Hausdorff measure of escaping and Julia sets for bounded-type functions of finite order

Abstract

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...

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