❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❘✉r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛
♣♦r
❍♦s❛♥❛ ▼❛r✐❛ ❞❡ ▲✐♠❛ ❘✐❜❡✐r♦
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❘✉r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛
†♣♦r
❍♦s❛♥❛ ▼❛r✐❛ ❞❡ ▲✐♠❛ ❘✐❜❡✐r♦
s♦❜ ♦r✐❡♥t❛çã♦ ❞❛
Pr♦❢❛✳ ❉r❛✳ ❇ár❜❛r❛ ❈♦st❛ ❞❛ ❙✐❧✈❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ❝♦♦r❞❡♥❛❞♦ ♣❡❧❛ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ♦❢❡rt❛❞♦ ♣❡❧❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❘✉r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❆❣♦st♦✴✷✵✶✹ ❘❡❝✐❢❡ ✲ P❊
† ❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛
♣♦r❍♦s❛♥❛ ▼❛r✐❛ ❞❡ ▲✐♠❛ ❘✐❜❡✐r♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ❝♦♦r❞❡♥❛❞♦ ♣❡❧❛ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ♦❢❡rt❛❞♦ ♣❡❧❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❘✉r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢❛✳ ❉r❛✳ ❇ár❜❛r❛ ❈♦st❛ ❞❛ ❙✐❧✈❛ ✲❯❋❘P❊ ✭❖r✐❡♥t❛❞♦r❛✮
Pr♦❢✳ ❉r✳ ❆✐rt♦♥ ❚❡♠✐st♦❝❧❡s ●♦♥ç❛❧✈❡s ❞❡ ❈❛str♦ ✲ ❯❋P❊
Pr♦❢✳ ❉r✳ ❆❞r✐❛♥♦ ❘❡❣✐s ▼❡❧♦ ❘♦❞r✐❣✉❡s ❞❛ ❙✐❧✈❛ ✲ ❯❋❘P❊
Pr♦❢❛✳ ❉r❛✳ ▼❛✐té ❑✉❧❡s③❛ ✲ ❯❋❘P❊
❉❡❞✐❝❛tór✐❛
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s✱ ♣♦r t✉❞♦✳
➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ♣r♦❢❡ss♦r❛ ❇ár❜❛r❛ ❈♦st❛✱ ♣❡❧♦ ❛♣♦✐♦✱ ♦r✐❡♥t❛çã♦✱ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡✱ ❞❡❞✐❝❛çã♦ ❡ ❝♦♥tr✐❜✉✐çã♦ ❞✉r❛♥t❡ ❛ ❡❧❛❜♦r❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❆s ✐♥ú♠❡r❛s r❡✉♥✐õ❡s ♣❛r❛ ❞✐s❝✉ssã♦ ❞❛s ❞❡♠♦♥str❛çõ❡s ❢♦r❛♠ ✐♠♣r❡s❝✐♥❞í✈❡✐s✳ ➚ ♣r♦❢❡ss♦r❛ ❇ár❜❛r❛✱ ♦ ♠❡✉ ♠❛✐s ♣r♦❢✉♥❞♦ r❡❝♦♥❤❡❝✐♠❡♥t♦✳
❆♦s ♣r♦❢❡ss♦r❡s q✉❡ ❝♦♠♣✉s❡r❛♠ ❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❡ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ❆✐rt♦♥ ❚❡♠✐st♦❝❧❡s ●✳ ❞❡ ❈❛str♦ q✉❡ ❛❝♦♠♣❛♥❤♦✉ ♠✐♥❤❛ tr❛❥❡tór✐❛ ❛❝❛❞ê♠✐❝❛ ❞❡s❞❡ ❛ ❣r❛❞✉❛çã♦ ♥❛ ❯❋P❊ ❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈♦✉✳
❆♦ ♣r♦❢❡ss♦r ❆♥t♦♥✐♦ ❈❛r❧♦s ❘♦❞r✐❣✉❡s ▼♦♥t❡✐r♦✱ ❞❛ ❯❋P❊✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❞❛❞♦ ❞❡st❡ ❛ ❣r❛❞✉❛çã♦✳
❆♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚✲❯❋❘P❊✳
❘❡s✉♠♦
◆❡st❛ ❞✐ss❡rt❛çã♦✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠♣❧❡①❛ ❞❡ ❣r❛✉ ♠❛✐♦r q✉❡ ♦✉ ✐❣✉❛❧ ❛ ✷ ❡ ❞❡♠♦♥str❛♠♦s ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❡❧❛s✳ ❆❧é♠ ❞✐st♦✱ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s sã♦ ❡st✉❞❛❞❛s✳ ❆❜♦r❞❛♠♦s ❛ ❢❛♠í❧✐❛F ❞❛s ❢✉♥çõ❡s q✉❛❞rát✐❝❛s ❡♠C❞❛❞❛s ♣♦rfc(z) =z2+c✱ ♦♥❞❡
cé ✉♠❛ ❝♦♥st❛♥t❡ ❝♦♠♣❧❡①❛ ❡ ❞❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦
❞♦s ♣❛râ♠❡tr♦sc t❛✐s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ J(fc) é ❝♦♥❡①♦✳ ▼♦str❛♠♦s q✉❡ ❡st❛ ❞❡✜♥✐çã♦ é ❡q✉✐✈❛❧❡♥t❡ à ór❜✐t❛ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ fc s❡r ❧✐♠✐t❛❞❛✳ ❉❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡ fc✱ ❝✉❥❛ ❢r♦♥t❡✐r❛ é ✐❣✉❛❧ ❛ J(fc) ❡ ✉t✐❧✐③❛♠♦s ♦s s♦❢t✇❛r❡s ▼❛t▲❛❜ ❡ ●❡♦●❡❜r❛ ♣❛r❛ ✈✐s✉❛❧✐③❛r ✉♠❛ ✜❣✉r❛ ❛♣r♦①✐♠❛❞❛ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡ fc ❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛✱ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✱ ✐t❡r❛çã♦ ❞❡ ❢✉♥çã♦ ❝♦♠♣❧❡①❛✳
❆❜str❛❝t
❆t t❤✐s ❞✐ss❡rt❛t✐♦♥✱ ✇❡ ♣r❡s❡♥t s♦♠❡ ❞❡✜♥✐t✐♦♥s ♦❢ ❏✉❧✐❛ s❡t ♦❢ ❛ ❝♦♠♣❧❡① ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ ❞❡❣r❡❡ ❣r❡❛t❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ ✷ ❛♥❞ ❞❡♠♦♥str❛t❡ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ t❤❡♠✳ ■♥ ❛❞❞✐t✐♦♥✱ s♦♠❡ t♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ❛r❡ st✉❞✐❡❞✳ ❲❡ ❛♣♣r♦❛❝❤❡❞ t❤❡ ❢❛♠✐❧② F ♦❢ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s ♦♥ C ❣✐✈❡♥ ❜② fc(z) = z2+c✱
✇❤❡r❡ c ✐s ❛ ❝♦♠♣❧❡① ❝♦♥st❛♥t ❛♥❞ ❞❡✜♥❡ ▼❛♥❞❡❧❜r♦t s❡t ❛s t❤❡ s❡t ♦❢ ♣❛r❛♠❡t❡rs c s✉❝❤ t❤❛t ❏✉❧✐❛ s❡t J(fc) ✐s ❝♦♥♥❡❝t❡❞✳ ❲❡ s❤♦✇ t❤✐s ❞❡✜♥✐t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♦r❜✐t ♦❢ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t ♦❢ fc t♦ ❜❡ ❧✐♠✐t❡❞✳ ❲❡ ❞❡✜♥❡ ✜❧❧❡❞ ❏✉❧✐❛ s❡t ♦❢ fc✱ ✇❤♦s❡ ❜♦r❞❡r ✐s ❡q✉❛❧ t♦J(fc)❛♥❞ ✉s❡ ▼❛t▲❛❜ ❛♥❞ ●❡♦●❡❜r❛ s♦❢t✇❛r❡s t♦ ✈✐❡✇ ❛♥ ❛♣♣r♦①✐♠❛t❡ ✜❣✉r❡ ♦❢ ✜❧❧❡❞ ❏✉❧✐❛ s❡t ♦❢fc ❛♥❞ ▼❛♥❞❡❧❜r♦t s❡t✳
❑❡②✇♦r❞s✿ ❏✉❧✐❛ s❡t✱ ▼❛♥❞❡❧❜r♦t s❡t✱ ✐t❡r❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ ❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛✿ ❞❡✜♥✐çã♦ ❡ ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛
✈✐s✉❛❧✐③❛r ✉♠❛ ✜❣✉r❛ ❛♣r♦①✐♠❛❞❛ ✸
✶✳✶ ❙✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛✿ ❞❡✜♥✐çã♦ ❡ ❡①❡♠♣❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✸ ❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✹ ▼ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ✈✐s✉❛❧✐③❛r ✉♠❛ ✜❣✉r❛ ❛♣r♦①✐♠❛❞❛ ❞♦
❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✹✳✶ ❈ó❞✐❣♦s ❞♦ ▼❛t❧❛❜ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✹✳✷ ❯t✐❧✐③❛♥❞♦ ♦ ●❡♦●❡❜r❛ ♣❛r❛ ♣❧♦t❛r ♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t
❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡fc ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷ ❆❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ✹✸ ✷✳✶ Pr♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✷ ❋✉♥çõ❡s q✉❛❞rát✐❝❛s ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
❆ ❱❛r✐á✈❡✐s ❝♦♠♣❧❡①❛s ✼✽
❆✳✶ ▲✐♠✐t❡s ❞❡ s❡q✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ❆✳✷ ❆❧❣✉♠❛s ♥♦çõ❡s ❞❛ t♦♣♦❧♦❣✐❛ ❞❡ C ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾
❆✳✸ ❋✉♥çã♦✱ ❧✐♠✐t❡ ❡ ❝♦♥t✐♥✉✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷
❆✳✸✳✶ ❋✉♥çã♦ ❝♦♠♣❧❡①❛ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❝♦♠♣❧❡①❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ❆✳✸✳✷ ■♠❛❣❡♠ ✐♥✈❡rs❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ❆✳✸✳✸ ▲✐♠✐t❡s ❞❡ ❢✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ❆✳✸✳✹ ❋✉♥çã♦ ❝♦♥tí♥✉❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ❆✳✹ ❋✉♥çã♦ ❛♥❛❧ít✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ❆✳✺ ❈♦♥❥✉❣❛çã♦ t♦♣♦❧ó❣✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ■t❡r❛çã♦ ❞❡ z0 = 0,9 + 0,1i ❡ w0 = 0,6 + 0,8i✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷ ■t❡r❛çã♦ ❞❡ p0 = 1,35 + 0,31i✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❆♥á❧✐s❡ ❣rá✜❝❛ ❞❛ ór❜✐t❛ ❞❡ z0 ∈R ♣❡❧❛ ❢✉♥çã♦f(z) =z2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ ❆♥á❧✐s❡ ❣rá✜❝❛ ❞❛ ór❜✐t❛ ❞❡ z0 ∈R ♣❡❧❛ ❢✉♥çã♦f(z) =z2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✺ Kc✱ ❝♦♠ c= 0,27334−0,00742i ❡q = 20 ✐t❡r❛çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✻ Kc✱ ❝♦♠ c= 0,27334−0,00742i ❡q = 100✐t❡r❛çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✼ ❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✱ q= 20 ✐t❡r❛çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✶✳✽ ❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✱ q= 100 ✐t❡r❛çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✶✳✾ Kc✱ ❝♦♠ c= 0,27334−0,00742i ❡q = 150✐t❡r❛çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✳✶✵ ❆♠♣❧✐❛çã♦ ❞❛ ✜❣✉r❛ ✶✳✾✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✳✶✶ ❆♠♣❧✐❛çã♦ ❞❛ ✜❣✉r❛ ✶✳✶✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✶✷ ❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✱ q= 150 ✐t❡r❛çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✶✳✶✸ ❆♠♣❧✐❛çã♦ ❞❛ ✜❣✉r❛ ✶✳✶✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✶✳✶✹ ▼❡♥✉ ✏❊①✐❜✐r✑ ❞♦ ●❡♦●❡❜r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✶✳✶✺ P✐♥❝❡❧ ❤♦r✐③♦♥t❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✶✳✶✻ P❧❛♥✐❧❤❛ ❞♦ ●❡♦●❡❜r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✶✳✶✼ ❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ♥♦ ●❡♦●❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✶✳✶✽ ▲✐st❛s ♥♦ ●❡♦●❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✶✳✶✾ ❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ♥♦ ●❡♦●❡❜r❛✱ ♥♦✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✶✳✷✵ ❆♠♣❧✐❛çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ♥♦ ●❡♦●❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✶✳✷✶ Kc✱ ❝♦♠ c= 0,27334−0,00742i ❡ ✾✾ ✐t❡r❛çõ❡s ♥♦ ●❡♦●❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✹✶ ✶✳✷✷ Kc✱ ❝♦♠ c= 0,27334−0,00742i ❡ ✾✾ ✐t❡r❛çõ❡s ♥♦ ●❡♦●❡❜r❛✱ ♥♦ ✷ ✳ ✳ ✹✷ ✶✳✷✸ ❈♦♥❥✉♥t♦s ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡c❡ ❛ ♣♦s✐çã♦
❞❡c❡♠ r❡❧❛çã♦ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✷✳✶ ❋✐❣✉r❛ ❞❡ ♦✐t♦ f−1
c (C) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✷✳✷ ■t❡r❛❞♦s ✐♥✈❡rs♦s ❞❡ ✉♠ ❝ír❝✉❧♦ C♣♦rfc✱ ❝♦♠ c=−0,3 + 0,3i✳ ❋♦♥t❡
❬✹❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✷✳✸ ■t❡r❛❞♦s ✐♥✈❡rs♦s ❞❡ ✉♠ ❝ír❝✉❧♦ C♣♦rfc✱ ❝♦♠ c=−0,9 + 0,5i✳ ❋♦♥t❡
❬✹❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻
■♥tr♦❞✉çã♦
❖ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ s✉r❣✐✉ ❝♦♠ ♦ ❡st✉❞♦ ❞❡ ✐t❡r❛çã♦ ❞❡ ❢✉♥çõ❡s ❝♦♠♣❧❡①❛s✳ ❖ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ●❛st♦♥ ▼❛✉r✐❝❡ ❏✉❧✐❛ ✭✶✽✾✸✲✶✾✼✽✮ ♣✉❜❧✐❝♦✉ s✉❛ ♦❜r❛ ♣r✐♠❛ ✏▼é♠♦✐r❡ s✉r ■✬✐tér❛t✐♦♥ ❞❡s ❢♦♥❝t✐♦♥s r❛t✐♦♥♥❡❧❧❡s✑ ❡♠ ✶✾✶✽ ♥♦ ✏❏♦✉r♥❛❧ ❞❡ ♠❛t❤é♠❛t✐q✉❡s ♣✉r❡s ❡t ❛♣♣❧✐q✉é❡s✑✱ ♦♥❞❡ ✐♥tr♦❞✉③✐✉ ♦ ❝♦♥❝❡✐t♦ ❞♦ ❝♦♥❥✉♥t♦ q✉❡ ❛t✉❛❧♠❡♥t❡ ❝♦♥❤❡❝❡♠♦s ♣♦r ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛✳ ❆ ♦❜r❛ ♦ t♦r♥♦✉ ❢❛♠♦s♦ ❡♥tr❡ ♦s ♠❛t❡♠át✐❝♦s ❞❛ é♣♦❝❛✱ s❡♥❞♦ ●❛st♦♥ ❏✉❧✐❛ ✉♠ ❞♦s ♣r❡❝✉rs♦r❡s ❞❛ ♠♦❞❡r♥❛ t❡♦r✐❛ ❞♦s s✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝♦♠♣❧❡①♦s✳
❙✐♠✉❧t❛♥❡❛♠❡♥t❡ ❛♦s tr❛❜❛❧❤♦s ❞❡ ●❛st♦♥ ❏✉❧✐❛ ❡ ❞❡ ❢♦r♠❛ ✐♥❞❡♣❡♥❞❡♥t❡✱ ♦ ♠❛t❡♠át✐❝♦ ❡ ❛strô♥♦♠♦ ❢r❛♥❝ês P✐❡rr❡ ❏♦s❡♣❤ ▲♦✉✐s ❋❛t♦✉ ✭✶✽✼✽✲✶✾✷✾✮ ❞❡s❡♥✈♦❧✈❡✉ ✈ár✐♦s tr❛❜❛❧❤♦s ♥❛ ár❡❛ ❞❡ ✐t❡r❛çã♦ ❞❡ ❢✉♥çõ❡s ❝♦♠♣❧❡①❛s✳ ❊♠ s✉❛ ❤♦♠❡♥❛❣❡♠✱ ♦ ❝♦♠♣❧❡♠❡♥t❛r ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛✱ ♥♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✱ é ❞❡♥♦♠✐♥❛❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❋❛t♦✉✳
❖s tr❛❜❛❧❤♦s ❞♦s ♠❛t❡♠át✐❝♦s ❏✉❧✐❛ ❡ ❋❛t♦✉ ❢♦r❛♠ r❡❧❛t✐✈❛♠❡♥t❡ ❡sq✉❡❝✐❞♦s ❛té ❛s ❞❡s❝♦❜❡rt❛s ❞♦ ♠❛t❡♠át✐❝♦ ❇❡♥♦✐t ❇✳ ▼❛♥❞❡❧❜r♦t ✭✶✾✷✹✲✷✵✶✵✮ ♥♦ ✜♥❛❧ ❞♦s ❛♥♦s s❡t❡♥t❛✳ ❊❧❡ ✉t✐❧✐③♦✉ ❝♦♠♣✉t❛❞♦r❡s ❣rá✜❝♦s ♣❛r❛ ✈✐s✉❛❧✐③❛r ❝♦♥❥✉♥t♦s ❞❡ ❏✉❧✐❛ q✉❡ ♣♦ss✉❡♠ ❢♦r♠❛s ❢r❛❝t❛✐s✳ ❊♠ ✶✾✽✵✱ ❛♦ ❡st✉❞❛r ♦s ❝♦♥❥✉♥t♦s ❞❡ ❏✉❧✐❛ ❞❡ ❢✉♥çõ❡s ❞❛ ❢♦r♠❛fc =z2+c✱ ❞❡♥♦♠✐♥♦✉ ❞❡ ❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣❛râ♠❡tr♦s
