Jarńık and Julia; a Diophantine analysis for parabolic rational maps
Jarńık and Julia; a Diophantine analysis for parabolic rational maps
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发表时间：
2013
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通讯作者：
B. Stratmann;M. Urbanski
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作者：
B. Stratmann;M. Urbanski
本文给出了抛物有理映射Julia集的丢番图分析。本文将数论中Dirichlet和Jarnik的两个定理推广到这些映射的迭代理论中。在这些结果的基础上，我们导出了与抛物有理映射自然相关的共形测度的“弱重分形分析”。本文的结果有助于进一步发展Sullivan著名的Kleinian群理论和有理映射理论之间的翻译词典。本文给出了抛物有理映射T：n → n的Julia集J（T）的丢番图分析。本文将Dirichlet和Jarnik的两个经典数论定理推广到有理映射的迭代理论。然后，我们表明，这些结果嵌入在共形措施的概念，他们承认一个'弱多重分形分析'的dimH（J（T））-共形措施，这是自然与动力系统（J（T），T）。此外，本文中的结果与[10]，[19]，[22]和[24]中获得的Kleinian群的结果相结合，为Sullivan着名的“Julia-Klein字典”[25]增加了另一个有趣的章节（参见[14]，[23]）。回想一下，对于抛物有理映射，众所周知J（T）= Jr（T）<$Jp（T），即Julia集J（T）允许不相交分解为径向Julia集Jr（T）和前抛物点的可数集Jp（T）：=<$ω∈Ω <$n∈N T −n（ω），其中Ω表示理性无关周期点的集合（[27]，[23]）。对于每个ω ∈ Ω，我们固定一个标准邻域B（ω，rω），粗略地说，考虑它的所有全纯逆迭代B（c（ω），rc（ω））。我们称这些球为典范球（参见第2节的精确定义）。本文的一个主要目的是Jarnik-Julia集的分形分析。对于ω ∈ Ω和σ > 0，这些集合是由J ω σ（T）：
In this paper we derive a Diophantine analysis for Julia sets of parabolic rational maps. We generalise two theorems of Dirichlet and Jarnik in number theory to the theory of iterations of these maps. On the basis of these results, we then derive a ‘weak multifractal analysis’ of the conformal measure naturally associated with a parabolic rational map. The results in this paper contribute to a further development of Sullivan’s famous dictionary translating between the theory of Kleinian groups and the theory of rational maps. 1 Statement of main results In this paper we derive a Diophantine analysis for Julia sets J(T ) of parabolic rational maps T : Ĉ → Ĉ . We generalise two classical number theoretical theorems of Dirichlet and Jarnik to the theory of iterations of rational maps. We then show that these results embed in the concept of conformal measures, where they admit a ‘weak multifractal analysis’ of the dimH(J(T )) -conformal measure which is naturally associated with the dynamical system (J(T ), T ) . Also, a combination of the results in this paper with those for Kleinian groups obtained in [10], [19], [22] and [24] adds another interesting chapter to Sullivan’s famous ‘Julia-Klein dictionary’ [25] (see also [14], [23]). Recall that for parabolic rational maps it is well-known that J(T ) = Jr(T )∪Jp(T ) , i.e. the Julia set J(T ) admits a disjoint decomposition into the radial Julia set Jr(T ) and the countable set of pre-parabolic points Jp(T ) := ⋃ ω∈Ω ⋃ n∈N T −n(ω) , where Ω denotes the set of rationally indifferent periodic points ([27], [23]). For each ω ∈ Ω , we fix a standard neighbourhood B(ω, rω) and consider, roughly speaking, all its holomorphic, inverse iterates B(c(ω), rc(ω)) . We call these balls canonical balls (see section 2, for the precise definition). A major aim of this paper will be the fractal analysis of the Jarnik-Julia sets. For ω ∈ Ω and σ > 0 , these sets are ‘ lim sup sets’ which are defined by J ω σ (T ) := ⋂