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◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t②

❏♦✐♥t ✇♦r❦ ✇✐t❤ ❆❧❡①❛♥❞❡r ❇♦r✐❝❤❡✈ ✭▼❛rs❡✐❧❧❡✮ ❛♥❞ ❑♦♥st❛♥t✐♥ ❋❡❞♦r♦✈s❦✐✐ ✭▼♦s❝♦✇✮

❚♦♣✐❝s ✐♥ ●❡♦♠❡tr✐❝ ❋✉♥❝t✐♦♥ ❚❤❡♦r②✱ ✶✻ ❋❡❜r✉❛r② ✷✵✶✽

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s

❉❡✜♥✐t✐♦♥ ❆ ❜♦✉♥❞❡❞ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥ G ⊂ C ✐s s❛✐❞ t♦ ❜❡ ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✱ ✐❢ t❤❡r❡ ❡①✐st ❢✉♥❝t✐♦♥s u, v ∈ H∞(G) s✉❝❤ t❤❛t t❤❡ ❡q✉❛❧✐t② z = u(z) v(z), ❤♦❧❞s ♦♥ ∂G ❛✳ ❡✳ ✐♥ s❡♥s❡ ♦❢ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣s✳ ❚❤❛t ♠❡❛♥s t❤❛t ϕ(ζ) = u(ϕ(ζ)) v(ϕ(ζ)), ❛✳❡✳ ♦♥ T ❢♦r s♦♠❡✭❛♥②✮ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣ ϕ : D → G✳ ◗✉❡st✐♦♥ ❍♦✇ ❧❛r❣❡ ❝❛♥ ❜❡ ✭❛❝❝❡ss✐❜❧❡✮ ❜♦✉♥❞❛r② ♦❢ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥❄

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s

❉❡✜♥✐t✐♦♥ ❆ ❜♦✉♥❞❡❞ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥ G ⊂ C ✐s s❛✐❞ t♦ ❜❡ ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✱ ✐❢ t❤❡r❡ ❡①✐st ❢✉♥❝t✐♦♥s u, v ∈ H∞(G) s✉❝❤ t❤❛t t❤❡ ❡q✉❛❧✐t② z = u(z) v(z), ❤♦❧❞s ♦♥ ∂G ❛✳ ❡✳ ✐♥ s❡♥s❡ ♦❢ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣s✳ ❚❤❛t ♠❡❛♥s t❤❛t ϕ(ζ) = u(ϕ(ζ)) v(ϕ(ζ)), ❛✳❡✳ ♦♥ T ❢♦r s♦♠❡✭❛♥②✮ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣ ϕ : D → G✳ ◗✉❡st✐♦♥ ❍♦✇ ❧❛r❣❡ ❝❛♥ ❜❡ ✭❛❝❝❡ss✐❜❧❡✮ ❜♦✉♥❞❛r② ♦❢ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥❄

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

❊①❛♠♣❧❡s

◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥s✿ ✉♥✐t ❞✐s❦ D✱ ◆❡✉♠❛♥♥✬s ♦✈❛❧ ✭✐♠❛❣❡ ♦❢ ❡❧❧✐♣s❡ ✇✐t❤ ❝❡♥t❡r ❛t ♦r✐❣✐♥ ✉♥❞❡r z → 1

z ✮✳

◆♦♥✲ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥s✿ ❡❧❧✐♣s❡✱ ♣♦❧②❣♦♥✳ Ga,b =

• z = x + iy : x2

a2 + y2 b2 < 1

• ,

b < a. ❇♦✉♥❞❛r② ♦❢ Ga,b z = Sa,b(z), Sa,b(z) = (a2 + b2)z − 2ab √ z2 − c2 c2 , c =

• a2 − b2.

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

▼♦t✐✈❛t✐♦♥s

P♦❧②❛♥❛❧②t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ◗✉❛❞r❛t✉tr❡ ❞♦♠❛✐♥s✳ ❙❝❤✇❛r③ ❢✉♥❝t✐♦♥s❀ ❯♥✐✈❛❧❡♥t ❢✉♥❝t✐♦♥s ✐♥ ♠♦❞❡❧ s✉❜s♣❛❝❡s ♦❢ ❍❛r❞② s♣❛❝❡

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

P♦❧②❛♥❛❧②t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥

❉❡✜♥✐t✐♦♥ ❲❡ ✇✐❧❧ s❛② t❤❛t ❢✉♥❝t✐♦♥ f ✐s ♥✲❛♥❛❧②t✐❝ ✐❢ f(z) = zn−1fn−1(z) + ... + zf1(z) + f0(z), ✇❤❡r❡ fk ❛r❡ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✳ X ✲ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ C An(X) := {f ∈ C(X) : f − ♥✲❛♥❛❧②t✐❝ ✐♥ Int X}, Pn(X) = ClosC(X){P : P − ♥✲❛♥❛❧②t✐❝ ♣♦❧②♥♦♠✐❛❧}. ◗✉❡st✐♦♥ ❋♦r ✇❤✐❝❤ X Pn(X) = An(X)?