ct❛✐s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❡ fc é ❝♦♥❡①♦✳
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❞❡✜♥✐r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠♣❧❡①❛ ❞❡ ❣r❛✉ n ≥ 2 ❝♦♠♦ ❛ ❢r♦♥t❡✐r❛ ❞❛ ❜❛❝✐❛ ❞❡ ❛tr❛çã♦ ❞❡ ✉♠ ♣♦♥t♦ ✜①♦
■♥tr♦❞✉çã♦
❛tr❛t♦r ❞❡ f✳ P❛r❛ ✐st♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s
❝♦♠♣❧❡①♦s✳ ❉❡✜♥✐r❡♠♦s t❛♠❜é♠ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦✱ ❝✉❥❛ ❢r♦♥t❡✐r❛ é ✐❣✉❛❧ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛✳ ❆❜♦r❞❛r❡♠♦s ❛ ❢❛♠í❧✐❛ F ❞❛s ❢✉♥çõ❡s q✉❛❞rát✐❝❛s ❞❛
❢♦r♠❛ f : C → C ❞❛❞❛s ♣♦r fc(z) = z2 +c✱ ♦♥❞❡ c é ✉♠❛ ❝♦♥st❛♥t❡ ❝♦♠♣❧❡①❛ ❡
❞❡✜♥✐r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✳ ◆❛ ú❧t✐♠❛ s❡çã♦✱ ♦s s♦❢t✇❛r❡s ▼❛t▲❛❜ ❡ ●❡♦●❡❜r❛ s❡rã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ✈✐s✉❛❧✐③❛r ✉♠❛ ✜❣✉r❛ ❛♣r♦①✐♠❛❞❛ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ♦✉tr❛ ❞❡✜♥✐çã♦ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠♣❧❡①❛ ❞❡ ❣r❛✉n≥2❡ ❡st✉❞❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s
t♦♣♦❧ó❣✐❝❛s✳ ▼♦str❛r❡♠♦s ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❛s ❞❡✜♥✐çõ❡s ❛❞♦t❛❞❛s✳ ◆❛ ú❧t✐♠❛ s❡çã♦ ♥♦ss♦ ❡st✉❞♦ s❡rá ✈♦❧t❛❞♦ às ❢✉♥çõ❡s q✉❛❞rát✐❝❛s ❡ ♠♦str❛r❡♠♦s q✉❡ ❛ ❞❡✜♥✐çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✱ ❛♣r❡s❡♥t❛❞❛ ♥♦ ❝❛♣ít✉❧♦ ✶✱ é ❡q✉✐✈❛❧❡♥t❡ à ór❜✐t❛ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ fc s❡r ❧✐♠✐t❛❞❛✳
❖ ❧❡✐t♦r ♥ã♦ ❢❛♠✐❧✐❛r✐③❛❞♦ ❝♦♠ ❛❧❣✉♥s tó♣✐❝♦s ❞❡ ❱❛r✐á✈❡✐s ❈♦♠♣❧❡①❛s ♣♦❞❡ ❝♦♥s✉❧t❛r ♦ ❛♣ê♥❞✐❝❡✱ ♦ q✉❛❧ ❢♦r♥❡❝❡ ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s s♦❜r❡✿ ❧✐♠✐t❡s ❞❡ s❡q✉ê♥❝✐❛s❀ ❛❧❣✉♠❛s ♥♦çõ❡s ❞❛ t♦♣♦❧♦❣✐❛ ❞❡C❀ ❢✉♥çã♦✱ ❧✐♠✐t❡ ❡ ❝♦♥t✐♥✉✐❞❛❞❡❀ ❢✉♥çã♦
❛♥❛❧ít✐❝❛ ❡ ❝♦♥❥✉❣❛çã♦ t♦♣♦❧ó❣✐❝❛✳
❈❛♣ít✉❧♦ ✶
❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛✿ ❞❡✜♥✐çã♦ ❡
♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛
✈✐s✉❛❧✐③❛r ✉♠❛ ✜❣✉r❛ ❛♣r♦①✐♠❛❞❛
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝♦♠♣❧❡①♦s ❡ ❞❡✜♥✐r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠♣❧❡①❛ ❞❡ ❣r❛✉n ≥2
❝♦♠♦ ❛ ❢r♦♥t❡✐r❛ ❞❛ ❜❛❝✐❛ ❞❡ ❛tr❛çã♦ ❞❡ ✉♠ ♣♦♥t♦ ✜①♦ ❛tr❛t♦r ❞❡ f✳ ❉❡✜♥✐r❡♠♦s
t❛♠❜é♠ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦✱ ❝✉❥❛ ❢r♦♥t❡✐r❛ é ✐❣✉❛❧ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛✳ ❆❜♦r❞❛r❡♠♦s ❛ ❢❛♠í❧✐❛ F ❞❛s ❢✉♥çõ❡s q✉❛❞rát✐❝❛s ❞❛ ❢♦r♠❛ f : C → C ❞❛❞❛s
♣♦r fc(z) = z2 + c✱ ♦♥❞❡ c é ✉♠❛ ❝♦♥st❛♥t❡ ❝♦♠♣❧❡①❛ ❡ ❞❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✳ ◆❛ ú❧t✐♠❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ✈✐s✉❛❧✐③❛r ✉♠❛ ✜❣✉r❛ ❛♣r♦①✐♠❛❞❛ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✱ ✉t✐❧✐③❛♥❞♦ ♦s s♦❢t✇❛r❡s ▼❛t▲❛❜ ❡ ●❡♦●❡❜r❛✳
❙✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝♦♠♣❧❡①♦s ❈❛♣ít✉❧♦ ✶
✶✳✶ ❙✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝♦♠♣❧❡①♦s
❉❡✜♥✐çã♦ ✶✳✶ ❯♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ ❝♦♥s✐st❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ X ❞❡ ❡st❛❞♦s
♣♦ssí✈❡✐s✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ✉♠❛ r❡❣r❛ q✉❡ ❞❡s❝r❡✈❡ ❝♦♠♦ ♦ s✐st❡♠❛ ❡✈♦❧✉✐ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✳ ◗✉❛♥❞♦ ♦ t❡♠♣♦ ❛ss✉♠❡ ✈❛❧♦r❡s ♥❛t✉r❛✐s ♦✉ ✐♥t❡✐r♦s✱ ♦ s✐st❡♠❛ ❞✐♥â♠✐❝♦ é ❞✐t♦ ❞✐s❝r❡t♦✳ ◗✉❛♥❞♦ ♦ t❡♠♣♦ ❛ss✉♠❡ ✈❛❧♦r❡s r❡❛✐s✱ ♦ s✐st❡♠❛ ❞✐♥â♠✐❝♦ é ❞✐t♦ ❝♦♥tí♥✉♦✳
❊st✉❞❛r❡♠♦s s✐st❡♠❛s ♦♥❞❡ ❛ r❡❣r❛ q✉❡ ♦ ❞❡s❝r❡✈❡ é ✉♠❛ ❢✉♥çã♦ f :X →X✳
◆❡st❛ ❞✐ss❡rt❛çã♦✱ ❞❡♥♦t❛r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣♦r N ❡ ♦
❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♥ã♦ ♥✉❧♦s ♣♦r N\ {0}✳
❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ X ❡ ✉♠❛ ❢✉♥çã♦ f : X → X✱ ❛ k ✲és✐♠❛ ✐t❡r❛❞❛ ❞❡ f é ❞❡✜♥✐❞❛ ♣♦r
f0(x) = x,
fk(x) = f(fk−1(x))✱ ♣❛r❛k ≥1.
❊♠ ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ ❞✐s❝r❡t♦✱ ❞❡♥♦t❡♠♦s ♦ t❡♠♣♦ ♣♦r k ❡ ♦ s✐st❡♠❛ é
❡s♣❡❝✐✜❝❛❞♦ ♣❡❧❛s ❡q✉❛çõ❡s
x(0) = x0,
x(k+ 1) = f(x(k)).
❯s❛♥❞♦ ❛ ♥♦t❛çã♦ x(k) =xk ❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ✐t❡r❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ f ♣♦❞❡♠♦s ❡s❝r❡✈❡r
x0 = f0(x0),
xk+1 = f(xk).
❙✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝♦♠♣❧❡①♦s ❈❛♣ít✉❧♦ ✶
❉❡st❛ ❢♦r♠❛✱
x0 = f0(x0)
x1 = f(x0)
x2 = f(x1) =f(f(x0)) = f2(x0)
x3 = f(x2) =f(f2(x0)) = f3(x0) ✳✳✳
xk = f(xk−1) =f(fk−1(x0)) =fk(x0)
❉❡✜♥✐çã♦ ✶✳✷ ❙❡❥❛♠ ❛ ❢✉♥çã♦ f : X → X ❡ x0 ∈ C✳ ❆ ór❜✐t❛ ❞❡ x0 ♣♦r f é ❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s x0, x1 =f(x0), x2 =f2(x0), x3 =f3(x0), . . . , xk=fk(x0), . . . ❉❡✜♥✐çã♦ ✶✳✸ ❙❡❥❛♠ ❛ ❢✉♥çã♦ f : X → X ❡ x0 ∈ C✳ ❆ ór❜✐t❛ ❞❡ x0 ♣♦r f é ❧✐♠✐t❛❞❛ s❡ ❡①✐st❡ M > 0 t❛❧ q✉❡ |fk(x
0)| ≤M✱ ♣❛r❛ t♦❞♦ k∈N✳
❆❜♦r❞❛r❡♠♦s s✐st❡♠❛s ❞✐♥â♠✐❝♦s ❞✐s❝r❡t♦s ♦♥❞❡ f é ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ ❞❡
✉♠❛ ✈❛r✐á✈❡❧ ❝♦♠♣❧❡①❛✳ ❊s♣❡❝✐✜❝❛♠❡♥t❡✱ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡ ❣r❛✉ n ≥ 2 ❝♦♠
❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s✱ ♦✉ s❡❥❛✱ f :C→C ❞❛❞❛ ♣♦r
f(z) =anzn+an−1zn−1+an−2zn−2+. . .+a2z2+a1z+a0. ❊st❛♠♦s ✐♥t❡r❡ss❛❞♦s ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ór❜✐t❛ ❞❡ ♣♦♥t♦sz0 ∈C✳
❊①❡♠♣❧♦ ✶✳✶ ❙❡❥❛ f : C → C ❞❛❞❛ ♣♦r f(z) = z2✳ ❱❛♠♦s ♦❜s❡r✈❛r ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❛ ór❜✐t❛ ❞❡ ♣♦♥t♦s ♣❡rt❡♥❝❡♥t❡s ❛ três r❡❣✐õ❡s ❞✐s❥✉♥t❛s q✉❡ ❞✐✈✐❞❡♠ ♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✳ ❙❡❥❛♠
R1 = {z ∈ C; |z| < 1}✱ R2 = {z ∈ C; |z| = 1} ❡ R3 = {z ∈ C; |z| > 1}✳ ❱❛♠♦s ❞❡t❡r♠✐♥❛r ♦s ❝✐♥❝♦ ♣r✐♠❡✐r♦s ❡❧❡♠❡♥t♦s ❞❛ ór❜✐t❛ ❞❡ z0 = 0,9 + 0,1i∈R1✱
❙✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝♦♠♣❧❡①♦s ❈❛♣ít✉❧♦ ✶
✈✐s✉❛❧✐③á✲❧♦s ♥♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦ ✭✜❣✉r❛ ✶✳✶✮ ❡ ❝❛❧❝✉❧❛r ♦ ♠ó❞✉❧♦ ❞❡ ❝❛❞❛ ✉♠ ❞❡❧❡s✳
z0 = 0,9 + 0,1i
z1 = f(z0) = 0,8 + 0,18i
z2 = f2(x0) = 0,6076 + 0,288i
z3 = f3(x0) = 0,28623376 + 0,3499776i
z4 = f4(x0) = −0,0405545551380224 + 0,200350808727552i
|z0| ∼= 0,91 > |z1|= 0,82 > |z2| ∼= 0,67 > |z3| ∼= 0,45 > |z4| ∼= 0,20 ❖❜s❡r✈❡ q✉❡ ❛ ❝❛❞❛ ✐t❡r❛çã♦ ❞❛ ❢✉♥çã♦f ♦❜t❡♠♦s ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ❝♦♠ ♠ó❞✉❧♦
♠❡♥♦r q✉❡ ♦ ❛♥t❡r✐♦r✳
❋✐❣✉r❛ ✶✳✶✿ ■t❡r❛çã♦ ❞❡z0 = 0,9 + 0,1i ❡ w0 = 0,6 + 0,8i✳
❙✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝♦♠♣❧❡①♦s ❈❛♣ít✉❧♦ ✶
❞❡ w0 = 0,6 + 0,8i∈R2 ❡ ♥❛ ✜❣✉r❛ ✶✳✷✱ ♦s q✉❛tr♦ ♣r✐♠❡✐r♦s ❡❧❡♠❡♥t♦s ❞❛ ór❜✐t❛ ❞♦ ♣♦♥t♦ p0 = 1,35 + 0,31i∈R3✳
❋✐❣✉r❛ ✶✳✷✿ ■t❡r❛çã♦ ❞❡ p0 = 1,35 + 0,31i✳
◆♦t❡ q✉❡ ❛ ❡s❝♦❧❤❛ ❞❛s r❡❣✐õ❡s ♥ã♦ ❢♦✐ ❛r❜✐trár✐❛✳ P♦❞❡♠♦s ♣r♦✈❛r q✉❡ ♣♦♥t♦s ♣❡rt❡♥❝❡♥t❡s ❛♦ ✐♥t❡r✐♦r ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❛✐♦ ✶ ♣♦ss✉❡♠ ór❜✐t❛s ❝✉❥❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s ❝♦♥✈❡r❣❡ ♣❛r❛ ❛ ♦r✐❣❡♠✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣♦♥t♦s s♦❜r❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♦r✐❣✐♥❛♠ s❡q✉ê♥❝✐❛s q✉❡ ♣❡r♠❛♥❡❝❡♠ s♦❜r❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ♣♦♥t♦s ♥♦ ❡①t❡r✐♦r ♦r✐❣✐♥❛♠ s❡q✉ê♥❝✐❛s q✉❡ t❡♥❞❡♠ ❛♦ ✐♥✜♥✐t♦✳ ❉❡ ❢❛t♦✱ ❝♦♠♦
fk(z) = z2k
❡ lim
k→∞
fk(z) = lim
k→∞
z2
k
= limk→∞|z|2 k
✱ ❛♥❛❧✐s❛♥❞♦ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❧✐♠✐t❡ ❞❛ ór❜✐t❛ ❞❡ ♣♦♥t♦s ❞♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦ ♣❡rt❡♥❝❡♥t❡s ❛R1✱R2 ❡R3✱ ♦❜t❡♠♦s✿
❙❡ |z|<1✱ ❡♥tã♦ fk(z)=|z|2k →0 q✉❛♥❞♦ k → ∞✳
❙❡ |z|= 1✱ ❡♥tã♦ fk(z)= 1 ♣❛r❛ t♦❞♦ k✳
❙✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝♦♠♣❧❡①♦s ❈❛♣ít✉❧♦ ✶
❙❡ |z|>1✱ ❡♥tã♦ fk(z)=|z|2k → ∞ q✉❛♥❞♦ k → ∞✳
❙❡ r❡str✐♥❣✐r♠♦s ♥♦ss❛ ❛t❡♥çã♦ ♣❛r❛ ♣♦♥t♦s r❡❛✐s✱ ♦✉ s❡❥❛✱ z0 ∈R ❡ ♦❜s❡r✈❛r♠♦s ❛s ✐t❡r❛❞❛s fk(z
0)✱ ♣♦❞❡♠♦s ❛♥❛❧✐s❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ór❜✐t❛ ❛tr❛✈és ❞❡ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ❣❡♦♠étr✐❝♦ ❞❡♥♦♠✐♥❛❞♦ ❛♥á❧✐s❡ ❣rá✜❝❛✳
■♥✐❝✐❛❧♠❡♥t❡ ❞❡✈❡♠♦s ❢❛③❡r ❡♠ ✉♠ ♠❡s♠♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ♦ ❣rá✜❝♦ ❞❡ f ❡ ❞❛
❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ g : R → R ❞❛❞❛ ♣♦r g(z) = z ✭✜❣✉r❛ ✶✳✸✮✳ ❆ ♣❛rt✐r ❞♦ ♣♦♥t♦
(z0,0) tr❛❝❡♠♦s ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ♣❛r❛❧❡❧♦ ❛♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s ✐♥t❡rs❡❝t❛♥❞♦ ♦ ❣rá✜❝♦ ❞❡f ♥♦ ♣♦♥t♦ (z0, f(z0))✳ ❈♦♠ ♦r✐❣❡♠ ♥❡st❡ ♣♦♥t♦✱ tr❛❝❡♠♦s ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ♣❛r❛❧❡❧♦ ❛♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ❛té ✐♥t❡rs❡❝t❛r ❛ r❡t❛ g(z) = z ♥♦ ♣♦♥t♦
(f(z0), f(z0))✳ ❊ ❛ ♣❛rt✐r ❞❡st❡ ♣♦♥t♦ tr❛❝❡♠♦s ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s ❛té ✐♥t❡rs❡❝t❛r ♦ ❣rá✜❝♦ ❞❡ f ♥♦ ♣♦♥t♦ (f(z0), f2(z0))✳ ❈♦♥t✐♥✉❛♥❞♦ ❡st❡ ♣r♦❝❡❞✐♠❡♥t♦ ♦❜t❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s s♦❜r❡ ♦ ❣rá✜❝♦ ❞❡ f
(z0, f(z0)), f(z0), f2(z0)
, f2(z0), f3(z0)
, f3(z0), f4(z0)
, . . .
❝✉❥❛ ♣r♦❥❡çã♦ s♦❜r❡ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ♥♦s ❢♦r♥❡❝❡ ❡❧❡♠❡♥t♦s ❞❛ ór❜✐t❛ ❞❡z0✳
❙✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝♦♠♣❧❡①♦s ❈❛♣ít✉❧♦ ✶
◆❛ ✜❣✉r❛ ✶✳✸ ♣♦❞❡♠♦s ✈❡r ♦s q✉❛tr♦ ♣r✐♠❡✐r♦s ❡❧❡♠❡♥t♦s ❞❛ ór❜✐t❛ ❞❡z0 = 0,8✳ ❖❜s❡r✈❡ q✉❡ ❛s ✜❣✉r❛s ✶✳✸ ❡ ✶✳✹ ♥♦s ❛❥✉❞❛♠ ❛ ❞❡s❝r❡✈❡r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ór❜✐t❛ ❞❡ ♣♦♥t♦s r❡❛✐s ♣❡❧❛ ❢✉♥çã♦f(z) =z2✳ ❆ ór❜✐t❛ ❞❡ ♣♦♥t♦s z
0 ❝♦♠ |z0|<1 ❝♦♥✈❡r❣❡ ♣❛r❛ ❛ ♦r✐❣❡♠✱ ❡♥q✉❛♥t♦ ❛ ór❜✐t❛ ❞❡ ♣♦♥t♦s ❝♦♠|z0|>1 ❡s❝❛♣❛ ♣❛r❛ ♦ ✐♥✜♥✐t♦✳ ❖s ♣♦♥t♦s ✵ ❡ ✶ ♣♦ss✉❡♠ ór❜✐t❛s ❝♦♥st❛♥t❡s✳ ◆♦t❡ q✉❡ ❡❧❡s s❛t✐s❢❛③❡♠f(z) =z✳
❋✐❣✉r❛ ✶✳✹✿ ❆♥á❧✐s❡ ❣rá✜❝❛ ❞❛ ór❜✐t❛ ❞❡ z0 ∈R ♣❡❧❛ ❢✉♥çã♦f(z) =z2✳ ❉❡✜♥✐çã♦ ✶✳✹ ❙❡❥❛♠ z∈C ❡ f ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠♣❧❡①❛✳ ❉✐③❡♠♦s q✉❡
❛✮ z é ✉♠ ♣♦♥t♦ ❞❡ ✜①♦ ❞❡ f s❡ f(z) =z✳
❜✮ z é ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❞❡ f s❡ ❡①✐st✐r ✉♠ ✐♥t❡✐r♦ p ≥1 t❛❧ q✉❡ fp(z) = z✳ ❖ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ p t❛❧ q✉❡ fp(z) =z é ❝❤❛♠❛❞♦ ♦ ♣❡rí♦❞♦ ❞❡ z✳
❙✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝♦♠♣❧❡①♦s ❈❛♣ít✉❧♦ ✶
❉❡✜♥✐çã♦ ✶✳✺ ❙❡❥❛♠ z ✉♠ ♣♦♥t♦ ♣❡r✐ó❞✐❝♦ ❞❡ f ❞❡ ♣❡rí♦❞♦ p ❡ λ= (fp)′(z)✳
❖ ♣♦♥t♦ z é ❞❡♥♦♠✐♥❛❞♦
superatrator se λ= 0;
atrator se 0≤ |λ|<1;
indif erente se |λ|= 1;
repulsor se |λ|>1.
