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

100%

Karpińska B.

Fundamenta Mathematicae

1999

tom 159

nr 3

269-287

The Julia set $J_λ$ of the exponential function $ E_λ:z → λ e^z$ for λ ∈ (0,1/e)$ is known to be a union of curves ("hairs") whose endpoints $C_λ$ are the only accessible points from the basin of attraction. We show that for λ as above the Hausdorff dimension of $C_λ$ is equal to 2 and we give estimates for the Hausdorff dimension of the subset of $C_λ$ related to a finite number of symbols. We also consider the set of endpoints for the sine family $F_λ:z → (1/(2i))λ (e^{iz}-e^{-iz})$ for λ ∈ (0,1) and prove that it has positive Lebesgue measure.

On the accessible points in the Julia sets of some entire functions

100%

Karpińska B.

Fundamenta Mathematicae

2003

tom 180

nr 1

89-98

We prove that for some families of entire functions whose Julia set is the complement of the basin of attraction every branch of a tree of preimages starting from this basin is convergent.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Karpińska B.

1 2003

1 1999

Wyniki wyszukiwania

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

On the accessible points in the Julia sets of some entire functions