Area and Hausdorff dimension of the set of accessible points of the Julia sets of λe^z and λ sin(z)

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Title

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Institute of Mathematics, Technical University of Warsaw, Pl. Politechniki 1, 00-661 Warszawa, Poland

The Julia set

J_{λ}

of the exponential function

E_{λ} : z \to λ e^{z}

for λ ∈ (0,1/e)

i s k n o w n \to b e a u n i o n o f c u r v e s (hairs) w h o s e e n d p \oint s

C_λ

a r e t h e o n l y a e s s i b \leq p \oint s o m t h e b a \sin o f a r a c t i o n . W e s h o w t \hat{f} or λ a s a b o v e t h e H a u s d or f f dim e n s i o n o f

C_λ

i s e q u a l \to 2 and w e g i v e e s t i m a t e s f or t h e H a u s d or f f dim e n s i o n o f t h e \subset o f

C_λ

r e l a t e d \to a f \in i t e ν m b e r o f s y m b o l s . W e a l s o c o n s r t h e s e t o f e n d p \oint s f or t h e \sin e f a m i l y

F_λ:z → (1/(2i))λ (e^{iz}-e^{-iz})!$! for λ ∈ (0,1) and prove that it has positive Lebesgue measure.

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