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❋❛t♦✉✬s ✇❡❜ ❛♥❞ ♥♦♥✲❡s❝❛♣✐♥❣ ❡♥❞♣♦✐♥ts ❱❛s✐❧✐❦✐ ❊✈❞♦r✐❞♦✉ ❉❡♣t✳ ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❙t❛t✐st✐❝s ❚❤❡ ❖♣❡♥ ❯♥✐✈❡rs✐t② ❚♦♣✐❝s ✐♥ ❝♦♠♣❧❡① ❞②♥❛♠✐❝s ❇❛r❝❡❧♦♥❛ ✲ ◆♦✈❡♠❜❡r ✷✸✱ ✷✵✶✺

❝♦♥s✐sts ♦❢ ❛♥ ✉♥❝♦✉♥t❛❜❧❡ ✉♥✐♦♥ ♦❢ ❝✉r✈❡s ✐♥ t❤❡ ❧❡❢t ❤❛❧❢✲♣❧❛♥❡ ✭❈❛♥✲ t♦r ❜♦✉q✉❡t✮✳ ❋❛t♦✉✬s ❢✉♥❝t✐♦♥ ❚❤❡ ❢✉♥❝t✐♦♥ f ( z ) = z + ✶ + e − z ✇❛s ✜rst st✉❞✐❡❞ ❜② ❋❛t♦✉ ✐♥ ✶✾✷✻✳ F ( f ) ❝♦♥s✐sts ♦❢ ♦♥❡ ✐♥✈❛r✐❛♥t ❝♦♠♣♦♥❡♥t ✭❇❛❦❡r ❞♦♠❛✐♥✮ ✇❤✐❝❤ ❝♦♥t❛✐♥s t❤❡ r✐❣❤t ❤❛❧❢✲♣❧❛♥❡✳

❋❛t♦✉✬s ❢✉♥❝t✐♦♥ ❚❤❡ ❢✉♥❝t✐♦♥ f ( z ) = z + ✶ + e − z ✇❛s ✜rst st✉❞✐❡❞ ❜② ❋❛t♦✉ ✐♥ ✶✾✷✻✳ F ( f ) ❝♦♥s✐sts ♦❢ ♦♥❡ ✐♥✈❛r✐❛♥t ❝♦♠♣♦♥❡♥t ✭❇❛❦❡r ❞♦♠❛✐♥✮ ✇❤✐❝❤ ❝♦♥t❛✐♥s t❤❡ r✐❣❤t ❤❛❧❢✲♣❧❛♥❡✳ J ( f ) ❝♦♥s✐sts ♦❢ ❛♥ ✉♥❝♦✉♥t❛❜❧❡ ✉♥✐♦♥ ♦❢ ❝✉r✈❡s ✐♥ t❤❡ ❧❡❢t ❤❛❧❢✲♣❧❛♥❡ ✭❈❛♥✲ t♦r ❜♦✉q✉❡t✮✳

❋❛t♦✉✬s ❢✉♥❝t✐♦♥ ❚❤❡ ❢✉♥❝t✐♦♥ f ( z ) = z + ✶ + e − z ✇❛s ✜rst st✉❞✐❡❞ ❜② ❋❛t♦✉ ✐♥ ✶✾✷✻✳ F ( f ) ❝♦♥s✐sts ♦❢ ♦♥❡ ✐♥✈❛r✐❛♥t ❝♦♠♣♦♥❡♥t ✭❇❛❦❡r ❞♦♠❛✐♥✮ ✇❤✐❝❤ ❝♦♥t❛✐♥s t❤❡ r✐❣❤t ❤❛❧❢✲♣❧❛♥❡✳ J ( f ) ❝♦♥s✐sts ♦❢ ❛♥ ✉♥❝♦✉♥t❛❜❧❡ ✉♥✐♦♥ ♦❢ ❝✉r✈❡s ✐♥ t❤❡ ❧❡❢t ❤❛❧❢✲♣❧❛♥❡ ✭❈❛♥✲ t♦r ❜♦✉q✉❡t✮✳