❈♦♥s✐❞❡r❛♥❞♦ ❛ ❢✉♥çã♦ ❞♦ ❊①❡♠♣❧♦ ✶✳✶✱ f : C → C ❞❛❞❛ ♣♦r f(z) = z2✱ ♦s
♣♦♥t♦s ✜①♦s ✜♥✐t♦s ♣♦❞❡♠ s❡r ❞❡t❡r♠✐♥❛❞♦s r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦z2 =z✱ ♦ q✉❡ ♥♦s ❞❛r z = 0 ❡ z = 1✳ ❉❡ f′(z) = 2z✱ s❡❣✉❡ q✉❡ |f′(0)| = 0 ❡ |f′(1)| = 2 > 1✳ ▲♦❣♦✱
z = 0 é ✉♠ ♣♦♥t♦ ✜①♦ ❛tr❛t♦r ❡z = 1 é ✉♠ ♣♦♥t♦ ✜①♦ r❡♣✉❧s♦r ❞❡ f✳
❈♦♠♦ ✈✐♠♦s ♥♦ ❊①❡♠♣❧♦ ✶✳✶✱ ♦s ♣♦♥t♦s ♣❡rt❡♥❝❡♥t❡s ❛♦ ✐♥t❡r✐♦r ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❛✐♦ ✶ ♣♦ss✉❡♠ ór❜✐t❛s ❝✉❥❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s ❝♦♥✈❡r❣❡ ♣❛r❛ ❛ ♦r✐❣❡♠✳ P♦❞❡♠♦s ✐♥t❡r♣r❡t❛r ❡st❛ r❡❣✐ã♦ ❝♦♠♦ s❡♥❞♦ ✉♠❛ ❜❛❝✐❛ ❞❡ ❛tr❛çã♦ ❝♦♠ ♣♦♥t♦ ❛tr❛t♦r z = 0✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣♦♥t♦s ♥♦ ❡①t❡r✐♦r ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦
♥❛ ♦r✐❣❡♠ ❡ r❛✐♦ ✶ ♦r✐❣✐♥❛♠ s❡q✉ê♥❝✐❛s q✉❡ t❡♥❞❡♠ ❛♦ ✐♥✜♥✐t♦✱ s❡♥❞♦ ❡st❛ r❡❣✐ã♦ ❛ ❜❛❝✐❛ ❞❡ ❛tr❛çã♦ ❞♦ ✐♥✜♥✐t♦✳
❖ ✐♥✜♥✐t♦ é t❛♠❜é♠ ✉♠ ♣♦♥t♦ ✜①♦ ❞❡ f(z) = z2✱ ❡ ❞❡✜♥✐♠♦s λ = 1
f′(∞) = 0✳
▲♦❣♦✱ ♦ ✐♥✜♥✐t♦ é ✉♠ ♣♦♥t♦ ✜①♦ ❛tr❛t♦r ❞❡f✳
❉❡✜♥✐çã♦ ✶✳✻ ❙❡❥❛ w ✉♠ ♣♦♥t♦ ✜①♦ ❛tr❛t♦r ❞❡ f✳ ❆ ❜❛❝✐❛ ❞❡ ❛tr❛çã♦ ❞❡ w✱
❞❡♥♦t❛❞❛ ♣♦r Af(w)✱ é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s ❝✉❥❛s ór❜✐t❛s t❡♥❞❡♠ à w✱ ♦✉ s❡❥❛✱ Af(w) =
z∈C; fk(z)→w q✉❛♥❞♦ k→ ∞ ✳ ❆ ❜❛❝✐❛ ❞❡ ❛tr❛çã♦ ❞♦ ✐♥✜♥✐t♦✱ Af(∞)✱ é ❞❡✜♥✐❞❛ ❞♦ ♠❡s♠♦ ♠♦❞♦✳
P❛r❛ ❛ ❢✉♥çã♦ f :C→C❞❛❞❛ ♣♦rf(z) =z2✱ t❡♠♦s Af(0) ={z ∈C; |z|<1} ❡
Af(∞) ={z ∈C; |z|>1}✳
❙✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝♦♠♣❧❡①♦s ❈❛♣ít✉❧♦ ✶
❖ t❡♦r❡♠❛ s❡❣✉✐♥t❡ s❡rá út✐❧ ♣❛r❛ ♠♦str❛r q✉❡ ❛ ❜❛❝✐❛ ❞❡ tr❛çã♦ é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✳
❚❡♦r❡♠❛ ✶✳✶ ✭❞♦ ♣♦♥t♦ ✜①♦ ❛tr❛t♦r✮ ❙❡❥❛♠ A ⊂ C ❡ w ✉♠ ♣♦♥t♦ ✜①♦ ❛tr❛t♦r
❞❛ ❢✉♥çã♦ f : A → A✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❜♦❧❛ ❛❜❡rt❛ ❞❡ ❝❡♥tr♦ ❡♠ w ❡ r❛✐♦ δ ♥❛
q✉❛❧ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦ é s❛t✐s❢❡✐t❛✿ s❡ z ∈Bδ(w)∩A✱ ❡♥tã♦ fk(z)∈Bδ(w)∩A ❡
fk(z)
→w q✉❛♥❞♦ k → ∞✳
❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ |f′
(w)| < 1✱ t♦♠❡ r = 1 +|f
′
(w)|
2 ✳ ❉❡st❛ ❢♦r♠❛✱
|f′(w)|< r <1✳ P♦r ❞❡✜♥✐çã♦✱ f′(w) = lim
z→w
f(z)−f(w)
z−w . P♦rt❛♥t♦✱
∀ε >0, ∃δ >0 t❛❧ q✉❡ 0<|z−w|< δ ⇒
f(z)−f(w)
z−w −f
′(w)
< ε.
P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱
f(z)−f(w)
z−w
−|f
′(w)|6
f(z)−f(w)
z−w −f
′(w)
< ε ❧♦❣♦✱
f(z)−f(w)
z−w
< ε+|f
′(w)|.
❚♦♠❡ ε =r− |f′(w)|>0✳ ❈♦♠ ✐st♦✱
f(z)−f(w)
z−w
< r ♣❛r❛ t♦❞♦ z ∈A\ {w} ❝♦♠ |z−w|< δ.
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣❛r❛ z∈A ❝♦♠ |z−w|< δ✱ t❡♠♦s
|f(z)−w| ≤r|z−w|< rδ, ♦♥❞❡ 0< r <1. ✭✶✳✶✮
■st♦ s✐❣♥✐✜❝❛ q✉❡f(z)❡stá ♠❛✐s ♣ró①✐♠♦ ❞❡w❞♦ q✉❡z✳ P♦rt❛♥t♦✱f(z)∈Bδ(w)∩A✳ ❖❜s❡r✈❡ q✉❡ f2(z) = f(f(z))✱ ❝♦♠ f(z)∈B
δ(w)∩A✳ ❆♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✶✳✶ ❞✉❛s ✈❡③❡s✱ ♦❜t❡♠♦s
|f2(z)−w|=|f(f(z))−w| ≤r|f(z)−w| ≤r.r|z−w|< r2.δ, ♦♥❞❡ 0< r <1.
❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❈❛♣ít✉❧♦ ✶
▲♦❣♦✱ f2(z)∈B
δ(w)∩A✳
P♦r ✐♥❞✉çã♦ ❡♠ k✱ ♣❛r❛ z ∈ A ❝♦♠ |z −w| < δ✱ t❡♠♦s |fk(z)−w| < rk.δ✱ ♦♥❞❡ 0 < r < 1✳ P♦rt❛♥t♦✱ fk(z) ∈ B
δ(w)∩A✳ ◗✉❛♥❞♦ k → ∞✱ rk → 0 ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ fk(z)→w✳
❚❡♦r❡♠❛ ✶✳✷ ✭❞♦ ♣♦♥t♦ ✜①♦ r❡♣✉❧s♦r✮ ❙❡❥❛♠A⊂C❡w✉♠ ♣♦♥t♦ ✜①♦ r❡♣✉❧s♦r
❞❛ ❢✉♥çã♦ f :A→A✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❜♦❧❛ ❛❜❡rt❛ ❞❡ ❝❡♥tr♦ ❡♠ w ❡ r❛✐♦δ ♥❛ q✉❛❧
❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦ é s❛t✐s❢❡✐t❛✿ s❡ z ∈Bδ(w)∩A❡ z 6=w✱ ❡♥tã♦ ❡①✐st❡n0 ∈N\{0} t❛❧ q✉❡ fn(z)∈/ B
δ(w)∩A ♣❛r❛ t♦❞♦ n≥n0✳ ❉❡♠♦♥str❛çã♦✿ ❆♥á❧♦❣❛ ❛ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳
Pr♦♣♦s✐çã♦ ✶✳✶ Af(w) é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✳
❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ w é ✉♠ ♣♦♥t♦ ✜①♦ ❛tr❛t♦r ❞❡ f✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✶✱ ❡①✐st❡
✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ V ❝♦♥t❡♥❞♦w t❛❧ q✉❡ V ⊂Af(w) ✭s❡w=∞✱ ♣♦❞❡♠♦s t♦♠❛r
{z; |z| > r}✱ ♣❛r❛ r s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✮✳ ■st♦ ✐♠♣❧✐❝❛ q✉❡ Af(w) é ❛❜❡rt♦✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ z ∈ Af(w)✱ ❡♥tã♦ fk(z)∈ V ♣❛r❛ ❛❧❣✉♠ k✱ ❧♦❣♦ z ∈ f−k(V)✱ ♦ q✉❛❧ é ❛❜❡rt♦✳
✶✳✷ ❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛✿ ❞❡✜♥✐çã♦ ❡ ❡①❡♠♣❧♦
❊①✐st❡♠ ❢♦r♠❛s ❡q✉✐✈❛❧❡♥t❡s ❞❡ ❞❡✜♥✐r ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❡ ✉♠❛ ❢✉♥çã♦ f✳
◆❡st❛ s❡çã♦ ❞❡✜♥✐r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ✉t✐❧✐③❛♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❜❛❝✐❛ ❞❡ ❛tr❛çã♦✳ ◆♦ ❝❛♣ít✉❧♦ ✷ ✈❡r❡♠♦s ♦✉tr❛ ❞❡✜♥✐çã♦ ❡ ❞❡♠♦♥str❛r❡♠♦s ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❡❧❛s✳
❉❡✜♥✐çã♦ ✶✳✼ ❙❡❥❛♠ f ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠♣❧❡①❛ ❞❡ ❣r❛✉ n ≥ 2 ❡ w ✉♠
♣♦♥t♦ ✜①♦ ❛tr❛t♦r ❞❡f✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❡f✱ ❞❡♥♦t❛❞♦ ♣♦rJ(f)✱ é ❛ ❢r♦♥t❡✐r❛
❞❛ ❜❛❝✐❛ ❞❡ ❛tr❛çã♦ ❞❡ w✳ ❖ ❝♦♠♣❧❡♠❡♥t❛r ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ é ❞❡♥♦♠✐♥❛❞♦
❝♦♥❥✉♥t♦ ❞❡ ❋❛t♦✉✱ ❋✭❢✮✳
❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❈❛♣ít✉❧♦ ✶
P❛r❛ ❛ ❢✉♥çã♦ f : C → C ❞❛❞❛ ♣♦r f(z) = z2✱ ✈✐♠♦s q✉❡ Af(0) =
{z ∈ C; |z| < 1} ❡ Af(∞) = {z ∈ C; |z| > 1}✳ P♦rt❛♥t♦✱ ∂Af(0) = ∂Af(∞) =