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

P♦❧②❛♥❛❧②t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥

❉❡✜♥✐t✐♦♥ ❲❡ ✇✐❧❧ s❛② t❤❛t ❢✉♥❝t✐♦♥ f ✐s ♥✲❛♥❛❧②t✐❝ ✐❢ f(z) = zn−1fn−1(z) + ... + zf1(z) + f0(z), ✇❤❡r❡ fk ❛r❡ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✳ X ✲ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ C An(X) := {f ∈ C(X) : f − ♥✲❛♥❛❧②t✐❝ ✐♥ Int X}, Pn(X) = ClosC(X){P : P − ♥✲❛♥❛❧②t✐❝ ♣♦❧②♥♦♠✐❛❧}. ◗✉❡st✐♦♥ ❋♦r ✇❤✐❝❤ X Pn(X) = An(X)?

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

P♦❧②❛♥❛❧②t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥

❚❤❡♦r❡♠ ✭▼❡r❣❡❧②❛♥ ✶✾✺✷✮ P1(X) = A1(X) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s❡t C \ X ✐s ❝♦♥♥❡❝t❡❞✳ ❚❤❡♦r❡♠ ✭❈❛r♠♦♥❛ ✶✾✽✺✮ ■❢ C \ X ✐s ❝♦♥♥❡❝t❡❞✱ t❤❡♥ Pm(X) = Am(X) ❢♦r ❛♥② m ≥ 2✳ ❚❤❡♦r❡♠ ✭❈❛r♠♦♥❛✱ P❛r❛♠♦♥♦✈✱ ❋❡❞♦r♦✈s❦✐② ✷✵✵✷✮ ▲❡t X ❜❡ ❈❛r❛t❤❡♦❞♦r② ❝♦♠♣❛❝t s❡t✱ m ≥ 2✳ ❲❡ ❤❛✈❡ Pm(X) = Am(X) ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② ❜♦✉♥❞❡❞ ❝♦♠♣♦♥❡♥t ♦❢ C \ X ✐s ♥♦t ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✳ ❋♦r ❛r❜✐tr❛r② ❝♦♠♣❛❝t X ❛♥s✇❡r ♠❛② ❞❡♣❡♥❞ ♦♥ m✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

P♦❧②❛♥❛❧②t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥

❚❤❡♦r❡♠ ✭▼❡r❣❡❧②❛♥ ✶✾✺✷✮ P1(X) = A1(X) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s❡t C \ X ✐s ❝♦♥♥❡❝t❡❞✳ ❚❤❡♦r❡♠ ✭❈❛r♠♦♥❛ ✶✾✽✺✮ ■❢ C \ X ✐s ❝♦♥♥❡❝t❡❞✱ t❤❡♥ Pm(X) = Am(X) ❢♦r ❛♥② m ≥ 2✳ ❚❤❡♦r❡♠ ✭❈❛r♠♦♥❛✱ P❛r❛♠♦♥♦✈✱ ❋❡❞♦r♦✈s❦✐② ✷✵✵✷✮ ▲❡t X ❜❡ ❈❛r❛t❤❡♦❞♦r② ❝♦♠♣❛❝t s❡t✱ m ≥ 2✳ ❲❡ ❤❛✈❡ Pm(X) = Am(X) ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② ❜♦✉♥❞❡❞ ❝♦♠♣♦♥❡♥t ♦❢ C \ X ✐s ♥♦t ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✳ ❋♦r ❛r❜✐tr❛r② ❝♦♠♣❛❝t X ❛♥s✇❡r ♠❛② ❞❡♣❡♥❞ ♦♥ m✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

