Hausdorff measure of escaping and Julia sets for bounded-type functions of finite order
Hausdorff measure of escaping and Julia sets for bounded-type functions of finite order
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DOI：
10.1017/s0143385711000745
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发表时间：
2011-02
影响因子：
0.9
通讯作者：
J. Peter
中科院分区：
文献类型：
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作者：
J. Peter
本文证明了ρ阶有界型超越整函数的逃逸集和Julia集随着ρ→∞变得‘小’.更确切地说，它们的Hausdorff测度关于规范函数hγ(T)=t2g(1/t)γ是无穷的，其中g是某个指数映射的线性化子和γ≥(logρ(F)+k1)/c的逆，但对于足够大的ρ，存在阶为ρ的有界型函数fρ，使得当fρ关于hγ‘的逃逸集和Julia集的Hausdorff测度关于hγ’≤‘为零时，ρ(logρK2)/c.
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 hγ(t)=t2g(1/t)γ, where g is the inverse of a linearizer of some exponential map and γ≥(log ρ(f)+K1)/c, but for ρ large enough, there exists a function fρ of bounded type with order ρ such that the Hausdorff measures of the escaping set and the Julia set of fρ with respect to hγ′ are zero whenever γ′ ≤(log ρ−K2)/c.