Rigidity of tame rational functions
Rigidity of tame rational functions
复制标题
驯服理性函数的刚性
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
M. Urbanski
中科院分区：
文献类型：
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作者：
F. Przytycki;M. Urbanski
引入并建立了驯服有理函数的一些基本性质。这些函数的类包含了所有在Julia集中没有递归临界点的有理函数。对于驯服的非例外函数，我们证明了Lipschitz共轭性、导数在周期轨道上的模的相同谱和共形共轭性是相互等价的。我们还证明了如下刚性结果：如果h是Borel可测可逆映射，它共轭两个驯服函数f和g a. e。若h将共形测度mf转换为与mg等价的测度，则h从满测度mf的集合扩张到相应Julia集的邻域的共形同胚.这扩展了D。全纯扩张排斥子的沙利文刚性定理。我们还提供了几行证明E。普拉多定理证明了两个广义类多项式映射在零Teichm-uller距离处全纯共轭.
We introduce and establish some basic properties of the tame rational functions. The class of these functions contains all the rational functions with no recurrent critical points in their Julia sets. For tame non-exceptional functions we prove that the Lipschitz conjugacy, the same spectra of moduli of derivatives at periodic orbits and con-formal conjugacy are mutually equivalent. We prove also the following rigidity result: If h is a Borel measurable invertible map which conjugates two tame functions f and g a.e. and if h transports conformal measure m f to a measure equivalent to m g then h extends from a set of full measure m f to a conformal homeomorphism of neighbourhoods of respective Julia sets. This extends D. Sullivan's rigidity theorem for holomorphic expanding repellers. We provide also a few lines proof of E. Prado's theorem that two generalized polynomial-like maps at zero Teichm uller's distance are holomorphically conjugate.