♦ r❡❛❧✱ s❡ ρ < 1✱ ❡♥tã♦ 0 s❡rá ❝❤❛♠❛❞♦ ❞❡ ♣♦♥t♦ ✜①♦ ❛tr❛t♦r ❡✱ s❡ ρ > 1✱ ❡♥tã♦ 0 s❡rá ❝❤❛♠❛❞♦ ❞❡ ♣♦♥t♦ ✜①♦ r❡♣✉❧s♦r✳
✸✳ ◆♦ ❝❛s♦ ♥❡✉tr♦✱ ✐st♦ é✱ ρ = 1✱ ❛ ❞✐♥â♠✐❝❛ é ♠❛✐s ❝♦♠♣❧✐❝❛❞❛✳ ❊①✐st❡♠ ❞♦✐s s✉❜❝❛s♦s ❞❡♣❡♥❞❡♥t❡s ❞♦ â♥❣✉❧♦ ♣♦❧❛r ψ✳ ❊s❝r❡✈❛♠♦s
ψ = 2πr.
❉❡✈❡♠♦s ❛♥❛❧✐s❛r ❞♦✐s ❞✐❢❡r❡♥t❡s ❝❛s♦s❀ q✉❛♥❞♦ r é r❛❝✐♦♥❛❧ ❡ q✉❛♥❞♦ r é ✐rr❛❝✐♦♥❛❧✳
✸✳✶✲ ❱❛♠♦s s✉♣♦r ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡r é r❛❝✐♦♥❛❧✱ ♦✉ s❡❥❛✱r = p
q, ♦♥❞❡p, q ∈Z✳ ❊♥tã♦✱
Lq α(re
iθ) = re(2πipq)q+iθ
= re2πip+iθ
= reiθ
❉❡ss❛ ❢♦r♠❛✱ ❝❛❞❛ ♣♦♥t♦ z0 6= 0 é ♣❡r✐ó❞✐❝♦✱ ❝♦♠ ♣❡rí♦❞♦ q ♣❛r❛ Lα✳ ◆♦t❡ q✉❡
t♦❞♦s ♦s ♣♦♥t♦s ♥❛ ór❜✐t❛ ❞❡ z0 ❡stã♦ ♥♦ ❝ír❝✉❧♦ ❝♦♠ ❝❡♥tr♦ 0 ❡ r❛✐♦ |z0|✱ ❝♦♠♦
♥❛ ✜❣✉r❛ ✶✳
❋✐❣✉r❛ ✶✿ ❉✐♥â♠✐❝❛ ❞❡ Lα.
✸✳✷✲ ◗✉❛♥❞♦r é ✐rr❛❝✐♦♥❛❧✱ ❛ ❞✐♥â♠✐❝❛ ❞❡Lα é ❞✐❢❡r❡♥t❡✳ ◆ã♦ ❡①✐st❡♠ ♣♦♥t♦s ♣❡r✐ó❞✐✲
❝♦s✳ P❛r❛ ♠♦str❛r ✐ss♦✱ ✈❛♠♦s s✉♣♦r q✉❡ ❡①✐st❛♠ ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s✳ ❉❡ss❛ ❢♦r♠❛ ❡①✐st❡ k ∈ N, t❛❧ q✉❡ Lαk(z0) = z0, ♦✉ s❡❥❛✱ reiθ = Lkα(reiθ) = re2πikr+iθ. ❆ss✐♠✱ ♦❜t❡♠♦sθ+ 2πm=i2πkr+θ,♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ r= m
k, q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳
◆❛ ✈❡r❞❛❞❡ ♣♦❞❡♠♦s ♠♦str❛r q✉❡ s❡ r ❢♦r ✐rr❛❝✐♦♥❛❧ ❡ρ= 1, ❛ ór❜✐t❛ ❞❡z0 é ✉♠ s✉❜❝♦♥✲
❥✉♥t♦ ❞❡♥s♦ ❞♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦|z0|✳ ❊st❡ r❡s✉❧t❛❞♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❡ ❏❛❝♦❜✐✳ P❛r❛
❞❡♠♦♥str❛r t❛❧ r❡s✉❧t❛❞♦✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ ❛ ór❜✐t❛ ❞❡z0 ❡♥tr❛ ❡♠ q✉❛❧q✉❡r s✉❜❛r❝♦ ❞❡
r❛✐♦ |z0| ❡ ❝♦♠♣r✐♠❡♥t♦ε✳
P❛r❛ ❡♥❝♦♥tr❛r t❛❧ ♣♦♥t♦✱ ❡s❝♦❧❤❛ k > 2π|z0|
ε ✳ ❖s ♣♦♥t♦s z0, z1, . . . , zk ♣❡rt❡♥❝❡♠ ❛ ❝✐r✲
❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦|z0|❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ sã♦ t♦❞♦s ❞✐st✐♥t♦s✳ ❯♠❛ ✈❡③ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦
❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ é2π|z0|❡
2π|z0|
k < ε✱ s❡❣✉❡ q✉❡✱ ♣❡❧♦ ♠❡♥♦s ❞♦✐s ♣♦♥t♦s ❞❛ ❧✐st❛ ❞❡ ♣♦♥t♦s z′
is ❛♥t❡r✐♦r✱ ❡stã♦ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ♠❡♥♦r ❞♦ q✉❡ ε✳
❙❡❥❛♠ zj ❡ zl, j > l ❞♦✐s ❞❡ss❡s ♣♦♥t♦s ❝✉❥❛ ❞✐stâ♥❝✐❛ é ♠❡♥♦r ❞♦ q✉❡ ε ❡ ❝♦♥s✐❞❡r❡ ❛
❢✉♥çã♦ Lj−l
α ✳ ❚❡♠♦s
Lj−l
α =e
2πir(j−l)z 0,
❡♥tã♦ Lj−l
α s✐♠♣❧❡s♠❡♥t❡ r♦t❛❝✐♦♥❛ ♦s ♣♦♥t♦s ♣♦r ✉♠ â♥❣✉❧♦ 2πr(j −l), ♣♦✐s
Lj−l
α (zl) = Lαj−l(Llα(z0))
= Lj
α(z0) =zj,
♦✉ ❛✐♥❞❛✱ ❡st❛ ❢✉♥çã♦ r♦t❛❝✐♦♥❛ ♣♦♥t♦s ❞♦ ❝ír❝✉❧♦ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ♠❡♥♦r ❞♦ q✉❡ε.
❖s ♣♦♥t♦s Lj−l
α (z0), L
2(j−l)
α (z0), . . . , Ln(j
−l)
α (z0), . . .✱ sã♦ ❛rr✉♠❛❞♦s ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡
t❛❧ ❢♦r♠❛ q✉❡ ♣♦♥t♦s ❝♦♥s❡❝✉t✐✈♦s ♥ã♦ t❡♥❤❛♠ ❞✐stâ♥❝✐❛ ♠❛✐♦r q✉❡ε✳ ❙❡❣✉❡ q✉❡ ❛ ór❜✐t❛ ❞❡ z0 ❞❡✈❡ ❡♥tr❛r ❡♠ ❛❧❣✉♠ s✉❜❛r❝♦ ❝✉❥♦ t❛♠❛♥❤♦ é ♠❡♥♦r q✉❡ ε✳ P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛
ór❜✐t❛ ❞❡ z0 é ❞❡♥s❛✳ ❆ss✐♠✱ ♦❜t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
Pr♦♣♦s✐çã♦ ✶✵✳ ❙✉♣♦♥❤❛ Lα(z) =αz ♦♥❞❡ α=ρe2πir✳
✶✮ ❙❡ ρ <1✱ ❡♥tã♦ t♦❞❛s ❛s ór❜✐t❛s t❡♥❞❡♠ ❛♦ ♣♦♥t♦ ✜①♦ ❛tr❛t♦r 0.
✷✮ ❙❡ρ >1✱ ❡♥tã♦ t♦❞❛s ❛s ór❜✐t❛s t❡♥❞❡♠ ❛♦ ✐♥✜♥✐t♦✱ ❝♦♠ ❛ ❡①❝❡çã♦ ❞♦ 0✱ q✉❡ é ✉♠ ♣♦♥t♦ ✜①♦ r❡♣✉❧s♦r✳
✸✮ ❙❡ ρ= 1✱ ❡♥tã♦✿
❛✮ ❙❡ r é r❛❝✐♦♥❛❧✱ t♦❞❛s ❛s ór❜✐t❛s sã♦ ♣❡r✐ó❞✐❝❛s✳
❜✮ ❙❡ r é ✐rr❛❝✐♦♥❛❧✱ ❝❛❞❛ ór❜✐t❛ é ❞❡♥s❛ ❡♠ ✉♠ ❝ír❝✉❧♦ ❝❡♥tr❛❞♦ ♥❛
♦r✐❣❡♠✳
✹ ❈á❧❝✉❧♦ ❞❡ ❋✉♥çõ❡s ❈♦♠♣❧❡①❛s
P❛r❛ ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ F(z)✱ ❞❡✜♥✐♠♦s ❛ ❞❡r✐✈❛❞❛ F′
(z) ❡①❛t❛♠❡♥t❡ ❝♦♠♦ ♥♦ ❝❛s♦ r❡❛❧✿
F′
(z0) = lim
z→z0
F(z)−F(z0)
z−z0
.
