Entire functions with Julia sets of positive measure

For a transcendental entire function f of finite order in the Eremenko-Lyubich class B, we give conditions under which the Lebesgue measure of the escaping set I(f ) of f is zero. This complements the recent work of Aspenberg and Bergweiler, in which they give conditions on entire functions in the same class with escaping sets of positive Lebesgue measure. We will construct an entire function in the Eremenko-Lyubich class to show that the condition given by Aspenberg and Bergweiler is essentially sharp. Furthermore, we adapt our idea of proof to certain infinite-order entire functions. Under some restrictions to the growth of these entire functions, we show that the escaping sets have zero Lebesgue measure. This generalizes a result of Eremenko and Lyubich.2010 Mathematics Subject Classification. 37F10 (primary), 30D05 (secondary).

Section: Introduction and Main Resultsmentioning

confidence: 82%

Lebesgue measure of escaping sets of entire functions

Cui¹

2018

“…Bergweiler and Chyzhykov [4] gave conditions ensuring that the Julia set and the escaping set of a transcendental entire function of completely regular growth have positive measure. These conditions are satisfied for the functions (1). In fact, they are also satisfied if one allows arg(b j +1 ) = arg(b j ) + π for some j ∈ {1, .…”

Section: Introduction and Resultsmentioning

confidence: 90%

“…, q − 1} or arg(b q ) = arg(b 1 ) + π . Further criteria for Julia sets and (fast) escaping sets to have positive measure are given in [1,3].…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Sixsmith [14] proved that if f (z) = q j =1 a j exp(ω j q z), where q ≥ 2, a j ∈ C\{0}, and ω q = exp(2πi/q), then J (f ) ∩ A(f ) has positive measure. Sixsmith remarked without proof that his result remains true for f (z) = q j =1 a j exp(b j z), ( 1 ) where q ≥ 3, a j , b j ∈ C\{0}, arg(b j ) < arg(b j +1 ) < arg(b j ) + π for j ∈ {1, . .…”

Section: Introduction and Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measure

Wolff

2020

“…Note that if f ∈ B then ρ(f ) ≥ 1/2 (see for example [1] for an argument). We examine the Hausdorff measure of escaping and Julia sets of functions f ∈ B of finite order with respect to certain gauge functions.…”

Section: Main Results and Outlinementioning

confidence: 99%

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

Peter¹

2011

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...