P♦❧②❛♥❛❧②t✐❝ ❛♣♣r♦①✐♠❛t✐♦♥

❚❤❡♦r❡♠ ✭▼❡r❣❡❧②❛♥ ✶✾✺✷✮ P1(X) = A1(X) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s❡t C \ X ✐s ❝♦♥♥❡❝t❡❞✳ ❚❤❡♦r❡♠ ✭❈❛r♠♦♥❛ ✶✾✽✺✮ ■❢ C \ X ✐s ❝♦♥♥❡❝t❡❞✱ t❤❡♥ Pm(X) = Am(X) ❢♦r ❛♥② m ≥ 2✳ ❚❤❡♦r❡♠ ✭❈❛r♠♦♥❛✱ P❛r❛♠♦♥♦✈✱ ❋❡❞♦r♦✈s❦✐② ✷✵✵✷✮ ▲❡t X ❜❡ ❈❛r❛t❤❡♦❞♦r② ❝♦♠♣❛❝t s❡t✱ m ≥ 2✳ ❲❡ ❤❛✈❡ Pm(X) = Am(X) ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② ❜♦✉♥❞❡❞ ❝♦♠♣♦♥❡♥t ♦❢ C \ X ✐s ♥♦t ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✳ ❋♦r ❛r❜✐tr❛r② ❝♦♠♣❛❝t X ❛♥s✇❡r ♠❛② ❞❡♣❡♥❞ ♦♥ m✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

◗✉❛❞r❛t✉r❡ ❞♦♠❛✐♥s

❉❡✜♥✐t✐♦♥ ❆ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Ω ✐s ❛ ❝❧❛ss✐❝❛❧ q✉❛❞r❛t✉r❡ ❞♦♠❛✐♥ ✐❢ t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts {zk} ⊂ Ω s✉❝❤ t❤❛t

• Ω

f(z)dxdy =

n

• j=1

nj−1

• s=0

ajsf (s)(zj) ❢♦r ❡✈❡r② s✉♠♠❛❜❧❡ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ f✳ ❊✈❡r② q✉❛❞r❛t✉r❡ ❞♦♠❛✐♥ ✐s ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✳ ▼♦r❡♦✈❡r ♥♦❞❡s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ♣♦❧❡s ♦❢ ❢✉♥❝t✐♦♥ u/v

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

❆♥② ❜♦✉♥❞❛r② ♦❢ q✉❛❞r❛t✉r❡ ❞♦♠❛✐♥ ✭❡✈❡♥ ✐♥ ✇✐❞❡ s❡♥s❡✮ Ω ❛❞♠✐ts ❛ ♦♥❡✲s✐❞❡❞ ❙❝❤✇❛r③ ❢✉♥❝t✐♦♥ z = S(z) ♦♥ ∂Ω, S ∈ C(Ω), S − ❛♥❛❧②t✐❝ ✐♥ Ω \ K. ❚❤❡♦r❡♠ ✭❙❛❦❛✐ ✶✾✾✶✮ ■❢ Ω ❛❞♠✐ts ❛ ♦♥❡✲s✐❞❡❞ ❙❝❤✇❛r③ ❢✉♥❝t✐♦♥✱ t❤❡♥ ∂Ω ❝♦♥s✐sts ♦❢ ✜♥✐t❡❧② ♠❛♥② ❛♥❛❧②t✐❝ ❝✉r✈❡s✳ ❚❤❡ ❢✉♥❝t✐♦♥ u/v s❡❡♠s t♦ ❜❡ ❛ r❛t❤❡r ✇❡❛❦ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❝♦♥❝❡♣t ♦❢ ❛ ♦♥❡✲s✐❞❡❞ ❙❝❤✇❛r③ ❢✉♥❝t✐♦♥✱ s✐♥❝❡ ✇❡ ❛r❡ ❞❡❛❧✐♥❣ ✇✐t❤ t❤❡ ❡q✉❛❧✐t② ♦❢ ❛♥❣✉❧❛r ❜♦✉♥❞❛r② ✈❛❧✉❡s ♦♥❧② ❢♦r ❛❧♠♦st ❛❧❧ ♣♦✐♥ts ♦♥ T✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