❉❡✜♥✐çã♦ ✶✶✳ ❯♠❛ ❢✉♥çã♦ é ❝❤❛♠❛❞❛ ❛♥❛❧ít✐❝❛ ♥✉♠❛ r❡❣✐ã♦ R✱ s❡ ❡❧❛ é ❞❡r✐✈á✈❡❧ ❡♠ ❝❛❞❛
♣♦♥t♦ ❞❡ R; f é ❛♥❛❧ít✐❝❛ ♥✉♠ ♣♦♥t♦ z0 s❡f é ❛♥❛❧ít✐❝❛ ♥✉♠❛ r❡❣✐ã♦ ❝♦♥t❡♥❞♦z0.
❉❡✜♥✐çã♦ ✶✷✳ ❙❡❥❛F ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ ❝♦♠ ♣♦♥t♦ ✜①♦ ❡♠z0✱ ✐st♦ é✱ F(z0) =z0✳ ❊♥tã♦✿
✶✳ ❖ ♣♦♥t♦ ✜①♦ é ❛tr❛t♦r s❡ |F′(z
0)|<1.
✷✳ ❖ ♣♦♥t♦ ✜①♦ é r❡♣✉❧s♦r s❡ |F′(z
0)|>1.
✸✳ ❖ ♣♦♥t♦ ✜①♦ é ♥❡✉tr♦ s❡ |F′
(z0)|= 1.
❖❜s❡r✈❛çã♦ ✶✸✳
✶✳ ❊①✐st❡ ✉♠❛ ❞✐❢❡r❡♥ç❛ s✐❣♥✐✜❝❛t✐✈❛ ❡♥tr❡ ♦ ❝❛s♦ ❝♦♠♣❧❡①♦ ❡ ♦ ❝❛s♦ r❡❛❧✳ P❛r❛ ♣♦♥t♦s ✜①♦s ♥❡✉tr♦s✱ t❡♠♦s ♥♦✈❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✳ ◆ã♦ s♦♠❡♥t❡ ♣♦❞❡♠♦s t❡r F′
(z0) =±1, ❝♦♠♦ t❛♠❜é♠ F′(z0) =eiθ✳ ❊st❡s t✐♣♦s ❞❡
♣♦♥t♦s ✜①♦s ♥❡✉tr♦s t❡♠ ✈✐③✐♥❤❛♥ç❛ ❝✉❥❛ ❞✐♥â♠✐❝❛ ♣♦❞❡ s❡r ❡①tr❡♠❛♠❡♥t❡ ❝♦♠♣❧✐❝❛❞❛✳
✷✳ ❊st✉❞❛r❡♠♦s ❛ ♥♦çã♦ ❞❡ ❛tr❛çã♦ ❡ r❡♣✉❧sã♦ ♣❛r❛ ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s ❞❡ ♣❡rí♦❞♦ n ❝♦♥s✐❞❡r❛♥❞♦|(Fn)′(z
0)| ❛♦ ✐♥✈és ❞❡ |F′(z0)|✳
❚❡♦r❡♠❛ ✶✹✳ ✭❞♦ P♦♥t♦ ❋✐①♦ ❆tr❛t♦r✮ ❙✉♣♦♥❤❛ q✉❡ z0 ✉♠ ♣♦♥t♦ ✜①♦ ❛tr❛t♦r ♣❛r❛ ❛
❢✉♥çã♦ ❝♦♠♣❧❡①❛ F✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠ ❞✐s❝♦ D ❞❛ ❢♦r♠❛ |z − z0| < δ ❡♠ z0✱ ♥♦ q✉❛❧ ❛
s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦ é s❛t✐s❢❡✐t❛✿ s❡z ∈D✱ ❡♥tã♦Fn(z)∈D✱ ❡ ♠❛✐s ❛✐♥❞❛✱Fn(z)→z
0 q✉❛♥❞♦
n→ ∞✳
❉❡♠♦♥str❛çã♦✳ ❯♠❛ ✈❡③ q✉❡ |F′(z
0)| < 1✱ ❞❡✈❡♠♦s ❡♥❝♦♥tr❛r δ > 0 ❡ µ < 1 t❛✐s q✉❡ s❡
|z−z0|< δ✱ ❡♥tã♦
F(z)−F(z0)
z−z0
= |F(z)−F(z0)|
|z−z0|
< µ <1.
❖ ❞✐s❝♦ D s❡rá ♦ ❞✐s❝♦ ❞❡ r❛✐♦ δ ❝❡♥tr❛❞♦ ❡♠ z0✳ ❆ss✐♠✱|F(z)−z0| =|F(z)−F(z0)|<
µ|z−z0|❡♠D✳ ❉❛í✱ ❝♦♥❝❧✉í♠♦s q✉❡F(z)❡stá ♠❛✐s ♣ró①✐♠♦ ❞❡z0❞♦ q✉❡z✳ ❆ss✐♠✱ ❛♣❧✐❝❛♥❞♦
❡st❡ r❡s✉❧t❛❞♦n ✈❡③❡s✱ ♦❜t❡♠♦s |Fn(z)−z
0|< µn|z−z0|.
▲♦❣♦✱ q✉❛♥❞♦ n→ ∞✱ Fn(z)
→z0✳
❚❡♦r❡♠❛ ✶✺✳ ✭❞♦ P♦♥t♦ ❋✐①♦ ❘❡♣✉❧s♦r✮ ❙❡❥❛ z0 ✉♠ ♣♦♥t♦ ✜①♦ r❡♣✉❧s♦r ♣❛r❛ ❛ ❢✉♥çã♦
❝♦♠♣❧❡①❛ F✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ❞✐s❝♦ D ❞❛ ❢♦r♠❛ |z −z0| < δ, ❝♦♠ ❝❡♥tr♦ ❡♠ z0 ❡♠ q✉❡ ❛
s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦ é s❛t✐s❢❡✐t❛✿ s❡ z ∈D ❡ z 6=z0✱ ❡♥tã♦ ❡①✐st❡ n∈N t❛❧ q✉❡ Fn(z)6∈D.
❉❡♠♦♥str❛çã♦✳ ❆♥á❧♦❣❛ ❛ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳
Pr✐♥❝í♣✐♦ ❞❛ ❆♣❧✐❝❛çã♦ ❞❡ ❋r♦♥t❡✐r❛✿ ❙✉♣♦♥❤❛ q✉❡ R é ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ ♥♦
♣❧❛♥♦ ❡ Qc(z) = z2+c, c ∈ C✳ ❙❡ z0 é ✉♠ ♣♦♥t♦ ✐♥t❡r✐♦r ❞❡ R✱ ❡♥tã♦ Qc(z0) é ✉♠ ♣♦♥t♦
✐♥t❡r✐♦r ❞❡ Qc(R)✳
❖❜s❡r✈❛çã♦ ✶✻✳ ❖ ♣r✐♥❝í♣✐♦ é ♦❜❡❞❡❝✐❞♦ ♣♦r q✉❛❧q✉❡r ❢✉♥çã♦ ❛♥❛❧ít✐❝❛✳
❊①❡♠♣❧♦ ✶✼✳ ❙✉♣♦♥❤❛z0 6= 0✱z0 =r0eiθ0 ❡ r1 < r0 < r2 ❡ θ1 < θ0 < θ2✳ ❈♦♥s✐❞❡r❡ ❛ r❡❣✐ã♦
W ❞❛❞❛ ♣♦r✿
W ={reiθ; r
1 < r0 < r2 ❡θ1 < θ0 < θ2}.
❋✐❣✉r❛ ✷✿ ❘❡❣✐ã♦ W.
◆♦t❡ q✉❡ z0 ❡stá ❝♦♥t✐❞♦ ♥❛ r❡❣✐ã♦ W✳ ❊♠ ❡s♣❡❝✐❛❧✱ s❡ z0 = 0✱ ❛ r❡❣✐ã♦ é ✉♠❛ s❡çã♦ ❞❡
✉♠ ♣❡q✉❡♥♦ ❞✐s❝♦ ❝❡♥tr❛❞♦ ❡♠ 0✳ ❆ ❢✉♥çã♦ Q0(z) =z2✱ ❧❡✈❛ ❛ r❡❣✐ã♦ W ♥❛ r❡❣✐ã♦ Q0(W)
❞❛❞❛ ♣♦r✿
Q0(W) = {reiθ;r12 < r < r22 ❡2θ1 < θ <2θ2}.
❋✐❣✉r❛ ✸✿ Q0 r♦t❛❝✐♦♥❛ ❡ ❡①♣❛♥❞❡ ❛ r❡❣✐ã♦ W.