{z ∈C; |z|= 1}✳ ▲♦❣♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❡ f é ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ ❡♠ ✵
❡ r❛✐♦ ✶ ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❋❛t♦✉ ❞❡ f ❡ ❛ ✉♥✐ã♦ ❞✐s❥✉♥t❛ ❡♥tr❡ ♦ ✐♥t❡r✐♦r ❡ ♦ ❡①t❡r✐♦r
❞❡st❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳
❆✐♥❞❛ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❊①❡♠♣❧♦ ✶✳✶✱ ♦❜s❡r✈❡ q✉❡ ♦s ♣♦♥t♦sz ∈Ct❛✐s q✉❡ |z| ≤1
♣♦ss✉❡♠ ór❜✐t❛s ❧✐♠✐t❛❞❛s✱ t❛♠❜é♠ ❞❡♥♦♠✐♥❛❞❛s ♣r✐s✐♦♥❡✐r❛s ❡ ♦s ♣♦♥t♦s t❛✐s q✉❡
|z| > 1 ♣♦ss✉❡♠ ór❜✐t❛s ✐❧✐♠✐t❛❞❛s✱ ♦✉ s❡❥❛✱ ór❜✐t❛s q✉❡ ❡s❝❛♣❛♠ ♣❛r❛ ♦ ✐♥✜♥✐t♦✳
❉❡st❛ ❢♦r♠❛✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿
❉❡✜♥✐çã♦ ✶✳✽ ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠♣❧❡①❛ ❞❡ ❣r❛✉ n ≥ 2✳ ❖
❝♦♥❥✉♥t♦ ♣r✐s✐♦♥❡✐r♦ ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❡s❝❛♣❡ ♣❛r❛ ❛ ❢✉♥çã♦ f sã♦ ❞❡✜♥✐❞♦s✱
r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r
K(f) = z ∈C; |fk(z)|9∞ quando k → ∞ ✱ ❡
E(f) = z ∈C; |fk(z)| → ∞ quando k → ∞ ✳
❖❜s❡r✈❡ q✉❡ C é ❛ ✉♥✐ã♦ ❞✐s❥✉♥t❛ ❡♥tr❡ K(f) ❡ E(f)✳ ■st♦ ♥♦s ❧❡✈❛ ❛ ✉♠❛
❞✐❝♦t♦♠✐❛✿ ♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦ é ❞✐✈✐❞✐❞♦ ❡♠ ❞♦✐s s✉❜❝♦♥❥✉♥t♦s ❝✉❥❛ ✐♥t❡rs❡çã♦ é ✈❛③✐❛✳ ◆♦t❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ E(f) = Af(∞) ❡ ♥♦ ❊①❡♠♣❧♦ ✶✳✶✱ ■♥t(K(f)) = Af(0)✳ ◆❛ ú❧t✐♠❛ s❡çã♦ ❞❡st❡ ❝❛♣ít✉❧♦ ✉t✐❧✐③❛r❡♠♦s ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ✈✐s✉❛❧✐③❛r ✉♠❛ ✜❣✉r❛ ❛♣r♦①✐♠❛❞❛ ❞♦ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ❛ s❡❣✉✐r✳
❉❡✜♥✐çã♦ ✶✳✾ ❙❡❥❛f ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠♣❧❡①❛ ❞❡ ❣r❛✉n ≥2✳ ❖ ❝♦♥❥✉♥t♦
❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❛ ❢✉♥çã♦ f é ❞❡✜♥✐❞♦ ♣♦r
K(f) = z ∈C; |fk(z)|9∞ quando k → ∞ ✳
❖❜s❡r✈❛çã♦ ✶✳✶ ❈♦♠♦ J(f) = ∂Af(∞) = ∂E(f) = ∂K(f)✱ s❡❣✉❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❡ f é ❛ ❢r♦♥t❡✐r❛ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦✳
❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ❈❛♣ít✉❧♦ ✶
❆✜r♠❛çã♦ ✶✳✶ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ sã♦ ❢❡❝❤❛❞♦s✳
❉❡♠♦♥str❛çã♦✿ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ é ❢❡❝❤❛❞♦ ♣♦r s❡r ❢r♦♥t❡✐r❛ ❞❡ ✉♠ ❝♦♥❥✉♥t♦✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✱ Af(∞) é ❛❜❡rt♦✳ ❈♦♠♦ Af(∞) = E(f) ❡ K(f) sã♦ ❝♦♠♣❧❡♠❡♥t❛r❡s✱ s❡❣✉❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ é ❢❡❝❤❛❞♦✳
P❛r❛ ❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ f(z) = z2 t❡♠♦s✱ J(f) = {z ∈ C; |z| = 1} ❡
K(f) = {z ∈C; |z| ≤1}✳
◆♦t❡ q✉❡ ♥❡st❡ ❡①❡♠♣❧♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ é ✉♠ ♦❜❥❡t♦ ❞❛ ❣❡♦♠❡tr✐❛ ❡✉❝❧✐❞✐❛♥❛✱ ♥ã♦ s❡♥❞♦✱ ♣♦rt❛♥t♦✱ ✉♠ ❢r❛❝t❛❧✳ ◆♦ ❡♥t❛♥t♦✱ ❡st❡ é ✉♠ ❝❛s♦ ❡s♣❡❝✐❛❧✱ ♣♦✐s ❛ ♠❛✐♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s ❞❡ ❏✉❧✐❛ é ✉♠ ❢r❛❝t❛❧✳
❆✜r♠❛çã♦ ✶✳✷ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ é s✐♠étr✐❝♦ ❡♠ r❡❧❛çã♦ à ♦r✐❣❡♠✳
❉❡♠♦♥str❛çã♦✿ ❉❡✈❡♠♦s ♠♦str❛r q✉❡z ∈J(f)s❡✱ ❡ s♦♠❡♥t❡ s❡✱−z ∈J(f)✳ ❙❡❥❛
z ∈C✳ ❉❡ fk
c(z) =fck−1(fc(z)) =fck−1(z2+c) = fck−1((−z)2 +c) =fck−1(fc(−z)) =
fk
c(−z)✱ s❡❣✉❡ q✉❡ fck(z) → ∞ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ fck(−z) → ∞✳ ▲♦❣♦✱ z ∈ K(fc) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ −z ∈ K(fc)✳ P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡ fc ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ s✉❛ ❢r♦♥t❡✐r❛✱J(f)✱ sã♦ s✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦ à ♦r✐❣❡♠✳
✶✳✸ ❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t
❙❡❥❛ F ❛ ❢❛♠í❧✐❛ ❞❛s ❢✉♥çõ❡s q✉❛❞rát✐❝❛s ❞❛ ❢♦r♠❛ fc : C → C ❞❛❞❛s ♣♦r
fc(z) =z2+c✱ ♦♥❞❡ cé ✉♠❛ ❝♦♥st❛♥t❡ ❝♦♠♣❧❡①❛✳
❱❡r❡♠♦s ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦ q✉❡ ✏t♦❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ é t♦♣♦❧♦❣✐❝❛♠❡♥t❡ ❝♦♥❥✉❣❛❞❛ ❛ ❛❧❣✉♠ ♠❡♠❜r♦ ❞❛ ❢❛♠í❧✐❛ q✉❛❞rát✐❝❛F ✑ ❬Pr♦♣♦s✐çã♦ ✷✳✾❪✳ ■st♦ s✐❣♥✐✜❝❛
q✉❡ ❛♦ ❡st✉❞❛r♠♦s ♦s ❝♦♥❥✉♥t♦s ❞❡ ❏✉❧✐❛ ❞❡ fc✱ ❝♦♠ c∈C✱ ❡st✉❞❛♠♦s ♦s ❝♦♥❥✉♥t♦s
❞❡ ❏✉❧✐❛ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s q✉❛❞rát✐❝❛s✳
❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ❈❛♣ít✉❧♦ ✶
❉❡✜♥✐çã♦ ✶✳✶✵ ❖ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t M é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣❛râ♠❡tr♦s c t❛✐s
q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ J(fc) é ❝♦♥❡①♦✱ ✐st♦ é
M={c∈C; J(fc) é ❝♦♥❡①♦}.
❆ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ♥ã♦ é út✐❧ ♣❛r❛ ✜♥s ❝♦♠♣✉t❛❝✐♦♥❛✐s✳ ❈♦♠ ❡st❡ ♦❜❥❡t✐✈♦ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ✉♠❛ ❞❡✜♥✐çã♦ ❡q✉✐✈❛❧❡♥t❡ ✭❚❡♦r❡♠❛ ✷✳✸✮✳
❉❡✜♥✐çã♦ ✶✳✶✶ ❖ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t M é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣❛râ♠❡tr♦s c t❛✐s
q✉❡ ❛ ór❜✐t❛ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ fc é ❧✐♠✐t❛❞❛✳
M=c∈C; {fck(0)}k≥1 é ❧✐♠✐t❛❞♦
❖❜s❡r✈❛çã♦ ✶✳✷ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❆✜r♠❛çã♦ ✷✳✺✱ ❛ ór❜✐t❛ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ s❡r ❧✐♠✐t❛❞❛ é ❡q✉✐✈❛❧❡♥t❡ à ór❜✐t❛ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ t❡♥❞❡r ❛♦ ✐♥✜♥✐t♦✳ ❉❡st❛ ❢♦r♠❛✱ t❡♠♦s ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s✿
M=c∈C; {fck(0)}k≥1 é ❧✐♠✐t❛❞♦ =c∈C; fck(0)9∞ q✉❛♥❞♦ k → ∞ .