▼♦❞❡❧ s✉❜s♣❛❝❡s ♦❢ ❍❛r❞② s♣❛❝❡

Ps❡✉❞♦❝♦♥t✐♥✉❛t✐♦♥ ❆ ❞♦♠❛✐♥ G ✐s ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣ f ♦❢ t❤❡ ✉♥✐t ❞✐s❝ D ♦♥t♦ G ❛❞♠✐ts ❛ ◆❡✈❛♥❧✐♥♥❛✲t②♣❡ ♣s❡✉❞♦❝♦♥t✐♥✉❛t✐♦♥✱ s♦ t❤❛t t❤❡r❡ ❡①✐st t✇♦ ❢✉♥❝t✐♦♥s f1, f2 ∈ H∞(C \ D) s✉❝❤ t❤❛t f(ζ) = f1(ζ)/f2(ζ) ❢♦r ❛✳❡✳ ζ ∈ T✳ ▼♦❞❡❧ s✉❜s♣❛❝❡s ♦❢ ❍❛r❞② s♣❛❝❡✱ Θ✲✐♥♥❡r ❢✉♥❝t✐♦♥s ✐♥ D✱ KΘ := (ΘH2)⊥ = H2 ⊖ ΘH2. P❛r❛♠❡tr✐③❛t✐♦♥ ▲❡t G ❜❡ ❛ ❜♦✉♥❞❡❞ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥ ❛♥❞ ❧❡t f ❜❡ s♦♠❡ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣ ❢r♦♠ D ♦♥t♦ G✳ ■❢ G ✐s ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ✐♥♥❡r ❢✉♥❝t✐♦♥ Θ s✉❝❤ t❤❛t f ∈ KΘ✳ ❘❡❝✐♣r♦❝❛❧❧②✱ ✐❢ Θ ✐s ❛♥ ✐♥♥❡r ❢✉♥❝t✐♦♥✱ t❤❡♥ ❛♥② ❜♦✉♥❞❡❞ ✉♥✐✈❛❧❡♥t ❢✉♥❝t✐♦♥ ❢r♦♠ t❤❡ s♣❛❝❡ KΘ ♠❛♣s D ❝♦♥❢♦r♠❛❧❧② ♦♥t♦ s♦♠❡ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

▼♦❞❡❧ s✉❜s♣❛❝❡s ♦❢ ❍❛r❞② s♣❛❝❡

Ps❡✉❞♦❝♦♥t✐♥✉❛t✐♦♥ ❆ ❞♦♠❛✐♥ G ✐s ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣ f ♦❢ t❤❡ ✉♥✐t ❞✐s❝ D ♦♥t♦ G ❛❞♠✐ts ❛ ◆❡✈❛♥❧✐♥♥❛✲t②♣❡ ♣s❡✉❞♦❝♦♥t✐♥✉❛t✐♦♥✱ s♦ t❤❛t t❤❡r❡ ❡①✐st t✇♦ ❢✉♥❝t✐♦♥s f1, f2 ∈ H∞(C \ D) s✉❝❤ t❤❛t f(ζ) = f1(ζ)/f2(ζ) ❢♦r ❛✳❡✳ ζ ∈ T✳ ▼♦❞❡❧ s✉❜s♣❛❝❡s ♦❢ ❍❛r❞② s♣❛❝❡✱ Θ✲✐♥♥❡r ❢✉♥❝t✐♦♥s ✐♥ D✱ KΘ := (ΘH2)⊥ = H2 ⊖ ΘH2. P❛r❛♠❡tr✐③❛t✐♦♥ ▲❡t G ❜❡ ❛ ❜♦✉♥❞❡❞ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥ ❛♥❞ ❧❡t f ❜❡ s♦♠❡ ❝♦♥❢♦r♠❛❧ ♠❛♣♣✐♥❣ ❢r♦♠ D ♦♥t♦ G✳ ■❢ G ✐s ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ✐♥♥❡r ❢✉♥❝t✐♦♥ Θ s✉❝❤ t❤❛t f ∈ KΘ✳ ❘❡❝✐♣r♦❝❛❧❧②✱ ✐❢ Θ ✐s ❛♥ ✐♥♥❡r ❢✉♥❝t✐♦♥✱ t❤❡♥ ❛♥② ❜♦✉♥❞❡❞ ✉♥✐✈❛❧❡♥t ❢✉♥❝t✐♦♥ ❢r♦♠ t❤❡ s♣❛❝❡ KΘ ♠❛♣s D ❝♦♥❢♦r♠❛❧❧② ♦♥t♦ s♦♠❡ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

▼♦❞❡❧ s✉❜s♣❛❝❡s ♦❢ ❍❛r❞② s♣❛❝❡

❋❛❝t♦r✐③❛t✐♦♥ ♦❢ Θ✳ Θ(z) = αB(z)S(z), B(z) :=

∞

• n=1

an an · z − an anz − 1, S(z) = exp

• −
• T

ζ + z ζ − z dµS(ζ)

• .