◆♦ ❝❛s♦ c6= 0✱ Qc ❧❡✈❛ r❡❣✐õ❡s ❞♦ t✐♣♦ W ❡♠ r❡❣✐õ❡s✱ ❛s q✉❛✐s sã♦ tr❛♥s❧❛çõ❡s ❞❡Q0(W)
♣❡❧♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ c✳
✺ ❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛
❯♠ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s✱ ❣❡r❛❞♦s ♣❡❧❛ ✐t❡r❛çã♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ ❞♦ t✐♣♦ f : C → C✱ ♦♥❞❡ C r❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ◆❛ ✈❡r❞❛❞❡✱ ♦ ❝♦rr❡t♦ s❡r✐❛ ❝♦♥❥✉♥t♦s ❞❡ ❋❛t♦✉✲❏✉❧✐❛✱ ♣♦✐s ❢♦r❛♠ ♦s ❞♦✐s ♠❛t❡♠át✐❝♦s ❢r❛♥❝❡s❡s P✐❡rr❡ ❋❛t♦✉ ❡ ●❛st♦♥ ❏✉❧✐❛ q✉❡ ✐♥tr♦❞✉③✐r❛♠ ♦s ♠ét♦❞♦s ✐t❡r❛t✐✈♦s ♥♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s ♣❛r❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❛ ❣❡♦♠❡tr✐❛ ❢r❛❝t❛❧✳
❱❡r❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ é ♦ ❧✉❣❛r ♦♥❞❡ t♦❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❝❛ót✐❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ ♦❝♦rr❡✳ ◆❡st❡ ♣♦♥t♦✱ r❡❝♦♠❡♥❞❛♠♦s ❛♦s ❧❡✐t♦r❡s ♥ã♦ ❢❛♠✐❧✐❛r✐③❛❞♦s ❝♦♠ ♦s ❝♦♥❝❡✐t♦s ❞❡ s✐st❡♠❛s ❝❛ót✐❝♦s✱ ✉♠❛ ❧❡✐t✉r❛ ♣r❡❧✐♠✐♥❛r ❞❡ ❬P❘❊❈■❖❙❖ ❡t ❛❧✳ ✲ ✷✵✵✾❪✳ ❆ s❡❣✉✐r✱ ❝♦♥s✐❞❡r❛r❡♠♦sQc(z) = z2+c, ♦♥❞❡ z ❡ c sã♦ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳
✻ ❆ ❉✐♥â♠✐❝❛ ❞❛ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛
❙❡❥❛♠ Q0(z) =z2 ❡ z0 =reiθ✳ ❆ ór❜✐t❛ ❞❡z0 é ❞❛❞❛ ♣♦r✿
z0 = reiθ
z1 = r2ei2θ
z2 = r4ei4θ
✳✳✳
zn = r2
n
ei2nθ
❚❡♠♦s ✸ ❝❛s♦s✿ ✶✳ ❙❡r <1,❡♥tã♦ r2n
→0 q✉❛♥❞♦ n→ ∞.■st♦ é✱|Qn
0(z0)| →0q✉❛♥❞♦ n→ ∞✳ ❖❜s❡r✈❡
q✉❡Q0(0) = 0✱ ♦✉ s❡❥❛✱0é ♣♦♥t♦ ✜①♦ ❞❡Q0 ❡Q′0(0) = limz→0
Q0(z)−Q0(0)
z−0 = limz→0
z2
z = 0✳
▲♦❣♦ z = 0 é ♣♦♥t♦ ✜①♦ ❛tr❛t♦r✳ ✷✳ ❙❡ r >1,❡♥tã♦ |Qn
0(z0)| → ∞ q✉❛♥❞♦ n→ ∞.
✸✳ ❙❡ r = 1, ❡♥tã♦ q✉❛♥❞♦ |z0| = 1✱ |Q0(z0)| = 1. ❉❛í✱ Q0 ♠❛♥té♠ t♦❞❛s ❛s ✐♠❛❣❡♥s ❞❡
♣♦♥t♦s ❞♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦ ♥♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦✳
◆♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦✱ ❛ ❛♣❧✐❝❛çã♦ q✉❛❞rát✐❝❛ ❡q✉✐✈❛❧❡ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ❞✉♣❧✐❝❛çã♦✳ ❉❛❞♦
z0 = eiθ ♥♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦✱ t❡♠♦s Q0(z0) = ei2θ✱ ❡♥tã♦ Q0 s✐♠♣❧❡s♠❡♥t❡ ❞♦❜r❛ ♦ ❛r✲
❣✉♠❡♥t♦ θ✳ ❈♦♠♦ ♥♦ ❝❛s♦ r❡❛❧✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ Q0 é ❝❛ót✐❝❛ ♥♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦ ✭✈❡r
❬P❘❊❈■❖❙❖ ❡t ❛❧✳ ✲ ✷✵✵✾❪✮✳ ❱❛♠♦s ♠♦str❛r q✉❡ ♦s ♣♦♥t♦s ❝✉❥❛ ♥♦r♠❛ é1 sã♦ ❝❛ót✐❝♦s✳ P❛r❛ ♠♦str❛r q✉❡ ♦s ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s sã♦ ❞❡♥s♦s✱ ❞❡✈❡♠♦s ♣r♦❞✉③✐r ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❡♠ q✉❛❧q✉❡r ❛r❝♦ ❞❛ ❢♦r♠❛ θ1 < θ < θ2✱ ✐st♦ é✱ ♣r❡❝✐s❛♠♦s ❡♥❝♦♥tr❛r n ❡ θ,t❛✐s q✉❡
Qn
0(eiθ) = eiθ, θ1 < θ < θ2.
▼❛sQn
0(eiθ) = ei2
nθ
,♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ei2nθ
=eiθ,♦✉ s❡❥❛✱2nθ =θ+2kπ, ♣❛r❛ ❛❧❣✉♠ k, n.
❆ss✐♠✱
θ = 2kπ
2n−2, k, n∈Z.
❙❡ ✜①❛r♠♦s n ❡ t♦♠❛r♠♦s k, t❛❧ q✉❡ 0 ≤ k < 2n
− 1, ❡♥tã♦ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s
❝♦♠ ❛r❣✉♠❡♥t♦s 2kπ
2n−1 ❡stã♦ ❞✐str✐❜✉í❞♦s ✉♥✐❢♦r♠❡♠❡♥t❡ ❛♦ ❧♦♥❣♦ ❞♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦ ❝♦♠
❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ 2π
2n−1 ❡♥tr❡ ♣♦♥t♦s ❝♦♥s❡❝✉t✐✈♦s✱ ✐st♦ s❡❣✉❡ ❞♦ ❢❛t♦ q✉❡ ❡st❡s ♣♦♥t♦s
sã♦ ❛s(2n
−1)′s r❛í③❡s ❞❛ ✉♥✐❞❛❞❡✳
❋✐❣✉r❛ ✹✿ P♦♥t♦s ♥♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦ ❝♦♠ ❛r❣✉♠❡♥t♦sθ= 2kπ 23−1.
❆❣♦r❛✱ ❛ ❡s❝♦❧❤❛ ❞❡ n, t❛❧ q✉❡
2π
2n−1 < θ2−θ1
❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ ❝♦♠ ❛r❣✉♠❡♥t♦ 2kπ
2n−1❡♥tr❡ θ1 ❡ θ2✳ ❈♦♠♦ n
❢♦✐ ♦❜t✐❞♦ r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦Qn
0(eiθ) = eiθ, s❡q✉❡ q✉❡z0 é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❞❡ ♣❡rí♦❞♦
n.
▼♦str❡♠♦s ❛❣♦r❛ ❛ tr❛♥s✐t✐✈✐❞❛❞❡✳ ◗✉❛❧q✉❡r ❛r❝♦ ❝♦♠ θ1 < θ < θ2, ♥♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦
t❡♠ ❝♦♠♣r✐♠❡♥t♦ ❛✉♠❡♥t❛❞♦ ♣❡❧❛Qn
0 ♣♦r ✉♠ ❢❛t♦r ❞❡2n✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r
n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ ♣❛r❛ q✉❡ ❛ ✐♠❛❣❡♠ ❞♦ ❛r❝♦ ❝♦♠ θ1 < θ < θ2, s♦❜ Qn0, ❝✉❜r❛ t♦❞♦
♦ ❝ír❝✉❧♦✳
■st♦ t❛♠❜é♠ ♠♦str❛ ❛ ❞❡♣❡♥❞ê♥❝✐❛ s❡♥s✐t✐✈❛✱ ✉♠❛ ✈❡③ q✉❡ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ♣♦♥t♦s ♣ró①✐♠♦s q✉❡ sã♦ ❡✈❡♥t✉❛❧♠❡♥t❡ ❧❡✈❛❞♦s ❡♠ ♣♦♥t♦s ❞✐❛♠❡tr❛❧♠❡♥t❡ ♦♣♦st♦s ♥♦ ❝ír❝✉❧♦✳ ❊st❛ ❛♥á❧✐s❡ ♣♦❞❡ s❡r r❡s✉♠✐❞❛ ♥♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
❚❡♦r❡♠❛ ✶✽✳ ❆ ❛♣❧✐❝❛çã♦ q✉❛❞rát✐❝❛ Q0(z) = z2 é ❝❛ót✐❝❛ ♥♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦✳ ❙❡ |z| <1,
❡♥tã♦ |Qn
0(z)| →0, q✉❛♥❞♦ n→ ∞ ❡ s❡ |z|>1, ❡♥tã♦ |Qn0(z)| → ∞, q✉❛♥❞♦ n → ∞✳
❆ ❛♣❧✐❝❛çã♦ q✉❛❞rát✐❝❛ é ❡①tr❡♠❛♠❡♥t❡ s❡♥s✐t✐✈❛ às ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ♥♦ s❡❣✉✐♥t❡ s❡♥t✐❞♦✳ ❙❡❥❛ z0 ✉♠ ♣♦♥t♦ ♥♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦✱ ❞❛❞❛ q✉❛❧q✉❡r ❜♦❧❛ ❛❜❡rt❛ ❝♦♠ ❝❡♥tr♦ ❡♠ z0 s❡♠♣r❡
❝♦♥s❡❣✉✐♠♦s ❡♥❝♦♥tr❛r
W ={reiθ; r
1 < r < r2 ❡θ1 < θ < θ2},
❝♦♠ r1 <1< r2,❝♦♥t✐❞♦ ♥❛ ❜♦❧❛✳
❆ ✐♠❛❣❡♠ ❞❡ W é ❞❛❞❛ ♣♦r
Q0(W) = {reiθ; r12 < r < r22 ❡2θ1 < θ <2θ2}.