❚❤❡ ❢❛st ❡s❝❛♣✐♥❣ s❡t ✱ ✱ ❝♦♥s✐sts ♦❢ t❤❡ ♣♦✐♥ts t❤❛t ❣♦ t♦ ✐♥✜♥✐t② ❛s q✉✐❝❦❧② ❛s ♣♦ss✐❜❧❡ ✉♥❞❡r ✐t❡r❛t✐♦♥✳ ❛s q✉✐❝❦❧② ❛s ♣♦ss✐❜❧❡ ❚❤❡ ❡s❝❛♣✐♥❣ s❡t ❚❤❡ s❡t I ( f ) = { z ∈ C : f n ( z ) → ∞ ❛s n → ∞} ✐s ❝❛❧❧❡❞ t❤❡ ❡s❝❛♣✐♥❣ s❡t ✳

❚❤❡ ❡s❝❛♣✐♥❣ s❡t ❚❤❡ s❡t I ( f ) = { z ∈ C : f n ( z ) → ∞ ❛s n → ∞} ✐s ❝❛❧❧❡❞ t❤❡ ❡s❝❛♣✐♥❣ s❡t ✳ ❚❤❡ ❢❛st ❡s❝❛♣✐♥❣ s❡t ✱ A ( f ) ⊂ I ( f ) ✱ ❝♦♥s✐sts ♦❢ t❤❡ ♣♦✐♥ts t❤❛t ❣♦ t♦ ✐♥✜♥✐t② ❛s q✉✐❝❦❧② ❛s ♣♦ss✐❜❧❡ ✉♥❞❡r ✐t❡r❛t✐♦♥✳ A ( f ) = { z ∈ C : f n ( z ) → ∞ ❛s q✉✐❝❦❧② ❛s ♣♦ss✐❜❧❡ } .

❝♦♥s✐sts ♦❢ t❤❡ ❝✉r✈❡s ✐♥ ❡①❝❡♣t ❢♦r s♦♠❡ ♦❢ t❤❡✐r ❡♥❞♣♦✐♥ts✳ ❚❤❡ ❡s❝❛♣✐♥❣ s❡t ❋♦r ❋❛t♦✉✬s ❢✉♥❝t✐♦♥✿ • I ( f ) ❝♦♥s✐sts ♦❢ t❤❡ ❇❛❦❡r ❞♦♠❛✐♥ ❛♥❞ t❤❡ ❝✉r✈❡s ✐♥ J ( f ) ❡①❝❡♣t ❢♦r s♦♠❡ ♦❢ t❤❡✐r ❡♥❞♣♦✐♥ts❀

❚❤❡ ❡s❝❛♣✐♥❣ s❡t ❋♦r ❋❛t♦✉✬s ❢✉♥❝t✐♦♥✿ • I ( f ) ❝♦♥s✐sts ♦❢ t❤❡ ❇❛❦❡r ❞♦♠❛✐♥ ❛♥❞ t❤❡ ❝✉r✈❡s ✐♥ J ( f ) ❡①❝❡♣t ❢♦r s♦♠❡ ♦❢ t❤❡✐r ❡♥❞♣♦✐♥ts❀ • A ( f ) ❝♦♥s✐sts ♦❢ t❤❡ ❝✉r✈❡s ✐♥ J ( f ) ❡①❝❡♣t ❢♦r s♦♠❡ ♦❢ t❤❡✐r ❡♥❞♣♦✐♥ts✳