❋❡❞♦r♦✈❦✐②✱ ❇✳ ▲❡t Θ ❜❡ ❛♥ ✐♥♥❡r ❢✉♥❝t✐♦♥ ✐♥ D✳ ❚❤❡ s♣❛❝❡ KΘ ❝♦♥t❛✐♥s ❛ ❜♦✉♥❞❡❞ ✉♥✐✈❛❧❡♥t ❢✉♥❝t✐♦♥s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❝♦♥❞✐t✐♦♥s s❛t✐s✜❡❞✿ ✐✮ Θ ❤❛s ③❡r♦ ✐♥ D❀ ✐✐✮ Θ = S ✐s ❛ s✐♥❣✉❧❛r ✐♥♥❡r ❢✉♥❝t✐♦♥ ❛♥❞ ♠❡❛s✉r❡ µS ✐s s✉❝❤ t❤❛t µS(E) > 0 ❢♦r s♦♠❡ ❈❛r❧❡s♦♥ s❡t E ⊂ T✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t

• T log dist(ζ, E)dζ > −∞✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

❇♦✉♥❞❛r✐❡s ♦❢ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥s

▲❡t Θ ❜❡ ❛ ❇❧❛s❤❦❡ ♣r♦❞✉❝t✳ ■❢ f ∈ KΘ✱ t❤❡♥ f(z) =

∞

• n=1

cn 1 − anz . ✭✶✮ ❆❧♠♦st ✉♥r❡❝t✐✜❛❜❧❡ ❜♦✉♥❞❛r②✳ ❚❤❡♦r❡♠ ✭❋❡❞♦r♦✈s❦✐② ✷✵✵✻✮ ❋♦r ❛♥② α ∈ (0, 1) t❤❡r❡ ❡①✐sts ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ ✇✐t❤ ❜♦✉♥❞❛r② ✐♥ t❤❡ ❝❧❛ss C1 ❜✉t ♥♦t ✐♥ t❤❡ ❝❧❛ss C1,α✳ ❚❤❡♦r❡♠ ✭❇❛r❛♥♦✈✱ ❋❡❞♦r♦✈s❦✐② ✷✵✶✶✮ ❚❤❡r❡ ❡①✐ts ❛♥ ✉♥✐✈❛❧❡♥t ✭✐♥ D✮ ❢✉♥❝t✐♦♥ f ♦❢ t❤❡ ❢♦r♠ ✭✶✮ s✉❝❤ t❤❛t f ′ ∈ Hp ❢♦r ❡✈❡r② p > 1✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

❇♦✉♥❞❛r✐❡s ♦❢ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥s

▲❡t Θ ❜❡ ❛ ❇❧❛s❤❦❡ ♣r♦❞✉❝t✳ ■❢ f ∈ KΘ✱ t❤❡♥ f(z) =

∞

• n=1

cn 1 − anz . ✭✶✮ ❆❧♠♦st ✉♥r❡❝t✐✜❛❜❧❡ ❜♦✉♥❞❛r②✳ ❚❤❡♦r❡♠ ✭❋❡❞♦r♦✈s❦✐② ✷✵✵✻✮ ❋♦r ❛♥② α ∈ (0, 1) t❤❡r❡ ❡①✐sts ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ ✇✐t❤ ❜♦✉♥❞❛r② ✐♥ t❤❡ ❝❧❛ss C1 ❜✉t ♥♦t ✐♥ t❤❡ ❝❧❛ss C1,α✳ ❚❤❡♦r❡♠ ✭❇❛r❛♥♦✈✱ ❋❡❞♦r♦✈s❦✐② ✷✵✶✶✮ ❚❤❡r❡ ❡①✐ts ❛♥ ✉♥✐✈❛❧❡♥t ✭✐♥ D✮ ❢✉♥❝t✐♦♥ f ♦❢ t❤❡ ❢♦r♠ ✭✶✮ s✉❝❤ t❤❛t f ′ ∈ Hp ❢♦r ❡✈❡r② p > 1✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

