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

Dimension properties of the boundaries of exponential basins

“…for a large constant C > 0, where z R is the centre of the square Q, that is z R = R + iR. Now we prove (8). By Lemma 1, for every z ∈ H s , s ∈ Z and j = 1, .…”

“…In fact, it follows from the proof in [11] that dim H (∂U ) = dim H (∂U ∩ I(f )) = 2. On the other hand, we have 1 < dim H (∂U ∩ J bd (f )) dim H (∂U \ I(f )) < 2 (see [8,15]). A refined analysis of the dimension of the subsets of I(f ) according to the speed of convergence to ∞ can be found in [10].…”

Hausdorff measure of hairs without endpoints in the exponential family

“…Karpińska [8,Theorem 1.1] proved the surprising result that J (E)\C has Hausdorff dimension 1. Of course, together with McMullen's result this implies that C has Hausdorff dimension 2, a result she had proved already earlier [7,Theorem 1].…”

Hausdorff measure of Julia sets in the exponential family

“…Devaney and Krych [4] proved that J (E λ ) is a 'Cantor bouquet' for 0 < λ < 1/e, that is, it is homeomorphic to the product of a Cantor set and the line [0, ∞). Schleicher and Zimmer [17] proved that I(E λ ) consists of curves for all λ. Karpińska [7,8] proved the following dimension paradox: for 0 < λ < 1/e, the Hausdorff dimension of the endpoints of the curves that form the Cantor bouquet is 2, but the Hausdorff dimension of the curves without endpoints is 1. Urbański and Zdunik [19] showed that the Hausdorff dimension of the set of non-escaping points in the Julia set of a hyperbolic exponential map is always less than 2.…”