❯♠❛ ✈❡③ q✉❡r1 <1< r2✱ s❡❣✉❡ q✉❡r12 < r1 < r < r2 < r22✳ ❆ss✐♠✱ ❛ r❡❣✐ã♦ ❛❜r❛♥❣✐❞❛ ♣♦r
Q0(W)é ♠❛✐♦r q✉❡ ❛ ❞❡W✳ ❈♦♥t✐♥✉❛♥❞♦ ❡st❡ ♣r♦❝❡ss♦ ✈❡r❡♠♦s q✉❡Qn0(W)❝r❡s❝❡ ❝♦♥❢♦r♠❡
n ❝r❡s❝❡✳
❊♠ ❛❧❣✉♠ ♠♦♠❡♥t♦ ♦ â♥❣✉❧♦ ♣♦❧❛rθ ❞❡Qn
0(W)❡①❝❡❞❡2π✳ ▲♦❣♦✱ ♣❛r❛n s✉✜❝✐❡♥t❡♠❡♥t❡
❣r❛♥❞❡ t❡r❡♠♦s
r21n < r < r22n.
▼❛✐s ❛✐♥❞❛✱ ❛♣❧✐❝❛çõ❡s ❞❡ Q0 ♠♦str❛♠ q✉❡ ♦ r❛✐♦ ✐♥t❡r♥♦ ❞❡Qn0(W) t❡♥❞❡ ❛0, ❡♥q✉❛♥t♦
♦ r❛✐♦ ❡①t❡r♥♦ t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦✳ ❆ss✐♠✱ ♦❜t❡♠♦s ∞
[
n=0
Qn0(W) = C− {0}.
■st♦ é✱ ❛s ór❜✐t❛s ❞♦s ♣♦♥t♦s ❡♠W ❛❧❝❛♥ç❛♠ q✉❛❧q✉❡r ♣♦♥t♦ ❞❡C✱ ❡①❝❡t♦ ♦0q✉❡ é ♣♦♥t♦ ✜①♦✳ ❊st❛ é ❛ ❞❡♣❡♥❞ê♥❝✐❛ s❡♥s✐t✐✈❛ ❡①tr❡♠❛✿ ❛ ✉♠❛ ♣r♦①✐♠✐❞❛❞❡ ❛r❜✐trár✐❛ ❞❡ q✉❛❧q✉❡r ♣♦♥t♦ ❞♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦✱ ❡①✐st❡ ✉♠ ♣♦♥t♦✱ ❝✉❥❛ ór❜✐t❛✱ ✐♥❝❧✉✐ q✉❛❧q✉❡r ♦✉tr♦ ♣♦♥t♦ ❞♦ ♣❧❛♥♦✱ ❡①❝❡t♦ ❛ ♦r✐❣❡♠✳
❉❡✜♥✐çã♦ ✶✾✳ ❆ ór❜✐t❛ ❞❡ z s♦❜ Qc é ❧✐♠✐t❛❞❛✱ s❡ ❡①✐st❡ k, t❛❧ q✉❡ |Qnc(z)| < k, ♣❛r❛ t♦❞♦
n✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❛ ór❜✐t❛ ♥ã♦ é ❧✐♠✐t❛❞❛✳
❊①❡♠♣❧♦ ✷✵✳ ◆♦ ❝❛s♦ Q0✱ s♦♠❡♥t❡ ♦s ♣♦♥t♦s ✐♥t❡r✐♦r❡s ❡ ♥♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦✱ t❡♠ ór❜✐t❛s
❧✐♠✐t❛❞❛s✳
❉❡✜♥✐çã♦ ✷✶✳ ❆ ór❜✐t❛ ❞❡z s♦❜Qc é s✉♣❡rs❡♥s✐t✐✈❛✱ s❡ q✉❛❧q✉❡r ❜♦❧❛ ❛❜❡rt❛ B ❝❡♥tr❛❞❛ ❡♠
z t❡♠ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿
∞ [
n=0
Qn
c(B)é t♦❞♦ C, ❝♦♠ ❛ ❡①❝❡çã♦ ❞❡ ♥♦ ♠á①✐♠♦ ✉♠ ♣♦♥t♦✳
❊①❡♠♣❧♦ ✷✷✳ ◆♦ ❝❛s♦Q0, ♦s ♣♦♥t♦s ❞♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦ t❡♠ ór❜✐t❛s s✉♣❡rs❡♥s✐t✐✈❛s✳
❉❡✜♥✐çã♦ ✷✸✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡ Qc é ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❝✉❥❛s ór❜✐t❛s
sã♦ ❧✐♠✐t❛❞❛s✳ ❖ ❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❡Qc é ❛ ❢r♦♥t❡✐r❛ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦✳
◆♦t❛çã♦✿ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ é ❞❡♥♦t❛❞♦ ♣♦r Kc ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣♦r
Jc✳
❊①❡♠♣❧♦ ✷✹✳ ◆♦ ❝❛s♦ Q0, ♦❜t❡♠♦s K0 = {z; |z| ≤ 1} ❡ J0 = {z; |z| = 1}. ◆♦t❡ q✉❡ ♦s
♣♦♥t♦s ❡♠ J0 sã♦ ❛q✉❡❧❡s ♥♦s q✉❛✐s ❛s ór❜✐t❛s sã♦ s✉♣❡rs❡♥s✐t✐✈❛s ❡ q✉❡Q0 é ❝❛ót✐❝❛ ❡♠ J0.
✼ ❆ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ❈❛ót✐❝❛
❚❡♦r❡♠❛ ✷✺✳ ❆ ❢✉♥çã♦ q✉❛❞rát✐❝❛ Q−2(z) = z2 −2 ❡♠ C−[−2,2] é ❝♦♥❥✉❣❛❞❛ ❞❛ ❢✉♥çã♦
q✉❛❞rát✐❝❛ Q0(z) = z2 ❡♠ {z; |z| > 1}✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s❡ z ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✈❛❧♦
❢❡❝❤❛❞♦ [−2,2]✱ ❡♥tã♦ ❛ ór❜✐t❛ ❞❡ z s♦❜ Q−2 t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦✳
❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ H(z) = z+ 1
z ❞❡✜♥✐❞❛ ♥❛ r❡❣✐ã♦ R= {z; |z| >1}✳ H é
✐♥❥❡t♦r❛ ❡♠ R, ♣♦✐s s❡ H(z) =H(w) s❡❣✉❡ q✉❡
z+ 1
z =w+
1
w
❡♥tã♦
z =w✱ ♦✉ zw = 1.
❆❣♦r❛ ♦❜s❡r✈❡ q✉❡✱ s❡ |z| >1 ❡ zw = 1✱ ❡♥tã♦ t❡♠♦s q✉❡ |w| = 1
|z| <1✱ ♦✉ s❡❥❛✱ w6∈ R✳
P♦rt❛♥t♦✱ H é ✐♥❥❡t♦r❛ ❡♠R.
❚❛♠❜é♠✱H❧❡✈❛R❡♠C−[−2,2]✳ P❛r❛ ✈❡r ✐ss♦✱ ❡s❝♦❧❤❡♠♦sw∈C−[−2,2]❡ r❡s♦❧✈❡♠♦s
H(z) = w ♣❛r❛ ♦❜t❡r
z±=
w±√w2−4
2 .