❇♦✉♥❞❛r✐❡s ♦❢ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥s

❍❡❞❣❡❤♦❣ ❞♦♠❛✐♥s✳ ❚❤❡♦r❡♠ ✭▼❛③❛❧♦✈ ✷✵✶✺✮ ❚❤❡r❡ ❡①✐sts ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ ✇✐t❤ ✉♥r❡❝t✐✜❛❜❧❡ ❜♦✉♥❞❛r②✳ ❚❤❡♦r❡♠ ✭▼❛③❛❧♦✈ ✷✵✶✼✮ ❚❤❡r❡ ❡①✐sts ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ G s✉❝❤ t❤❛t dimH(∂G) = log2 3. ❋♦r ❛ ❣✐✈❡♥ ❜♦✉♥❞❡❞ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥ G ❧❡t ✉s ❞❡✜♥❡ t❤❡ s❡t ∂aG ⊂ ∂G✱ ✇❤✐❝❤ ❝♦♥s✐sts ♦❢ ❛❧❧ ♣♦✐♥ts ♦❢ ∂G ❜❡✐♥❣ ❛❝❝❡ss✐❜❧❡ ❢r♦♠ G ❜② s♦♠❡ ❝✉r✈❡✳ ◗✉❡st✐♦♥ ❍♦✇ ❧❛r❣❡ ❝❛♥ ❜❡ ❛❝❝❡ss✐❜❧❡ ❜♦✉♥❞❛r② ♦❢ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥❄

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

❇♦✉♥❞❛r✐❡s ♦❢ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥s

❍❡❞❣❡❤♦❣ ❞♦♠❛✐♥s✳ ❚❤❡♦r❡♠ ✭▼❛③❛❧♦✈ ✷✵✶✺✮ ❚❤❡r❡ ❡①✐sts ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ ✇✐t❤ ✉♥r❡❝t✐✜❛❜❧❡ ❜♦✉♥❞❛r②✳ ❚❤❡♦r❡♠ ✭▼❛③❛❧♦✈ ✷✵✶✼✮ ❚❤❡r❡ ❡①✐sts ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ G s✉❝❤ t❤❛t dimH(∂G) = log2 3. ❋♦r ❛ ❣✐✈❡♥ ❜♦✉♥❞❡❞ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥ G ❧❡t ✉s ❞❡✜♥❡ t❤❡ s❡t ∂aG ⊂ ∂G✱ ✇❤✐❝❤ ❝♦♥s✐sts ♦❢ ❛❧❧ ♣♦✐♥ts ♦❢ ∂G ❜❡✐♥❣ ❛❝❝❡ss✐❜❧❡ ❢r♦♠ G ❜② s♦♠❡ ❝✉r✈❡✳ ◗✉❡st✐♦♥ ❍♦✇ ❧❛r❣❡ ❝❛♥ ❜❡ ❛❝❝❡ss✐❜❧❡ ❜♦✉♥❞❛r② ♦❢ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥❄

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

▼❛✐♥ r❡s✉❧ts

❍❡❞❣❡❤♦❣ ✇✐t❤ ♥❡❡❞❧❡s ♦♥ ♥❡❡❞❧❡s✳ ❚❤❡♦r❡♠ ✭❇♦r✐❝❤❡✈✱ ❋❡❞♦r♦✈s❦✐②✱ ❇✳ ✷✵✶✽✮ ❋♦r ❡❛❝❤ β ∈ [1, 2] t❤❡r❡ ❡①✐sts ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ G s✉❝❤ t❤❛t dimH(∂aG) = β✳ ❚❤✐s ❞♦♠❛✐♥ ❤❛s t❤❡ ❢♦r♠ G = f(D)✱ ✇❤❡r❡ f ✐s s♦♠❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❢♦r♠ ✭✶✮ ✉♥✐✈❛❧❡♥t ✐♥ D✳ ❯♥✐✈❛❧❡♥t ❢✉♥❝t✐♦♥s ❢r♦♠ ❇❡r♥st❡✐♥ ❝❧❛ss ✭❝♦rr❡♣♦♥❞s t♦ t❤❡ ❝❛s❡ ✇❤❡♥ µS = δ1✮✳ ❚❤❡♦r❡♠ ✭❇♦r✐❝❤❡✈✱ ❋❡❞♦r♦✈s❦✐②✱ ❇✳ ✷✵✶✽✮ ❋♦r ❡❛❝❤ β ∈ [1, 2] t❤❡r❡ ❡①✐sts ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ G s✉❝❤ t❤❛t dimH(∂G) = β ❛♥❞ G = f(C+)✱ ✇❤❡r❡ f ✐s s♦♠❡ ✉♥✐✈❛❧❡♥t ❢✉♥❝t✐♦♥ ❢r♦♠ ❇❡r♥st❡✐♥ ❝❧❛ss B[0,1]✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