✸ ✷ ✶ ❙♣✐❞❡rs✬ ✇❡❜s ❘✐♣♣♦♥ ❛♥❞ ❙t❛❧❧❛r❞ s❤♦✇❡❞ t❤❛t ❢♦r ♠❛♥② tr❛♥s❝❡♥❞❡♥t❛❧ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s t❤❡ ❡s❝❛♣✐♥❣ s❡t ❤❛s ❛ str✉❝t✉r❡ ❝❛❧❧❡❞ ❛ s♣✐❞❡r✬s ✇❡❜ ✳ ❉❡✜♥✐t✐♦♥ ✶ ❆ s❡t E ✐s ❛♥ ✭✐♥✜♥✐t❡✮ s♣✐❞❡r✬s ✇❡❜ ✐❢✿ ✶✮ E ✐s ❝♦♥♥❡❝t❡❞ ❛♥❞ ✷✮ ∃ ❛ s❡q✉❡♥❝❡ ( G n ) , n ∈ N , ♦❢ ❜♦✉♥❞❡❞✱ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥s s✉❝❤ t❤❛t • G n ⊂ G n + ✶ , n ∈ N , • ∂ G n ⊂ E , n ∈ N , • ∪ n ∈ N G n = C .

❙♣✐❞❡rs✬ ✇❡❜s ❘✐♣♣♦♥ ❛♥❞ ❙t❛❧❧❛r❞ s❤♦✇❡❞ t❤❛t ❢♦r ♠❛♥② tr❛♥s❝❡♥❞❡♥t❛❧ ❡♥t✐r❡ ❢✉♥❝t✐♦♥s t❤❡ ❡s❝❛♣✐♥❣ s❡t ❤❛s ❛ str✉❝t✉r❡ ❝❛❧❧❡❞ ❛ s♣✐❞❡r✬s ✇❡❜ ✳ ❉❡✜♥✐t✐♦♥ ✶ ❆ s❡t E ✐s ❛♥ ✭✐♥✜♥✐t❡✮ s♣✐❞❡r✬s ✇❡❜ ✐❢✿ ✶✮ E ✐s ❝♦♥♥❡❝t❡❞ ❛♥❞ ✷✮ ∃ ❛ s❡q✉❡♥❝❡ ( G n ) , n ∈ N , ♦❢ ❜♦✉♥❞❡❞✱ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥s s✉❝❤ t❤❛t • G n ⊂ G n + ✶ , n ∈ N , • ∂ G n ⊂ E , n ∈ N , • ∪ n ∈ N G n = C . G ✸ G ✷ G ✶

❙♣✐❞❡rs✬ ✇❡❜s • ❘✐♣♣♦♥ ❛♥❞ ❙t❛❧❧❛r❞ s❤♦✇❡❞ t❤❛t ✇❤❡♥ I ( f ) ❝♦♥t❛✐♥s ❛ ❙❲ t❤❡♥ ✐t ✐s ❛ ❙❲✳ • ■♥ ♠♦st ❡①❛♠♣❧❡s ✇❡ s❤♦✇ t❤❛t A ( f ) ✐s ❛ ❙❲ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t I ( f ) ✐s ❛ ❙❲✳ • ❚❤❡r❡ ❡①✐sts ❛ ❝♦♠♣❧✐❝❛t❡❞ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❢♦r ✇❤✐❝❤ I ( f ) ✐s ❛ ❙❲ ✇❤❡r❡❛s A ( f ) ✐s ♥♦t✱ ❞✉❡ t♦ ❘✐♣♣♦♥ ❛♥❞ ❙t❛❧❧❛r❞✳

