The finer geometry and dynamics of the hyperbolic exponential family
The finer geometry and dynamics of the hyperbolic exponential family
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双曲指数族的更精细的几何和动力学
DOI：
10.1307/mmj/1060013195
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发表时间：
2003
影响因子：
0.9
通讯作者：
A. Zdunik
中科院分区：
文献类型：
--
作者：
M. Urbanski;A. Zdunik
McMullen [Mc]证明了在fλ的前向迭代下逃逸到无穷远点集的Hausdorff维数等于2.本文深入研究了Julia集J（fλ）的补集的几何（分形）结构和动力学结构，并将其记为Jr（fλ）.虽然我们的结果适用于所有具有吸引周期圈的函数fλ，但我们在假设λ ∈（0，1/e）的情况下进行了详细的分析，并在第6节中简要讨论了一般情况。(In在即将发表的一篇论文中，我们以同样的精神处理了一大类非双曲函数fλ，包括λ∈ [1/e，∞）的情况。由于ef是周期性的，周期为2πi，因此很自然地要识别相差2kπi的点，并考虑（而不是f）定义在高度为2π的条带P上的映射F，这是我们的主要技术设备。利用映射F和紧性的概念，证明了F的概率共形测度m（指数大于1）和f的σ -有限共形测度的存在唯一性.这个强大的工具使我们能够证明集合Jr（fλ）的Hausdorff维数λ小于2，Jr（f λ）的hλ维Hausdorff测度在每个水平条上是正的和有限的，并且Jr（fλ）的hλ维packing测度在Jr（fλ）的每个点上是局部无限的。λ < 2的事实特别表明，有理函数迭代理论中所证明的双曲维数和豪斯多夫维数相等，在超越整函数的上下文中是不成立的。转向动力学，我们证明了与共形测度m等价的Borel概率F-不变遍历测度的存在性和唯一性。我们首先应用M的方法。Martens证明了存在一个与测度m等价的σ -有限F -不变保守遍历测度，并检验了这个测度是有限的。我们的论文组织如下。在第二节中，我们证明了对任意λ，集合Jbd（fλ）= {z ∈ J（fλ）：{fn（z）}的Hausdorff维数是有界的}
McMullen [Mc] proved that the Hausdorff dimension of the set of points escaping to infinity under forward iterates of fλ is equal to 2. In this paper we thoroughly investigate the geometric (fractal) and dynamical structure of the complement (in the Julia setJ(fλ)) of this set, which will be denoted in the sequel by Jr(fλ). Although our results apply to all functions fλ with attracting periodic cycles, we perform our analysis in great detail assuming that λ ∈ (0,1/e) and treat the general case briefly in Section 6. (In a forthcoming paper we treat in the same spirit a large class of nonhyperbolic functions fλ, including the case when λ∈ [1/e,∞).) Sincef is periodic with period 2πi, it is natural to identify points that differ by 2kπi and to consider (instead of f ) the mapF, our main technical device, defined on some stripP of height 2π. Armed with the mapF and the concept of tightness, we prove the existence and uniqueness of a probability conformal measure m (with an exponent greater than 1) for F and aσ -finite conformal measure for f. This powerful tool enables us in turn to prove that λ, the Hausdorff dimension of the setJr(fλ), is less than 2, that the hλ-dimensional Hausdorff measure of Jr(fλ) is positive and finite on each horizontal strip, and that the hλ-dimensional packing measure of Jr(fλ) is locally infinite at each point of Jr(fλ). The fact thathλ < 2 shows in particular that the equality of the hyperbolic dimension and the Hausdorff dimension, conjectured in the theory of iteration of rational functions, fails in the context of transcendental entire functions. Turning toward dynamics, we prove the existence and uniqueness of a Borel probabilityF -invariant ergodic measure equivalent with the conformal measure m. We do this by applying first the method of M. Martens to show the existence of a σ -finite F -invariant conservative ergodic measure equivalent with the measure m and then checking that this measure is finite. Our paper is organized as follows. In Section 2 we prove that, for every λ, the Hausdorff dimension of the set Jbd(fλ) = {z ∈ J(fλ) : {f n(z)} is bounded}