▼❛✐♥ r❡s✉❧ts

❍❡❞❣❡❤♦❣ ✇✐t❤ ♥❡❡❞❧❡s ♦♥ ♥❡❡❞❧❡s✳ ❚❤❡♦r❡♠ ✭❇♦r✐❝❤❡✈✱ ❋❡❞♦r♦✈s❦✐②✱ ❇✳ ✷✵✶✽✮ ❋♦r ❡❛❝❤ β ∈ [1, 2] t❤❡r❡ ❡①✐sts ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ G s✉❝❤ t❤❛t dimH(∂aG) = β✳ ❚❤✐s ❞♦♠❛✐♥ ❤❛s t❤❡ ❢♦r♠ G = f(D)✱ ✇❤❡r❡ f ✐s s♦♠❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❢♦r♠ ✭✶✮ ✉♥✐✈❛❧❡♥t ✐♥ D✳ ❯♥✐✈❛❧❡♥t ❢✉♥❝t✐♦♥s ❢r♦♠ ❇❡r♥st❡✐♥ ❝❧❛ss ✭❝♦rr❡♣♦♥❞s t♦ t❤❡ ❝❛s❡ ✇❤❡♥ µS = δ1✮✳ ❚❤❡♦r❡♠ ✭❇♦r✐❝❤❡✈✱ ❋❡❞♦r♦✈s❦✐②✱ ❇✳ ✷✵✶✽✮ ❋♦r ❡❛❝❤ β ∈ [1, 2] t❤❡r❡ ❡①✐sts ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ G s✉❝❤ t❤❛t dimH(∂G) = β ❛♥❞ G = f(C+)✱ ✇❤❡r❡ f ✐s s♦♠❡ ✉♥✐✈❛❧❡♥t ❢✉♥❝t✐♦♥ ❢r♦♠ ❇❡r♥st❡✐♥ ❝❧❛ss B[0,1]✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥s ♦❢ ✜♥✐t❡ ♦r❞❡r

▲❡t RUn ❜❡ ❛ s❡t ♦❢ ❛❧❧ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ♦❢ ❞❡❣r❡❡ n ✇❤✐❝❤ ✐s ✉♥✐✈❛❧❡♥t ✐♥ D✳ ❲❡ ❦♥♦✇ t❤❛t R(D) ✐s ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ ❢♦r R ∈ RUn✳ ▲❡t γ0 = lim sup

n→∞

sup

R∈RUn,R∞≤1

log ℓ(R) log n , ℓ(R) := 1 2π

• T

|R′(ζ)||dζ|. ❚❤❡♦r❡♠ ✭❇❛r❛♥♦✈✱ ❋❡❞♦r♦✈s❦✐② ✷✵✶✸✮ Bb(1) < γ0 ≤ 1/2. Bb(1) ✐s t❤❡ ✐♥t❡❣r❛❧ ♠❡❛♥s s♣❡❝tr✉♠ ❢♦r ❜♦✉♥❞❡❞ ✉♥✐✈❛❧❡♥t ❢✉♥❝t✐♦♥s✳ ■t ✐s ❦♥♦✇♥ t❤❛t ✭❙♠✐r♥♦✈✱ ❇❡❧②❛❡✈✱ ❙❤✐♠♦r✐♥✱ ❍❡❞❡♥♠❛❧♠✮ 0, 23 < Bb(1) ≤ 0, 46.

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥s ♦❢ ✜♥✐t❡ ♦r❞❡r

▲❡t RUn ❜❡ ❛ s❡t ♦❢ ❛❧❧ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ♦❢ ❞❡❣r❡❡ n ✇❤✐❝❤ ✐s ✉♥✐✈❛❧❡♥t ✐♥ D✳ ❲❡ ❦♥♦✇ t❤❛t R(D) ✐s ❛ ◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥ ❢♦r R ∈ RUn✳ ▲❡t γ0 = lim sup

n→∞

sup

R∈RUn,R∞≤1

log ℓ(R) log n , ℓ(R) := 1 2π

• T

|R′(ζ)||dζ|. ❚❤❡♦r❡♠ ✭❇❛r❛♥♦✈✱ ❋❡❞♦r♦✈s❦✐② ✷✵✶✸✮ Bb(1) < γ0 ≤ 1/2. Bb(1) ✐s t❤❡ ✐♥t❡❣r❛❧ ♠❡❛♥s s♣❡❝tr✉♠ ❢♦r ❜♦✉♥❞❡❞ ✉♥✐✈❛❧❡♥t ❢✉♥❝t✐♦♥s✳ ■t ✐s ❦♥♦✇♥ t❤❛t ✭❙♠✐r♥♦✈✱ ❇❡❧②❛❡✈✱ ❙❤✐♠♦r✐♥✱ ❍❡❞❡♥♠❛❧♠✮ 0, 23 < Bb(1) ≤ 0, 46.