❖s r❡s✉❧t❛❞♦s s❡❣✉✐♥t❡s s❡rã♦ út❡✐s ❛♦s ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❞❛ ♣ró①✐♠❛ s❡çã♦✳
▲❡♠❛ ✶✳✶ ❙❡❥❛♠ z ∈ C ❡ fc(z) = z2 +c✱ ♦♥❞❡ c é ✉♠❛ ❝♦♥st❛♥t❡ ❝♦♠♣❧❡①❛✳ ❙❡
|z|>|c| ❡ |z|>2✱ ❡♥tã♦ ❡①✐st❡ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ ε t❛❧ q✉❡ |fn
c(z)|>(1 +ε)n|z|✳ ❉❡♠♦♥str❛çã♦✿ ✭■♥❞✉çã♦ ❡♠ n✮
❉❡ |z| > 2✱ s❡❣✉❡ q✉❡ ❡①✐st❡ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ ε ❝♦♠ |z| = 2 + ε✳
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ |z| −1 = 1 +ε✳
P❛r❛ n= 0✱|f0
c(z)|=|z|>(1 +ε)0|z|✳ P❛r❛ n= 1✱|fc(z)|=|z2 +c|✳
❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ❈❛♣ít✉❧♦ ✶
▲♦❣♦✱|fc(z)|=|z2+c|>|z2| − |c|=|z|2− |c|>|z|2− |z|= (|z| −1)|z|= (1 +ε)|z|✳ P❛r❛ n= 2✱|f2
c(z)|=|fc(fc(z))|✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❝❛s♦ ❛♥t❡r✐♦r✱ |fc(z)|>(1 +ε)|z|>|z|= 2 +ε✳
❈♦♠♦|z|>|c|❡|z|>2✱ s❡❣✉❡ q✉❡|fc(z)|>|c|❡|fc(z)|>2✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ✉s❛r ♦ ❝❛s♦ ❛♥t❡r✐♦r ♣❛r❛ ♦ ❝♦♠♣❧❡①♦ fc(z)✳
|f2
c(z)|=|fc(fc(z))|>(1 +ε)|fc(z)|>(1 +ε).(1 +ε)|z|= (1 +ε)2|z|✳
❙✉♣♦♥❤❛ q✉❡ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛❧❣✉♠ n = k✳ ❖✉ s❡❥❛✱
|fk
c(z)|>(1 +ε)k|z|✳ ❈♦♠♦ |fk
c(z)|>c ❡ |fck(z)|>2✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
|fk+1
c (z)|=|fc(fck(z))|>(1 +ε).|fck(z)| H.I.
> (1 +ε).(1 +ε)k|z|= (1 +ε)k+1|z|✳ P♦rt❛♥t♦✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✱ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦n ∈N✳
Pr♦♣♦s✐çã♦ ✶✳✷ ❙❡❥❛♠z ∈C ❡ fc(z) =z2+c✱ ♦♥❞❡ c é ✉♠❛ ❝♦♥st❛♥t❡ ❝♦♠♣❧❡①❛✳
❙❡ |z| > |c| ❡ |z| > 2✱ ❡♥tã♦ lim
n→∞|f
n
c (z)| = ∞ ✭♦✉ s❡❥❛✱ ❛ ór❜✐t❛ ❞❡ z t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦✮✳
❉❡♠♦♥str❛çã♦✿ ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ▲❡♠❛ ✶✳✶✱ ❡①✐st❡ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ε t❛❧ q✉❡
|fn
c (z)|>(1 +ε)n|z|✳ ❈♦♠♦ nlim
→∞(1 +ε)
n
=∞✱ t❡♠♦s
lim
n→∞|f
n
c (z)|> lim
n→∞(1 +ε)
n
|z|=|z| lim
n→∞(1 +ε)
n=
∞.
P♦rt❛♥t♦✱ ❛ ór❜✐t❛ ❞❡ z t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦ s❡ |z|>c ❡ |z|>2✳
❈♦r♦❧ár✐♦ ✶✳✶ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❡ fc ❡stá ❝♦♥t✐❞♦ ♥♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ ✵ ❡ r❛✐♦
rc = max{|c|, 2}✳
❉❡♠♦♥str❛çã♦✿ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❉❡✜♥✐çã♦ ✶✳✼✱J(fc) = ∂Af(∞)✱ ♦♥❞❡Af(∞) é ❛❜❡rt♦ ✭Pr♦♣♦s✐çã♦ ✶✳✶✮✳ ▲♦❣♦✱ ♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦ q✉❡ sã♦ ✐t❡r❛❞♦s ♣❛r❛
❈♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ❈❛♣ít✉❧♦ ✶
♦ ✐♥✜♥✐t♦ ♥ã♦ ♣❡rt❡♥❝❡♠ ❛ J(fc)✳ ❉❡st❛ ❢♦r♠❛✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✷ ✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❡fc ❡stá ❝♦♥t✐❞♦ ♥♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ ✵ ❡ r❛✐♦ rc = max{|c|, 2}✳
❈♦r♦❧ár✐♦ ✶✳✷ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡ fc ❡stá ❝♦♥t✐❞♦ ♥♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ ✵ ❡ r❛✐♦ rc = max{|c|, 2}✳
❉❡♠♦♥str❛çã♦✿ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✷✱ ♦s ♣♦♥t♦s ❢♦r❛ ❞♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ ✵ ❡ r❛✐♦
rc sã♦ ✐t❡r❛❞♦s ♣❛r❛ ♦ ✐♥✜♥✐t♦✱ ❧♦❣♦ ♥ã♦ ♣❡rt❡♥❝❡♠ ❛ Kc✳
❆✜r♠❛çã♦ ✶✳✸ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡ fc é ❝♦♠♣❛❝t♦✳
❉❡♠♦♥str❛çã♦✿ P❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ Kc é ❧✐♠✐t❛❞♦ ❡ ♣❡❧❛ ❆✜r♠❛çã♦ ✶✳✶✱ é ❢❡❝❤❛❞♦✳
❈♦r♦❧ár✐♦ ✶✳✸ ❙❡❥❛♠ z ∈ C✱ fc(z) = z2 +c✱ ♦♥❞❡ c é ✉♠❛ ❝♦♥st❛♥t❡ ❝♦♠♣❧❡①❛
❡ rc = max{|c|, 2}✳ ❙❡ ♣❛r❛ ❛❧❣✉♠ n ∈N ❢♦r s❛t✐s❢❡✐t❛ ❛ ❡①♣r❡ssã♦ |fcn(z)|> rc✱ ❡♥tã♦ ❛ ór❜✐t❛ ❞❡ z t❡♥❞❡ ♣❛r❛ ♦ ✐♥✜♥✐t♦✳
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛n0 ∈Nt❛❧ q✉❡ |fcn0(z)|> rc✳
❈♦♠♦ rc = max{|c|, 2}✱ s❡❣✉❡ q✉❡ rc > |c| ❡ rc > 2✳ P♦rt❛♥t♦✱ |fcn0(z)| > |c| ❡ |fn0
c (z)| > 2✳ ▲♦❣♦✱ w = fn
0
c (z) s❛t✐s❢❛③ ❛s ❤✐♣ót❡s❡s ❞❛ Pr♦♣♦s✐çã♦ ✶✳✷✳ ❆ss✐♠✱
lim
n→∞|f
n
c(w)|= limn
→∞|f
n c(f
n0
c (z))|= limn
→∞|f
n+n0
c (w)|=∞✳
P♦rt❛♥t♦✱ ❛ ór❜✐t❛ ❞❡ w t❡♥❞❡ ♣❛r❛ ♦ ✐♥✜♥✐t♦✳ ❖ ♠❡s♠♦ ❛❝♦♥t❡❝❡ ❝♦♠ ❛ ór❜✐t❛
❞❡z✱ ✉♠❛ ✈❡③ q✉❡ ❛ ór❜✐t❛ ❞❡ z ❝♦♥té♠ ❛ ór❜✐t❛ ❞❡ w✳
❈♦r♦❧ár✐♦ ✶✳✹ ❖ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t ♣❡rt❡♥❝❡ ❛♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ 2✳
❉❡♠♦♥str❛çã♦✿ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❖❜s❡r✈❛çã♦ ✶✳✷✱
M=c∈C; fck(0)9∞ q✉❛♥❞♦ k → ∞ .