❈♦♠♦z+z− = 1✱ s❡❣✉❡ q✉❡z+♦✉z−❡stá ❡♠R,♦✉ ❛♠❜♦s ❡stã♦ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✉♥✐tár✐❛✳ ◆❡st❡ ú❧t✐♠♦ ❝❛s♦✱ ♣♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ w=H(z+) =H(z−)∈[−2,2]✳
▲♦❣♦✱ H é ❜✐❥❡t♦r❛ ❡♠ C−[−2,2]✳ ❋✐♥❛❧♠❡♥t❡✱
H(Q0(z)) = z2+
1
z2
= z2+ 2 + 1
z2 −2
=
z+1
z
2
−2
= Q−2(H(z)), ∀z
P♦rt❛♥t♦ Q−2 ❡♠ C−[−2,2] é ❝♦♥❥✉❣❛çã♦ ❞❡ Q0 ❡♠ R✳ ❯♠❛ ✈❡③ q✉❡ t♦❞❛s ❛s ór❜✐t❛s
❞❡ Q0 t❡♥❞❡♠ ❛♦ ✐♥✜♥✐t♦ ❡♠ R✱ s❡❣✉❡ q✉❡ t♦❞❛s ❛s ór❜✐t❛s ❞❡ Q−2 t❡♥❞❡♠ ❛♦ ✐♥✜♥✐t♦ ❡♠
C−[−2,2]✳ ■st♦ ❝♦♠♣❧❡t❛ ❛ ♣r♦✈❛✳ ❈♦r♦❧ár✐♦ ✷✻✳ J−2 =K−2 = [−2,2]✳
❉❡♠♦♥str❛çã♦✳ ❖ ❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ é ♦ ✐♥t❡r✈❛❧♦ [−2,2] ♥❛ r❡t❛ r❡❛❧✳ ❙✉❛ ❢r♦♥t❡✐r❛ ♥♦ ♣❧❛♥♦ ❞❡ ❆r❣❛♥❞✲●❛✉ss é ❡❧❡ ♣ró♣r✐♦✱ ❧♦❣♦ J−2 =K−2✳
❯s❛♥❞♦ ❛ ❝♦♥❥✉❣❛çã♦ H✱ t❛♠❜é♠ ♣♦❞❡♠♦s ✈❡r q✉❡ ❛ ór❜✐t❛ ❞❡ t♦❞♦ ♣♦♥t♦ ❡♠ J−2 é
s✉♣❡rs❡♥s✐t✐✈❛✳ ▼❛✐s ❛✐♥❞❛✱ s❡ ❝♦♥s✐❞❡r❛r♠♦s ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ ❞❡ q✉❛❧q✉❡r ❜♦❧❛ ❛❜❡rt❛ s♦❜r❡ ✉♠ ♣♦♥t♦ ❡♠J−2 s♦❜H✱ s❛❜❡♠♦s q✉❡ ❛ ❢✉♥çã♦Q0 ❧❡✈❛ ❡st❡ ❝♦♥❥✉♥t♦ ❡♠ t♦❞♦ ♦ ♣❧❛♥♦✳
❊♥tã♦✱ ✈✐❛ ❝♦♥❥✉❣❛çã♦✱ ♦ ♠❡s♠♦ ❛❝♦♥t❡❝❡ ❝♦♠ Q−2✳
✽ ❖ ❈♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r
◆❡st❛ s❡çã♦✱ ❝♦♥s✐❞❡r❛r❡♠♦s ❛ ❢✉♥çã♦ Qc(z) = z2+c, q✉❛♥❞♦|c| > 2✳ ▲❡♠❜r❡♠♦s q✉❡✱
♣❛r❛ ❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ r❡❛❧Qc(x) = x2+c, s❡c <−2✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r
✐♥✈❛r✐❛♥t❡✱ ♥♦ q✉❛❧ Qc é ❝❛ót✐❝♦ ✭✈❡r ❬P❘❊❈■❖❙❖ ❡t ❛❧✳ ✲ ✷✵✵✾❪✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱ q✉❛♥❞♦
c >2✱ t♦❞❛s ❛s ór❜✐t❛s ❞❡ Qc ♥❛ r❡t❛✱ sã♦ ♠✉✐t♦ ❢á❝❡✐s ❞❡ ❡♥t❡♥❞❡r✱ ♣♦✐s t❡♥❞❡♠ ❛♦ ✐♥✜♥✐t♦✳
❱❡r❡♠♦s q✉❡ ❤á ❛❧❣♦ s❡♠❡❧❤❛♥t❡ ♥♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✳
❚❡♦r❡♠❛ ✷✼✳ ❙✉♣♦♥❤❛ |c| > 2✳ ❊♥tã♦ Jc = Kc é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r✳ ▼❛✐s ❛✐♥❞❛✱ ❛
❛♣❧✐❝❛çã♦ Qc✱ q✉❛♥❞♦ r❡str✐t❛ ❛ Jc✱ é ❝♦♥❥✉❣❛❞❛ ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ♠✉❞❛♥ç❛ ❡♠ ❞♦✐s sí♠❜♦❧♦s✳
❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ s❡rá ♦♠✐t✐❞❛✱ ✉♠❛ ✈❡③ q✉❡ ❡♥✈♦❧✈❡ tó♣✐❝♦s ❛✈❛♥ç❛❞♦s ❞❡ ❛♥á❧✐s❡ ❝♦♠♣❧❡①❛✳ ◆♦ ❡♥t❛♥t♦✱ ♠♦str❛r❡♠♦s ❝♦♠♦ ❢✉♥❝✐♦♥❛ ❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r ♥♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✳
◆♦ss❛ ♣r✐♠❡✐r❛ ♦❜s❡r✈❛çã♦ é q✉❡ ♦ ❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡ Qc, q✉❛♥❞♦ |c| >2,
❡stá ✐♥t❡✐r❛♠❡♥t❡ ❝♦♥t✐❞♦ ♥♦ ❞✐s❝♦|z|<|c|✳
❚❡♦r❡♠❛ ✷✽✳ ✭❖ ❈r✐tér✐♦ ❞❡ ❊s❝❛♣❡✮ ❙✉♣♦♥❤❛|z| ≥ |c|>2✳ ❊♥tã♦✱ t❡♠♦s|Qn
c(z)| → ∞,
q✉❛♥❞♦ n → ∞✳
❉❡♠♦♥str❛çã♦✳ P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ t❡♠♦s✿
|Qc(z)| ≥ |z|2− |c|
≥ |z|2− |z|
= |z|(|z| −1).
❈♦♠♦ |z|>2✱ ❡①✐st❡ λ >0,t❛❧ q✉❡ |z| −1>1 +λ✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱
|Qc(z)|>(1 +λ)|z|.
❊♠ ♣❛rt✐❝✉❧❛r✱ |Qc(z)| > |z| ❡✱ ♣♦rt❛♥t♦✱ ❛♣❧✐❝❛♥❞♦ r❡♣❡t✐❞❛♠❡♥t❡ ♦ ❛r❣✉♠❡♥t♦ ❛❝✐♠❛✱
♦❜t❡♠♦s
|Qnc(z)|>(1 +λ) n
|z|.
❆ss✐♠✱ ❛s ór❜✐t❛s ❞❡ z t❡♥❞❡♠ ❛♦ ✐♥✜♥✐t♦ ❡ ✐st♦ ❝♦♠♣❧❡t❛ ❛ ❞❡♠♦♥str❛çã♦✳
❈♦r♦❧ár✐♦ ✷✾✳ ❙✉♣♦♥❤❛ |c|>2✳ ❊♥tã♦ ❛ ór❜✐t❛ ❞❡ 0 ❡s❝❛♣❛ ♣❛r❛ ♦ ✐♥✜♥✐t♦ s♦❜ Qc✳
❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡✱ s❡ |c| > 2✱ ❡♥tã♦ |Qc(0)| = |c| > 2✳ ▲♦❣♦✱ ❛ ór❜✐t❛ ❞❡ 0✱ ♦ ♣♦♥t♦
❝rít✐❝♦✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❡s❝❛♣❛ ♣❛r❛ ♦ ✐♥✜♥✐t♦✳ P♦❞❡♠♦s ❡♥tã♦✱ r❡✜♥❛r ♦ ❝r✐tér✐♦ ❞❡ ❡s❝❛♣❡✳
❈♦r♦❧ár✐♦ ✸✵✳ ❙✉♣♦♥❤❛ |z| > ♠❛①{|c|,2}✳ ❊♥tã♦✱ |Qn
c(z)| > (1 +λ)n|z| ❡ |Qnc(z)| → ∞,
q✉❛♥❞♦ n → ∞.
◆♦t❡ q✉❡ s❡ |Qk
c(z)|>♠❛①{|c|,2}✱ ♣❛r❛ ❛❧❣✉♠ k ≥0, ❡♥tã♦ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ❝♦r♦❧ár✐♦
❛♥t❡r✐♦r ❛Qk
c ♣❛r❛ ♦❜t❡r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳
❈♦r♦❧ár✐♦ ✸✶✳ ❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❛❧❣✉♠k ≥0 t❡♠♦s |Qk
c(z)|>♠❛①{|c|,2}✳ ❊♥tã♦ |Qkc+1|>
(1 +λ)|Qk
c(z)| ❡ |Qcn(z)| → ∞, q✉❛♥❞♦ n → ∞✳
❊st❡ ❝♦r♦❧ár✐♦ ♥♦s ❞á ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❝♦♠♣✉t❛r ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡ Qc
♣❛r❛ q✉❛❧q✉❡r c✳ ❉❛❞♦ ✉♠ ♣♦♥t♦ z s❛t✐s❢❛③❡♥❞♦ |z| ≤ |c|, ❝♦♠♣✉t❛♠♦s ❛ ór❜✐t❛ ❞❡ z✳ ❙❡✱
♣❛r❛ ❛❧❣✉♠ n✱ Qn
c(z) s❛✐ ❞♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ ♠❛①{|c|,2}✱ ❣❛r❛♥t✐♠♦s q✉❡ s✉❛ ór❜✐t❛ ❡s❝❛♣❛
❡✱ ♣♦rt❛♥t♦✱ z ♥ã♦ ❡stá ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ |Qn
c(z)| ♥✉♥❝❛
❡①❝❡❞❡ ❡st❛ ❢r♦♥t❡✐r❛✱ ❡♥tã♦✱ ♣♦r ❞❡✜♥✐çã♦✱ z ❡stá ❡♠Kc✳
❱❛♠♦s ✈♦❧t❛r ❛ ♥♦ss❛ ❛t❡♥çã♦ ♣❛r❛ ♦ ❝❛s♦ |c| >2✳ ❙❡❥❛ D ♦ ❞✐s❝♦ ❢❡❝❤❛❞♦ {z :|z| ≤ |c|}✳
❖ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡Qc é ❞❛❞♦ ♣♦r
\
n≥0
Q−n
c (D),♦♥❞❡ Q
−n
c (D) é ❛ ♣ré✲✐♠❛❣❡♠
❞❡D s♦❜ Qn
c✱ ✐st♦ é✱
Q−n
c (D) ={z :Qnc(z)∈D}.