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥s ♦❢ ✜♥✐t❡ ♦r❞❡r

❙♥❛❦❡ ❞♦♠❛✐♥ ❚❤❡♦r❡♠ ✭❇♦r✐❝❤❡✈✱ ❋❡❞♦r♦✈s❦✐②✱ ❇✳ ✷✵✶✽✮ ❋♦r ❡✈❡r② R ∈ RUn✱ R∞ ≤ 1 ✇❡ ❤❛✈❡ √n 6π ≤ ℓ(R) ≤ 6π √ n. ❙♦✱ γ0 = 1/2✳ ❚❤❡♦r❡♠ ✭❉♦❧③❤❡♥❦♦ ✶✾✼✽✱ ❙♣✐❥❦❡r ✶✾✾✶ ✭E = T✮✮ ▲❡t R ❜❡ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ♦❢ ❞❡❣r❡❡ n ✇✐t❤ ♣♦❧❡s ♦✉ts✐❞❡ D✳ ❋♦r ❛♥② ♠❡❛s✉r❛❜❧❡ s❡t E ⊂ T ♦❢ ♣♦s✐t✐✈❡ ♠❡s❛✉r❡ t❤❡ ❡st✐♠❛t❡

• T

|R′(ζ)||dζ| ≤ nR∞,E ❤♦❧❞s ❛♥❞ ✐s s❤❛r♣✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

◆❡✈❛♥❧✐♥♥❛ ❞♦♠❛✐♥s ♦❢ ✜♥✐t❡ ♦r❞❡r

❙♥❛❦❡ ❞♦♠❛✐♥ ❚❤❡♦r❡♠ ✭❇♦r✐❝❤❡✈✱ ❋❡❞♦r♦✈s❦✐②✱ ❇✳ ✷✵✶✽✮ ❋♦r ❡✈❡r② R ∈ RUn✱ R∞ ≤ 1 ✇❡ ❤❛✈❡ √n 6π ≤ ℓ(R) ≤ 6π √ n. ❙♦✱ γ0 = 1/2✳ ❚❤❡♦r❡♠ ✭❉♦❧③❤❡♥❦♦ ✶✾✼✽✱ ❙♣✐❥❦❡r ✶✾✾✶ ✭E = T✮✮ ▲❡t R ❜❡ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ♦❢ ❞❡❣r❡❡ n ✇✐t❤ ♣♦❧❡s ♦✉ts✐❞❡ D✳ ❋♦r ❛♥② ♠❡❛s✉r❛❜❧❡ s❡t E ⊂ T ♦❢ ♣♦s✐t✐✈❡ ♠❡s❛✉r❡ t❤❡ ❡st✐♠❛t❡

• T

|R′(ζ)||dζ| ≤ nR∞,E ❤♦❧❞s ❛♥❞ ✐s s❤❛r♣✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s

■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ♦❢ s♦♠❡ r❡s✉❧ts

P✉t ε = 10−9✳ ▲❡t {wn}∞

n=0 ❜❡ ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡✳ P✉t

an = wn+1−wn, Q±

n = conv{wn, wn+1, wn±2ian, wn+1±2ian},

T ±

n = conv{wn+1, wn+1 ± 2ian+1, wn+1 ± 2ian+1}.

❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t |wn| < 1✱ w0 = 0✱ 1 − ε < |an+1|/|an| < 1 + ε, | arg an+1an| ≤ ε, Q±

n ∩ Q± m = ∅, Q± n ∩ T ± m = ∅ ❢♦r |n − m| > 1✳

P✉t L = ∪n[wn, wn+1], ΩL = ∪nQ±

n ∪ T ± n .

❚❤❡r❡ ❡①✐sts ❛ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ f ✇❤✐❝❤ ✐s ✉♥✐✈❛❧❡♥t ✐♥ C+ ❛♥❞ L ⊂ f(C+) ⊂ ΩL✳

❨✉r✐✐ ❇❡❧♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❡✈❛♥❧✐♥♥❛ ❉♦♠❛✐♥s ✇✐t❤ ▲❛r❣❡ ❇♦✉♥❞❛r✐❡s