❙❦❡t❝❤ ♦❢ Pr♦♦❢✳ ■❞❡❛ ✿ ❯s❡ ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t ✇❤✐❝❤ ✐♠♣❧✐❡s ❚❤❡♦r❡♠ ✶✳ ▲❡t ❜❡ ❛ t✳❡✳❢✳ ❛♥❞ ❜❡ ❛ ♣♦s✐t✐✈❡ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t✿ ✭✶✮ ❛s ✱ ✭✷✮ t❤❡ ❞✐s❝ ✵ ❝♦♥t❛✐♥s ❛ ♣❡r✐♦❞✐❝ ❝②❝❧❡ ♦❢ ❢♦r ❛❧❧ ❈♦♥s✐❞❡r t❤❡ s❡t ❚❤❡♦r❡♠ ✷ ▲❡t ❜❡ ❛ t✳❡✳❢✳ ■❢ s❛t✐s✜❡s ✭✶✮✱ ✭✷✮ ❛♥❞ ❤❛s ❛ ❜♦✉♥❞❡❞ ❝♦♠♣♦♥❡♥t✱ t❤❡♥ ✐s ❛ ❙❲✳ ❋❛t♦✉✬s ✇❡❜ ❚❤❡♦r❡♠ ✶ ▲❡t f ( z ) = z + ✶ + e − z . ❚❤❡♥ I ( f ) ✐s ❛ ❙❲✳

▲❡t ❜❡ ❛ t✳❡✳❢✳ ❛♥❞ ❜❡ ❛ ♣♦s✐t✐✈❡ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t✿ ✭✶✮ ❛s ✱ ✭✷✮ t❤❡ ❞✐s❝ ✵ ❝♦♥t❛✐♥s ❛ ♣❡r✐♦❞✐❝ ❝②❝❧❡ ♦❢ ❢♦r ❛❧❧ ❈♦♥s✐❞❡r t❤❡ s❡t ❚❤❡♦r❡♠ ✷ ▲❡t ❜❡ ❛ t✳❡✳❢✳ ■❢ s❛t✐s✜❡s ✭✶✮✱ ✭✷✮ ❛♥❞ ❤❛s ❛ ❜♦✉♥❞❡❞ ❝♦♠♣♦♥❡♥t✱ t❤❡♥ ✐s ❛ ❙❲✳ ❋❛t♦✉✬s ✇❡❜ ❚❤❡♦r❡♠ ✶ ▲❡t f ( z ) = z + ✶ + e − z . ❚❤❡♥ I ( f ) ✐s ❛ ❙❲✳ ❙❦❡t❝❤ ♦❢ Pr♦♦❢✳ ■❞❡❛ ✿ ❯s❡ ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t ✇❤✐❝❤ ✐♠♣❧✐❡s ❚❤❡♦r❡♠ ✶✳

❚❤❡♦r❡♠ ✷ ▲❡t ❜❡ ❛ t✳❡✳❢✳ ■❢ s❛t✐s✜❡s ✭✶✮✱ ✭✷✮ ❛♥❞ ❤❛s ❛ ❜♦✉♥❞❡❞ ❝♦♠♣♦♥❡♥t✱ t❤❡♥ ✐s ❛ ❙❲✳ ❋❛t♦✉✬s ✇❡❜ ❚❤❡♦r❡♠ ✶ ▲❡t f ( z ) = z + ✶ + e − z . ❚❤❡♥ I ( f ) ✐s ❛ ❙❲✳ ❙❦❡t❝❤ ♦❢ Pr♦♦❢✳ ■❞❡❛ ✿ ❯s❡ ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t ✇❤✐❝❤ ✐♠♣❧✐❡s ❚❤❡♦r❡♠ ✶✳ ▲❡t f ❜❡ ❛ t✳❡✳❢✳ ❛♥❞ ( a n ) ❜❡ ❛ ♣♦s✐t✐✈❡ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t✿ ✭✶✮ a n → ∞ ❛s n → ∞ ✱ ✭✷✮ t❤❡ ❞✐s❝ D ( ✵ , a n ) ❝♦♥t❛✐♥s ❛ ♣❡r✐♦❞✐❝ ❝②❝❧❡ ♦❢ f , ❢♦r ❛❧❧ n ∈ N . ❈♦♥s✐❞❡r t❤❡ s❡t I ( f , ( a n )) = { z ∈ C : | f n ( z ) | ≥ a n , n ∈ N } .