■st♦ ❞❡✈❡✲s❡ ❛♦ ❢❛t♦ q✉❡✱ s❡z 6∈ \
n≥0
Q−n
c (D)✱ ❡♥tã♦∃k≥0,t❛❧ q✉❡ Qkc(z)6∈D❡✱ ♣♦rt❛♥t♦✱
♣❡❧♦ ❈♦r♦❧ár✐♦ ✸✶✱ ❛ ór❜✐t❛ ❞❡ z t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦✳ P♦rt❛♥t♦✱ ❜❛st❛ ❡♥t❡♥❞❡r ❛ ✐♥t❡rs❡❝çã♦
✐♥✜♥✐t❛ \
n≥0
Q−n
c (D), ♣❛r❛ ❡♥t❡♥❞❡rKc✳
P❛r❛ ✐ss♦✱ t♦♠❡♠♦s C ❞❡♥♦t❛♥❞♦ ♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ |c|,❝❡♥tr❛❞♦ ♥❛ ♦r✐❣❡♠✳ ◆♦t❡ q✉❡ C é
❛ ❢r♦♥t❡✐r❛ ❞❡D✳ ❆ ♣r✐♠❡✐r❛ q✉❡stã♦ é✿ ❝♦♠♦ é Q−1
c (C)? ❖❜t❡♠♦s ❡st❡ ❝♦♥❥✉♥t♦✱ s✉❜tr❛✐♥❞♦
c ❞❡ t♦❞♦ ♣♦♥t♦ ❡♠ C✳ ■st♦ t❡♠ ♦ ❡❢❡✐t♦ ❞❡ tr❛♥s❧❛çã♦ ❞♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ♣❛r❛
♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ ❡♠ −c✳ ❉❡♣♦✐s ❡①tr❛í♠♦s ❛ r❛í③ q✉❛❞r❛❞❛ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s ❞❡st❡ ♥♦✈♦
❝ír❝✉❧♦ ♦ q✉❡ ♥♦s ❢♦r♥❡❝❡ ✉♠❛ ❝✉r✈❛ ✏♦✐t♦✑✱ ✈❡r ✜❣✉r❛ ✺✳ ◆♦t❡ q✉❡Q−1
c (z)❡stá ❝♦♠♣❧❡t❛♠❡♥t❡
❝♦♥t✐❞❛ ♥♦ ✐♥t❡r✐♦r ❞❡ D✱ ♣♦✐s ❥á s❛❜❡♠♦s q✉❡ q✉❛❧q✉❡r ♣♦♥t♦ s♦❜r❡ ♦✉ ❢♦r❛ ❞❡ C é ❧❡✈❛❞♦
❧♦♥❣❡ ❞❛ ♦r✐❣❡♠ ♣♦r Qc✳ P❡❧♦ ♣r✐♥❝í♣✐♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ❢r♦♥t❡✐r❛✱ Q−c1(D) é ♣r❡❝✐s❛♠❡♥t❡ ❛
✜❣✉r❛ ♦✐t♦✱ ❥✉♥t♦ ❝♦♠ s❡✉s ❞♦✐s ✏❧ó❜✉❧♦s✑ ✐♥t❡r✐♦r❡s q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r I0 ❡ I1✳ ◆♦t❡ q✉❡
I0 ❡ I1 sã♦ s✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦ ❛ ♦r✐❣❡♠ ❡ q✉❡ Qc ❧❡✈❛ ❝❛❞❛ ✉♠ ❞❡❧❡s ❞❡ ✈♦❧t❛ ❡♠ D ❞❡
♠❛♥❡✐r❛ ❜✐❥❡t♦r❛✳
❋✐❣✉r❛ ✺✿ ❈♦♥str✉çã♦ ❞❡ I0 ❡ I1.
❈♦♠♦Qc é ✐♥❥❡t♦r❛ ❡♠I0 ❡I1✱ ❡Qc−1(C)❡stá ❝♦♥t✐❞❛ ♥♦ ✐♥t❡r✐♦r ❞❡D✱ s❡❣✉❡ q✉❡ Q−c2(C)
é ✉♠ ♣❛r ❞❡ ✜❣✉r❛s ♦✐t♦ ♠❡♥♦r❡s✱ ✉♠❛ ❝♦♥t✐❞❛ ❡♠I0 ❡ ❛ ♦✉tr❛ ❝♦♥t✐❞❛ ❡♠ I1✳ ◆♦✈❛♠❡♥t❡✱
✉t✐❧✐③❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ❢r♦♥t❡✐r❛✱ ♦❜t❡♠♦s q✉❡Q−2
c (D)❝♦♥s✐st❡ ❞❡ ❞✉❛s ✜❣✉r❛s
♦✐t♦ ❥✉♥t❛♠❡♥t❡ ❝♦♠ s❡✉s q✉❛tr♦ ❧ó❜✉❧♦s✳ ❉❡✜♥✐♠♦s✿
I00 = {z ∈I0/Qc(z)∈I0}
I01 = {z ∈I0/Qc(z)∈I1}
I10 = {z ∈I1/Qc(z)∈I0}
I11 = {z ∈I1/Qc(z)∈I1}
❆ss✐♠✱ Q−2
c (D) é ❝♦♥st✐t✉í❞❛ ♣♦r ✉♠❛ ✜❣✉r❛ ♦✐t♦ ❡♠I0 ❥✉♥t❛♠❡♥t❡ ❝♦♠ s❡✉s ❧ó❜✉❧♦s I00
❡I01 ❡ ♣♦r ♦✉tr❛ ✜❣✉r❛ ♦✐t♦ ❡♠ I1 ❝♦♠ ❧ó❜✉❧♦s I10 ❡ I11✱ ✈❡❥❛ ✜❣✉r❛ ✻✳
❋✐❣✉r❛ ✻✿ ❈♦♥str✉çã♦ ❞❡ I00✱ I01, I10 ❡I11.
❖❜s❡r✈❡ q✉❡ Q−n
c (D) ❝♦♥s✐st❡ ❞❡ 2n
−1 ✜❣✉r❛s ♦✐t♦ ❝♦♠ 2n ❧ó❜✉❧♦s ❧✐♠✐t❛❞♦s ♣♦r ❡st❛s
❝✉r✈❛s✳ ❙❡❥❛s0s1. . . sn ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ 0✬s ❡ 1✬s✳ ❉❡✜♥✐♠♦s
Is0s1...sn ={z ∈D/ z ∈Is0, Qc(z)∈Is1, . . . , Q
n
c(z)∈Isn}.
❯t✐❧✐③❛♥❞♦✲s❡ ❛r❣✉♠❡♥t♦s s✐♠✐❧❛r❡s àq✉❡❧❡s ❛♣❧✐❝❛❞♦s ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❞❛ ❈♦♥❥✉❣❛çã♦ ✭✈❡r ❬P❘❊❈■❖❙❖ ❡t ❛❧✳ ✲ ✷✵✵✾❪✮✱ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡Is0s1...sn é ❢❡❝❤❛❞♦ ❡ ❡stá ❝♦♥t✐❞♦ ♥♦ ✐♥t❡r✐♦r ❞❡ Is0s1...sn−1✳ ❆ss✐♠✱ ♦s Is0s1...sn ❢♦r♠❛♠ ✉♠❛ ✐♥t❡rs❡❝çã♦ ❞❡ ❝♦♥❥✉♥t♦s
❢❡❝❤❛❞♦s✱
\
n≥0
Is0s1...sn,
❛ q✉❛❧ é ♥ã♦ ✈❛③✐❛✳ ❉❛í✱ s❡z ∈ \
n≥0
Is0s1...sn✱ ❡♥tã♦ Q
k
c(z)∈D ♣❛r❛ t♦❞♦k✳
P♦rt❛♥t♦✱ z ∈ Kc✳ ❯♠❛ ✐♥t❡s❡çã♦ ✐♥✜♥✐t❛ ❞❛s ✜❣✉r❛s ♦✐t♦ ❡ s❡✉s ❧ó❜✉❧♦s é ❞❛❞❛ ♣♦r
\
n≥0
Is0s1...sn ❡ é ❝❤❛♠❛❞❛ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❞❡ Kc✳ ◆♦t❡ q✉❡ q✉❛✐sq✉❡r ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❞❡
Kc sã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞✐s❥✉♥t❛s✳
P♦r ♦✉tr♦ ❧❛❞♦✱ q✉❛❧q✉❡rz ∈Kc ❡stá ❡♠ ❛❧❣✉♠❛ ❞❡st❛s ❝♦♠♣♦♥❡♥t❡s✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱
♣♦❞❡♠♦s ❛ss♦❝✐❛r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡0✬s ❡ 1✬s ♣❛r❛ t❛❧ z ✈✐❛
S(z) =s0s1s2. . . , ❞❡s❞❡ q✉❡ z ∈
\
n≥0
Is0s1...sn.
