For a polynomial p with a repelling fixed point z 0 , we consider Poincaré functions of p at z 0 , i.e. entire functions L which satisfy L(0) = z 0 and p(L(z)) = L(p ′ (z 0 ) · z) for all z ∈ C. We show that if the component of the Julia set of p that contains z 0 equals {z 0 }, then the (fast) escaping set of L is a spider's web; in particular it is connected. More precisely, we classify all linearizers of polynomials with regards to the spider's web structure of the set of all points which escape faster than the iterates of the maximum modulus function at a sufficiently large point R.

We consider the Hausdorff measure of Julia sets and escaping sets of exponential maps with respect to certain gauge functions. We give conditions on the growth of the gauge function that imply that the measure is zero or infinity.

Abstract. The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these sets with respect to certain gauge functions.

Link to this article: http://journals.cambridge.org/abstract_S0143385711000745How to cite this article: JÖRN PETER (2013). Hausdorff measure of escaping and Julia sets for bounded-type functions of nite order.Abstract. We show that the escaping sets and the Julia sets of bounded-type transcendental entire functions of order ρ become 'smaller' as ρ → ∞. More precisely, their Hausdorff measures are infinite with respect to the gauge function h γ (t) = t 2 g(1/t) γ , where g is the inverse of a linearizer of some exponential map and γ ≥ (log ρ( f ) + K 1 )/c, but for ρ large enough, there exists a function f ρ of bounded type with order ρ such that the Hausdorff measures of the escaping set and the Julia set of f ρ with respect to h γ are zero whenever γ ≤ (log ρ − K 2 )/c. Hausdorff measure of Julia sets 285We examine the Hausdorff measure of escaping and Julia sets of functions f ∈ B of finite order with respect to certain gauge functions. By a gauge function, we mean an increasing function h : [0, ε) → R ≥0 (where ε > 0) which is continuous from the right and satisfies h(0) = 0. For an arbitrary set A ⊂ C, defineThen H h is a metric outer measure on all subsets of C, called the Hausdorff measure with respect to h. Following [14], we introduce the notation h 1 ≺ h 2 for gauge functions h 1 and h 2 whenever the quotient h 1 (t)/ h 2 (t) tends to 0 as t → 0. In the special case where h(t) = h s (t) := t s for some s > 0, H h s is the s-dimensional outer Hausdorff measure.This value s 0 is called the Hausdorff dimension of the set A, which we will denote by HD(A). Barański [2] and (independently) Schubert [15] showed that HD(J ( f )) = 2 whenever f ∈ B ρ . In fact, the stronger result HD(I ( f )) = 2 also holds. However, if the order of f is infinite, this need not be true anymore, as was shown by Stallard [16]. In [5], Bergweiler et al proved that if the order of f is infinite and M(r, f ) ≤ exp(exp((log r ) q+ε )) for large r , then HD(J ( f )) ≥ 1 + 1/q, and this estimate is sharp [16]. This suggests that the escaping set and the Julia set of a function f ∈ B ρ get 'smaller' as ρ increases. On the other hand, a result by Eremenko and Lyubich [8, Proposition 4, Theorem 7] implies that if f has finite order and a finite logarithmic singularity, then I ( f ) has zero two-dimensional Hausdorff measure; there are many functions satisfying this condition, so the usual s-dimensional Hausdorff measure is not suitable for distinguishing sizes of escaping sets of bounded-type entire functions with finite order, which is why we use more general gauge functions to measure them. This question was addressed for the exponential functions E λ (z) := λ exp(z) in [13]. Let λ ∈ (0, 1/e) and E λ (z) := λ exp(z) be the exponential map with parameter λ. The function E λ has exactly one real repelling fixed point β λ , that is, E λ (β λ ) = β λ and E λ (β λ ) > 1. A classical result due to Koenigs and Poincaré implies that there exists an entire function L λ satisfying L λ (0) = β λ , L λ (0) = 1 and E λ (L λ (z)) = L λ (β λ z) for a...

Abstract. We prove some results concerning the possible configurations of Herman rings for transcendental meromorphic functions. We show that one pole is enough to obtain cycles of Herman rings of arbitrary period and give a sufficient condition for a configuration to be realizable.