❚♦♠❡ c∈C t❛❧ q✉❡ |c|>2 ❡ z = 0✳ ❖❜s❡r✈❡ q✉❡ rc = max{|c|, 2}=|c|✳
▼ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❈❛♣ít✉❧♦ ✶
❈♦♠♦ fc(0) = 02 +c = c✱ t❡♠♦s |fc(0)| = |c| = rc✳ ▼❛s fc2(0) = fc(fc(0)) =
f(c) =c2+c✱ ❡ ♥❡st❡ ❝❛s♦✱
|f2
c(0)|=|c2+c|>|c2| − |c|=|c|2− |c|=|c|(|c| −1)
|c|>2
> |c|=rc. ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❈♦r♦❧ár✐♦ ✶✳✸✱ ❛ ór❜✐t❛ ❞❡z = 0 t❡♥❞❡ ♣❛r❛ ♦ ✐♥✜♥✐t♦✳
✶✳✹ ▼ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ✈✐s✉❛❧✐③❛r ✉♠❛
✜❣✉r❛ ❛♣r♦①✐♠❛❞❛ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛
♣r❡❡♥❝❤✐❞♦ ❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t
❉❛r❡♠♦s ❝♦♥t✐♥✉✐❞❛❞❡ ❛♦ ❡st✉❞♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ❞❛s ❢✉♥çõ❡s ♣❡rt❡♥❝❡♥t❡s ❛ ❢❛♠í❧✐❛F✱ ♦✉ s❡❥❛✱ ❢✉♥çõ❡s ❞❛ ❢♦r♠❛ fc :C→C ❞❛❞❛s ♣♦r fc(z) =z2+c✱ ♦♥❞❡
c é ✉♠❛ ❝♦♥st❛♥t❡ ❝♦♠♣❧❡①❛✳
❈♦♠♦ ✈❡r❡♠♦s ♥❡st❛ s❡çã♦✱ ♥ã♦ é ♣♦ssí✈❡❧ ❢❛③❡r ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡fc ❡ ♥❡♠ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✳ ❯t✐❧✐③❛r❡♠♦s ♦s s♦❢t✇❛r❡s ▼❛t▲❛❜ ❡ ●❡♦●❡❜r❛ ♣❛r❛ ✈✐s✉❛❧✐③❛r ✉♠❛ ✜❣✉r❛ ❛♣r♦①✐♠❛❞❛ ❞❡st❡s ❝♦♥❥✉♥t♦s✳
P❛r❛ ♣❧♦t❛r ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡ fc ✉t✐❧✐③❛r❡♠♦s ♦ t❡st❡ s❡❣✉✐♥t❡✳ ❚❡st❡ ♣❛r❛ ✈❡r✐✜❝❛r s❡ ✉♠ ♣♦♥t♦ z ♣❡rt❡♥❝❡ ❛ Kc
❉❡♥♦t❡♠♦s ♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ ✵ ❡ r❛✐♦ rc = max{|c|, 2} ♣♦r Brc(0)✳ ❉❡ ❛❝♦r❞♦
❝♦♠ ♦ ❈♦r♦❧ár✐♦ ✶✳✷✱ ♦s ♣♦♥t♦s ❢♦r❛ ❞❡st❡ ❝ír❝✉❧♦ ♥ã♦ ♣❡rt❡♥❝❡♠ ❛ Kc✳ ▲♦❣♦✱ ♣r❡❝✐s❛♠♦s ❛♣❡♥❛s ✈❡r✐✜❝❛r q✉❛✐s ♣♦♥t♦s ❞❡ Brc(0) ♣❡rt❡♥❝❡♠ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛
♣r❡❡♥❝❤✐❞♦✳ P❛r❛ ✐st♦✱ ✉t✐❧✐③❛r❡♠♦s ♦ ❈♦r♦❧ár✐♦ ✶✳✸✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❊s❝♦❧❤❡♠♦s ✉♠ ♣♦♥t♦ z ∈ Brc(0) ❡ ✈❡r✐✜❝❛♠♦s s❡ ♣❛r❛ ❛❧❣✉♠ k ∈ N\ {0}✱
|fk
c(z)| > rc✳ ❊♠ ❝❛s♦ ❛✜r♠❛t✐✈♦✱ ❛ ór❜✐t❛ ❞❡ z t❡♥❞❡ ♣❛r❛ ♦ ✐♥✜♥✐t♦ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ z /∈Kc✳
▼ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❈❛♣ít✉❧♦ ✶
◆♦ ❡♥t❛♥t♦✱ ❤á ✉♠ ♣r♦❜❧❡♠❛ ♥❡st❡ t❡st❡✳ ❙❡❥❛♠ z ∈ Brc(0) ❡ k0 ∈ N\ {0} t❛✐s q✉❡ |f
k0−1
c (z)| ≤ rc ❡ |fck0(z)| > rc✳ P❛r❛ ✈❛❧♦r❡s ❣r❛♥❞❡s ❞❡ k0 ♥ã♦ t❡r❡♠♦s ❝♦♠♦ r❡❛❧✐③❛r ❛s k0 ✐t❡r❛çõ❡s ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡
z /∈Kc✳
❈♦♠ 100 ✐t❡r❛çõ❡s ♣♦r ❡①❡♠♣❧♦✱ t❡r❡♠♦s ✉♠❛ ♣♦r❝❡♥t❛❣❡♠ ❞❡ ♣♦♥t♦s
❝❧❛ss✐✜❝❛❞♦s ❝♦rr❡t❛♠❡♥t❡ ❝♦♠♦ ♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛ Kc✱ ♦✉ s❡❥❛✱ ♣♦♥t♦s t❛✐s q✉❡
|fk0−1
c (z)| ≤ rc ❡ |fck0(z)| > rc✱ ♣❛r❛ ❛❧❣✉♠ k0 ∈ {1,2,3, . . . ,100}✳ ◆♦ ❡♥t❛♥t♦✱ s❡ ❛♣ós ❛s 100 ✐t❡r❛çõ❡s |fk
c(z)| ≤ rc ♣❛r❛ t♦❞♦ k ∈ {1,2,3, . . . ,100}✱ ♥ã♦ t❡r❡♠♦s ❝❡rt❡③❛ s❡ z ∈ Kc✱ ♣♦✐s ♣♦❞❡ ♦❝♦rr❡r ❞❡ |f100+j
c (z)| > rc ♣❛r❛ ❛❧❣✉♠
j ∈ N \ {0}✳ ◗✉❛♥t♦ ♠❛✐♦r ♦ ♥ú♠❡r♦ ❞❡ ✐t❡r❛çõ❡s✱ ♠❛✐♦r s❡rá ❛ ♣♦r❝❡♥t❛❣❡♠
❞❡ ♣♦♥t♦s ❝❧❛ss✐✜❝❛❞♦s ❝♦rr❡t❛♠❡♥t❡✳ P❛r❛ ♦❜t❡r ✉♠ r❡s✉❧t❛❞♦ ❝♦♠ t♦❞♦s ♦s ♣♦♥t♦s ❝❧❛ss✐✜❝❛❞♦s ❝♦rr❡t❛♠❡♥t❡ t❡rí❛♠♦s q✉❡ r❡❛❧✐③❛r ✐♥✜♥✐t❛s ✐♥t❡r❛çõ❡s✱ ♦ q✉❡ é ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ✐♠♣♦ssí✈❡❧✳ ❉❡st❛ ❢♦r♠❛✱ t❡r❡♠♦s ❛♣❡♥❛s ✉♠❛ ✜❣✉r❛ ❛♣r♦①✐♠❛❞❛ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦✳
❚❡st❡ ♣❛r❛ ✈❡r✐✜❝❛r s❡ ✉♠ ♣♦♥t♦ c ♣❡rt❡♥❝❡ ❛ M
❉❡♥♦t❡♠♦s ♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ ✵ ❡ r❛✐♦ 2 ♣♦r B2(0)✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❈♦r♦❧ár✐♦ ✶✳✹✱ ♦s ♣♦♥t♦s ❢♦r❛ ❞❡st❡ ❝ír❝✉❧♦ ♥ã♦ ♣❡rt❡♥❝❡♠ ❛ M✳ ▲♦❣♦✱ ♣r❡❝✐s❛♠♦s ❛♣❡♥❛s
✈❡r✐✜❝❛r q✉❛✐s ♣♦♥t♦s ❞❡ B2(0) ♣❡rt❡♥❝❡♠ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ▼❛♥❞❡❧❜r♦t✳ P❛r❛ ✐st♦✱ ✉t✐❧✐③❛r❡♠♦s ♦ ❈♦r♦❧ár✐♦ ✶✳✸✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❊s❝♦❧❤❡♠♦s ✉♠ ♣♦♥t♦ c ∈ B2(0) ❡ ✈❡r✐✜❝❛♠♦s s❡ ♣❛r❛ ❛❧❣✉♠ k ∈ N \ {0}✱
|fk
c(0)| > 2✳ ❊♠ ❝❛s♦ ❛✜r♠❛t✐✈♦✱ ❛ ór❜✐t❛ ❞❡ 0 t❡♥❞❡ ♣❛r❛ ♦ ✐♥✜♥✐t♦ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ c /∈ M✳
❖ t❡st❡ ❛❝✐♠❛ ❛♣r❡s❡♥t❛ ♦ ♠❡s♠♦ ♣r♦❜❧❡♠❛ ❞❡s❝r✐t♦ ♥♦ t❡st❡ ❛♥t❡r✐♦r✳
▼ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❈❛♣ít✉❧♦ ✶
✶✳✹✳✶ ❈ó❞✐❣♦s ❞♦ ▼❛t❧❛❜
❈ó❞✐❣♦s ✉t✐❧✐③❛♥❞♦ ♦s ❝♦♠❛♥❞♦s ❢♦r ❡ ✐❢ ♣❛r❛ ♣❧♦t❛r ♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡ fc
■♥✐❝✐❛❧♠❡♥t❡ ❞❡✈❡♠♦s ❞❡✜♥✐r✿
• ❖ ✈❛❧♦r ❞❛ ❝♦♥st❛♥t❡ c✳
• ❆ q✉❛♥t✐❞❛❞❡q ❞❡ ✐t❡r❛çõ❡s ❛ s❡r❡♠ r❡❛❧✐③❛❞❛s✳
• ❖ ✐♥t❡r✈❛❧♦ ❞❛ ♣❛rt❡ r❡❛❧✱ ♦ ✐♥t❡r✈❛❧♦ ❞❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❡ ♦ ❝♦♠♣r✐♠❡♥t♦
d ❞❛s s✉❜❞✐✈✐sõ❡s ❞❡ss❡s ✐♥t❡r✈❛❧♦s✳ P♦r ❡①❡♠♣❧♦✱ x = −2 : d : 2 s✐❣♥✐✜❝❛
q✉❡ s❡rã♦ ❝♦♥s✐❞❡r❛❞♦s ♦s s❡❣✉✐♥t❡s ✈❛❧♦r❡s ❞❡ x : −2,−2 +d,−2 +d+d =
−2 + 2d,−2 + 3d, ...,2−d,2✳ ❖ ♥ú♠❡r♦ d ❞❡✈❡ s❡r ❡s❝♦❧❤✐❞♦ ❞❡ ♠♦❞♦ q✉❡ ♦
✐♥t❡r✈❛❧♦ ❬✲✷✱ ✷❪ t❡♥❤❛ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✐♥t❡✐r❛ ❞❡ s✉❜✐♥t❡r✈❛❧♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦
d✳
• ❯♠❛ ❝♦r ♣❛r❛ ♦s ♣♦♥t♦s ♣❡rt❡♥❝❡♥t❡s ❛ Kc ❡ ♦✉tr❛ ♣❛r❛ ♦s ♣♦♥t♦s q✉❡ ♥ã♦ ♣❡rt❡♥❝❡♠✳
❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ ❞❡ fc✱ ♦♥❞❡c= 0,27334−0,00742i
❆ ✜❣✉r❛ ✶✳✺ ❢♦✐ ❣❡r❛❞❛ ❛tr❛✈és ❞♦ ❝ó❞✐❣♦ s❡❣✉✐♥t❡ s✉❜st✐t✉✐♥❞♦q= 100 ✐t❡r❛çõ❡s
♣♦r ✷✵✳ ❯♠❛ ♠❡❧❤♦r ❛♣r♦①✐♠❛çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛ ♣r❡❡♥❝❤✐❞♦ é ♦❜t✐❞❛ ❝♦♠ ✶✵✵ ✐t❡r❛çõ❡s ✭✜❣✉r❛ ✶✳✻✮✳ ◆♦ ❡♥t❛♥t♦✱ ♦ ❝♦♠♣✉t❛❞♦r ❧❡✈❛rá ♠❛✐s t❡♠♣♦ ♣❛r❛ ❣❡r❛r ❛ ✜❣✉r❛✳ ❖ ❝♦♠♣✉t❛❞♦r ♣♦❞❡ tr❛✈❛r ❞❡♣❡♥❞❡♥❞♦ ❞♦ ♥ú♠❡r♦ ❞❡ ✐t❡r❛çõ❡s ❡ ❞❛ s✉❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ ❡ ♠❡♠ór✐❛✳