❊st❛ s❡q✉ê♥❝✐❛ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ s❡q✉ê♥❝✐❛ ❡s♣❛ç♦ ❡♠ ❞♦✐s sí♠❜♦❧♦s ❡ ♠♦str❛♠♦s q✉❡ ❡①✲ ✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ♥❛t✉r❛❧ ❡♥tr❡ ❡❧❛ ❡ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡Kc✳ ❉❡ ❢❛t♦✱ ♣♦❞❡✲s❡ ♠♦str❛r
q✉❡ ❡st❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ♥❛ ✈❡r❞❛❞❡ é ❝♦♥tí♥✉❛✳ ❯♠ ❢❛t♦ ♠❛✐s ❞✐❢í❝✐❧ ❞❡ s❡r ❞❡♠♦♥str❛❞♦ é q✉❡ ❝❛❞❛ ✉♠❛ ❞❡st❛s ❝♦♠♣♦♥❡♥t❡s é ❞❡ ❢❛t♦ ✉♠ ú♥✐❝♦ ♣♦♥t♦✳ ■st♦ ♣♦❞❡ s❡r ❞❡♠♦♥str❛❞♦ ✉s❛♥❞♦ té❝♥✐❝❛s ♠❛✐s s♦✜st✐❝❛❞❛s ❞❡ ❛♥á❧✐s❡ ❝♦♠♣❧❡①❛✱ ❡ s❡rá ♦♠✐t✐❞♦✳ ❆ ✐❞é✐❛ ❜ás✐❝❛ ♣♦r trás ❞✐st♦ é q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ é ❛ ✐♥t❡rs❡❝çã♦ ❞❡ t♦❞❛s ❛s ❝✉r✈❛s ♦✐t♦ ❡ s❡✉s ❧ó❜✉❧♦s✳
❖❜s❡r✈❛çã♦ ✸✷✳
✶✳ ❆❝❡✐t❛♥❞♦ ♦ ❢❛t♦ q✉❡ q✉❛❧q✉❡r ❝♦♠♣♦♥❡♥t❡ ❞❡ Kc é ✉♠ ♣♦♥t♦✱ q✉❛♥❞♦|c|>2✱ ♦❜t❡♠♦s
q✉❡Kc =Jc✳
✷✳ Qc é t❛♠❜é♠ ❡①tr❡♠❛♠❡♥t❡ s❡♥s✐t✐✈❛ às ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ♥♦ s❡✉ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛✱
q✉❛♥❞♦ |c| > 2✳ ❉❛❞♦ z ∈ Kc ❡ ✉♠❛ ♣❡q✉❡♥❛ ❜♦❧❛ ❝♦♠ z ❡♠ s❡✉ ✐♥t❡r✐♦r✱ ♣♦❞❡♠♦s
❡s❝♦❧❤❡r k s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ t❛❧ q✉❡ ♦ ❧ó❜✉❧♦ Is0s1...sk❝♦♥t❡♥❞♦ z ❡stá t❛♠❜é♠ ❝♦♥t✐❞♦ ♥❡st❛ ❜♦❧❛✳ ▼❛s ❡♥tã♦✱ Ql
c ❛♣❧✐❝❛ ❡st❡ ❧ó❜✉❧♦ ❡♠ t♦❞♦ ♦ ❞✐s❝♦ D ❡ ✐t❡r❛çõ❡s
s✉❜s❡q✉❡♥t❡s ❡①♣❛♥❞❡♠D✳ ❋✐♥❛❧♠❡♥t❡✱ q✉❛❧q✉❡r ♣♦♥t♦ ♥♦ ♣❧❛♥♦ ❡stá ♥❛ ✐♠❛❣❡♠ ❞❡st❡
❧ó❜✉❧♦ s♦❜ ✉♠❛ ✐t❡r❛❞❛ ❞❡ ♦r❞❡♠ s✉✜❝✐❡♥t❡♠❡♥t❡ ❛❧t❛ ❞❡Qc✳ ❆ss✐♠Qc é s✉♣❡rs❡♥s✐t✐✈❛
❡♠ Jc✳
✸✳ ◆♦t❡ q✉❡ ♣❛r❛ |c|>2 ❛ ór❜✐t❛ ❞❡0 t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦✳
❯s❛♥❞♦ ♦ ❝r✐tér✐♦ ❞❡ ❡s❝❛♣❡✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✈ár✐♦s ❝♦♥❥✉♥t♦s ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦s❀ s✉❛s ❢♦r♠❛s ✈❛r✐❛♠ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ✈❛❧♦r ❞❡c✳ ❖s ❝♦♥❥✉♥t♦s ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦s ❛♣r❡s❡♥t❛♠ ❛
❛✉t♦✲s✐♠✐❧❛r✐❞❛❞❡✱ ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❢r❛❝t❛❧✳ ❆s ✜❣✉r❛s ✼ ❡ ✽ r❡♣r❡s❡♥t❛♠ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ♣❛r❛Qc ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡ c✳
❋✐❣✉r❛ ✼✿ ❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ♣❛r❛ c=−0.75 ❡c=−0.75 + 0.1i✳
❚❛♠❜é♠ ♣♦❞❡♠♦s ♦❜s❡r✈❛r✱ q✉❡ ♣❛r❛ ❝❡rt♦s ✈❛❧♦r❡s ❞❡ c✱ Jc♣❛r❡❝❡ s❡r ❛♣❡♥❛s ❛❧❣✉♥s
♣♦♥t♦s ✐s♦❧❛❞♦s ❡ ♣❛r❛ ♦✉tr♦s✱Jc ♣❛r❡❝❡ s❡r ✉♠❛ ♣❡ç❛ ú♥✐❝❛✳
❋✐❣✉r❛ ✽✿ ❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ♣❛r❛ c= 0.360284 + 0.100376i ❡ c=−0.255✳
✾ ❆ ❉✐❝♦t♦♠✐❛ ❋✉♥❞❛♠❡♥t❛❧
P♦❞❡♠♦s ♥♦t❛r✱ q✉❡ ♦s ❝♦♥❥✉♥t♦s ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦s✱ ❛♣❛r❡♥t❡♠❡♥t❡ ❡stã♦ ❡♠ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❝❧❛ss❡s✿ ❛q✉❡❧❡s q✉❡ sã♦ ❝♦♥❡①♦s ❡ ❛q✉❡❧❡s q✉❡ sã♦ t♦t❛❧♠❡♥t❡ ❞❡s❝♦♥❡①♦s✳ ❆♣❛r❡♥t❡♠❡♥t❡ ♥ã♦ ❤á ✉♠ ❝❛s♦ ✐♥t❡r♠❡❞✐ár✐♦✿ ♦✉ Kc ❝♦♥s✐st❡ ❞❡ ✉♠ ♣❡❞❛ç♦ ❝♦♥❡①♦ ♦✉ ❞❡
✐♥✜♥✐t♦s ♣❡❞❛ç♦s✳ ◆❛ r❡❛❧✐❞❛❞❡✱ ❡st❡ é ✉♠ ❞♦s r❡s✉❧t❛❞♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ❞✐♥â♠✐❝❛ ❝♦♠♣❧❡①❛✳ ❚❡♦r❡♠❛ ✸✸✳ ✭❆ ❉✐❝♦t♦♠✐❛ ❋✉♥❞❛♠❡♥t❛❧✮ ❙❡❥❛ Qc(z) =z2 +c✳ ❊♥tã♦✿
i) ❆ ór❜✐t❛ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ ✵ ❡s❝❛♣❛ ♣❛r❛ ♦ ✐♥✜♥✐t♦✱ ♥❡st❡ ❝❛s♦ Kc ❝♦♥s✐st❡ ❞❡
✐♥✜♥✐t❛s ❝♦♠♣♦♥❡♥t❡s ❞✐s❥✉♥t❛s✱ ♦✉
ii) ❆ ór❜✐t❛ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ ✵ é ❧✐♠✐t❛❞❛✱ ❡ ♥❡st❡ ❝❛s♦ Kc é ❝♦♥❡①♦✳
❏á ♠♦str❛♠♦s q✉❡ s❡ |c|>2✱ ❡♥tã♦ ❛ ór❜✐t❛ ❞❡ ✵ ❡s❝❛♣❛ ♣❛r❛ ♦ ✐♥✜♥✐t♦ ❡ Kc ❝♦♥s✐st❡ ❞❡
✐♥✜♥✐t❛s ❝♦♠♣♦♥❡♥t❡s✱ Kc é ♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r ❡ ❛ ❞✐♥â♠✐❝❛ ❞❡ Qc ❡♠ Kc é ❝♦♥❥✉❣❛❞❛
❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ♠✉❞❛♥ç❛✳ ❆ss✐♠✱ ❝♦♥s✐❞❡r❛♠♦s ❛q✉✐ ♦ ❝❛s♦ ❡♠ q✉❡|c| ≤ 2✳ ◆♦t❡ q✉❡ ❡①✐st❡ ✉♠❛ ❜♦❛ r❛③ã♦ ♣❛r❛ s❡♣❛r❛r ♦s ❞♦✐s ❝❛s♦s✳ ❏á ✈✐♠♦s q✉❡ K−2 é ♦ ✐♥t❡r✈❛❧♦ [−2,2] ♦ q✉❛❧ é
❝♦♥❡①♦ ❡ ❛ ór❜✐t❛ ❞❡ ✵ s♦❜Q−2 é ❡✈❡♥t✉❛❧♠❡♥t❡ ✜①❛✳ P♦rt❛♥t♦✱ |c|= 2 é ❡①❛t❛♠❡♥t❡ ♦ ♠❛✐♦r
✈❛❧♦r ♣❛r❛ ♦ q✉❛❧Kc ❞❡✈❡ s❡r ❝♦♥❡①♦✳
▲❡♠❜r❡✲s❡ q✉❡ é ♦ ❝r✐tér✐♦ ❞❡ ❡s❝❛♣❡ q✉❡ ♥♦s ❞á ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ♣❛r❛ ✉♠❛ ór❜✐t❛ ❡s❝❛♣❛r✱ q✉❛♥❞♦|c| ≤2✿ s❡ |Qn
c(0)| >2 ♣❛r❛ ❛❧❣✉♠ n✱ ❡♥tã♦ ❛ ór❜✐t❛ ❞❡ ✵ t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦✳
P❛r❛ ♣r♦✈❛r ❛ ❞✐❝♦t♦♠✐❛ ❢✉♥❞❛♠❡♥t❛❧✱ ✈❛♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ s✉♣♦r q✉❡|Qn
c(0)| → ∞q✉❛♥❞♦
n → ∞ ♣❛r❛ ❛❧❣✉♠ c ❝♦♠ |c| ≤ 2✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ♣r✐♠❡✐r❛ ✐t❡r❛çã♦ k, ♣❛r❛ ♦ q✉❛❧ |Qk
c(0)| > 2✳ ❙❡❥❛ ρ = |Qkc(0)| ❡ ❝♦♥s✐❞❡r❡ ♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ ρ✱ Cρ✱ ❝❡♥tr❛❞♦ ♥❛ ♦r✐❣❡♠✳ ❆
❛♣❧✐❝❛çã♦ q✉❛❞rát✐❝❛ Qc ❛♣❧✐❝❛ ♣♦♥t♦s ❞♦ ❝ír❝✉❧♦ ♥♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ ρ2, ❝❡♥tr❛❞♦ ❡♠ c ♥♦
♣❧❛♥♦✳
◆♦ ❡♥t❛♥t♦✱ ♣❡❧♦ ❝♦r♦❧ár✐♦ ✸✶✱ ❞♦ ❝r✐tér✐♦ ❞❡ ❡s❝❛♣❡✱ s❛❜❡♠♦s q✉❡ ❛s ór❜✐t❛s ❞❡ t♦❞♦s ♦s ♣♦♥t♦s ❡♠ Cρ ❡s❝❛♣❛♠ ♣❛r❛ ♦ ✐♥✜♥✐t♦ ❡✱ ♠❛✐s ❛✐♥❞❛✱ ❛ ✐♠❛❣❡♠ ❞♦ ❝ír❝✉❧♦ ✜❝❛ ♥♦ ❡①t❡r✐♦r
❞♦ ❞✐s❝♦ |z| ≤ ρ✳ ❙❡❥❛ Dρ = {z : |z| ≤ ρ}✳ ❖ ❝ír❝✉❧♦ Qc(Cρ) ❡♥✈♦❧✈❡ Cρ✱ ❝♦♠♦ ♠♦str❛❞♦ ❛
s❡❣✉✐r✱ ♥❛ ✜❣✉r❛ ✾✱ ✉♠❛ ✈❡③ q✉❡ ♦ ❝❡♥tr♦ ❞❡st❡ ❝ír❝✉❧♦ s❛t✐s❢❛③|c| ≤2< ρ✳ P❡❧♦ Pr✐♥❝í♣✐♦ ❞❛
❆♣❧✐❝❛çã♦ ❞❡ ❋r♦♥t❡✐r❛✱ ♦ ✐♥t❡r✐♦r ❞❡ Dρ é ❧❡✈❛❞♦ ♥♦ ✐♥t❡r✐♦r ❞❡ Qc(Dρ)✳ ❚❡♠♦s ❡♥tã♦✱ ✉♠❛
s✐t✉❛çã♦ s✐♠✐❧❛r ❛ ❞♦ ❝❛s♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r✳
❋✐❣✉r❛ ✾✿ ❖ ❝❛s♦ ♦♥❞❡ ❛ ór❜✐t❛ ❞❡ 0 ❡s❝❛♣❛✳
❈♦♥s✐❞❡r❡ ❛❣♦r❛ Q−1
c (Dρ)✳ ❖ ❝♦♥❥✉♥t♦ Q−c1(Dρ) ❡stá ❝♦♠♣❧❡t❛♠❡♥t❡ ❝♦♥t✐❞♦ ♥♦ ✐♥t❡r✐♦r
❞❡ Dρ, ✉♠❛ ✈❡③ q✉❡✱ Cρ é ❧❡✈❛❞❛ ♣❛r❛ ❢♦r❛ ❞❡ Dρ✳ ▼❛✐s ❛✐♥❞❛✱ ✵ ♣❡rt❡♥❝❡ ❛ Q−c1(Dρ) ✉♠❛
✈❡③ q✉❡ Qc(0) = c❡ |c|< ρ✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ Q−c1(Dρ)é ❧✐♠✐t❛❞♦ ♣♦r ✉♠❛ ❝✉r✈❛ s✐♠♣❧❡s
❢❡❝❤❛❞❛✳
❯♠❛ ✈❡③ q✉❡ k é ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣❛r❛ ♦ q✉❛❧ Qk
c(0) ∈ Cρ✱ s❡❣✉❡ q✉❡ ♣♦❞❡♠♦s r❡♣❡t✐r ♦
❛r❣✉♠❡♥t♦ ❛❝✐♠❛ k−1 ✈❡③❡s✳ ❊♥❝♦♥tr❛♠♦s
Q−c(k−1)(Dρ)⊂ · · · ⊂Q−c1(Dρ)⊂Dρ,
❝♦♠Q−j
c (Dρ)❝♦♥t✐❞♦ ♥♦ ✐♥t❡r✐♦r ❞❡Q
−(j−1)
c (Dρ),♣❛r❛j = 1, . . . , k−1✳ ▼❛✐s ❛✐♥❞❛✱Q−cj(Dρ)
é ❧✐♠✐t❛❞♦ ♣♦r ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ s✐♠♣❧❡s q✉❡ é ❧❡✈❛❞❛ ♣♦r Qc ❛ ❢r♦♥t❡✐r❛ ❞❡ Q−cj+1(Dρ)✱
❝♦♠♦ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ✶✵✳
❋✐❣✉r❛ ✶✵✿ Q−c(k−1)(Dρ)⊂ · · · ⊂Q−c1(Dρ)⊂Dρ✳
◆❛k✲és✐♠❛ ✐t❡r❛çã♦ ❤á ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦✳ ❖ ♣♦♥t♦ ❝rít✐❝♦0❛❣♦r❛ ♣❡rt❡♥❝❡ ❛ Q−k
c (Dρ)✳ ❯♠❛ ✈❡③ q✉❡ Qkc(0) ∈ Cρ✱ s❡❣✉❡ q✉❡ 0 ❡stá ♥❛ ❢r♦♥t❡✐r❛ ❞❡ Q−ck(Dρ)✳ ❆ss✐♠✱ ❛
❢r♦♥t❡✐r❛ ❞❡ Q−k
c (Dρ) é ✉♠❛ ✜❣✉r❛ ♦✐t♦✱ ❡ ❝❛❞❛ ✉♠ ❞♦s ❞♦✐s ❧ó❜✉❧♦s ♥♦ ✐♥t❡r✐♦r ❞❡st❛ ❝✉r✈❛
sã♦ ❧❡✈❛❞♦s ❞❡ ♠❛♥❡✐r❛ ✐♥❥❡t♦r❛ ♣♦rQc, ❡♠ t♦❞♦ ♦ ✐♥t❡r✐♦r ❞❡ Q−ck+1(Dρ)✳
❉❡ss❛ ❢♦r♠❛✱ ✈❡♠♦s q✉❡ ❛ ♣ré✲✐♠❛❣❡♠ ❞❡ Dρ ❡stá ♣r❡❝✐s❛♠❡♥t❡ ♥♦s ❞♦✐s ❧ó❜✉❧♦s✱ q✉❛♥❞♦
0❡stá ♥❛ ❢r♦♥t❡✐r❛ ❞❡ Q−k c (Dρ)✳
❖ ❛r❣✉♠❡♥t♦ ❛❣♦r❛ é s✐♠✐❧❛r ❛♦ ❞❛ s❡çã♦ ❛♥t❡r✐♦r✳ ❖ ❝♦♥❥✉♥t♦ Q−k−1
c (Dρ) ❝♦♥s✐st❡ ❞❡
✉♠ ♣❛r ❞❡ ✜❣✉r❛s ♦✐t♦ ❡ s❡✉s ❧ó❜✉❧♦s ❡stã♦ ♥♦ ✐♥t❡r✐♦r ❞♦ ❝♦♥❥✉♥t♦ Q−k
c (Dρ)✳ ❊♠ ❣❡r❛❧✱
Q−k−n
c (Dρ) ❝♦♥s✐st❡ ❞❡ 2n ✜❣✉r❛s ♦✐t♦ ❞✐s❥✉♥t❛s ❡ s❡✉s ❧ó❜✉❧♦s✱ ❡♥tã♦ ✈❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦
❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡❝♦♠♣õ❡✲s❡ ✐♥✜♥✐t❛♠❡♥t❡ ❡♠ ✐♥✜♥✐t❛s ❝♦♠♣♦♥❡♥t❡s ❞✐s❥✉♥t❛s✱ q✉❛♥❞♦ ❛ ór❜✐t❛ ❞❡ ✵ ❡s❝❛♣❛✳ ■st♦ ♣r♦✈❛ ❛ ♣❛rt❡i) ❞❛ ❉✐❝♦t♦♠✐❛ ❋✉♥❞❛♠❡♥t❛❧✳
❆ ú♥✐❝❛ ♦✉tr❛ ♣♦ss✐❜✐❧✐❞❛❞❡ é Qk
c(0) ♥✉♥❝❛ ♣❡rt❡♥ç❛ ❛♦ ❡①t❡r✐♦r ❞♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ ✷✳
◆❡ss❡ ❝❛s♦✱ ♣❛r❛ q✉❛❧q✉❡rρ > 2✱ Q−k
c (Dρ) ❡stá s❡♠♣r❡ ❡♠ ✉♠❛ r❡❣✐ã♦ ❝♦♥❡①❛ ❧✐♠✐t❛❞❛ ♣♦r
✉♠❛ ❝✉r✈❛ s✐♠♣❧❡s ❢❡❝❤❛❞❛✳ P❡❧❛ Pr♦♣r✐❡❞❛❞❡ ❞❡ ❈♦♥❡❝t✐✈✐❞❛❞❡✱ \
n≥0
Q−n
c (Dρ) é ✉♠ ❝♦♥❥✉♥t♦
❝♦♥❡①♦✳ ❈♦♠♦ s❡♠♣r❡✱ ❡st❡ é ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❡ ✐st♦ ❝♦♠♣❧❡t❛ ❛ ❞❡♠♦♥str❛çã♦✳
✶✵ ❖ ❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t
❆ ❉✐❝♦t♦♠✐❛ ❋✉♥❞❛♠❡♥t❛❧ ✐♥❞✐❝❛ q✉❡ ❡①✐st❡♠ ❛♣❡♥❛s ❞♦✐s t✐♣♦s ❜ás✐❝♦s ❞❡ ❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ♣❛r❛ Qc, ❛q✉❡❧❡s q✉❡ sã♦ ❝♦♥❡①♦s ❡ ❛q✉❡❧❡s q✉❡ ❝♦♥s✐st❡♠ ❞❡ ✐♥✜♥✐t❛s
