Non-uniform hyperbolicity in complex dynamics
Non-uniform hyperbolicity in complex dynamics
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DOI：
10.1007/s00222-008-0152-8
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发表时间：
2008-10
影响因子：
3.1
通讯作者：
J. Graczyk;S. Smirnov
中科院分区：
文献类型：
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作者：
J. Graczyk;S. Smirnov
我们说有理函数 F 满足指数为 α 的可求性条件，如果对于属于 Julia 集 J 的每个临界点 c 都存在一个正整数 nc ，并且 F 没有抛物线周期循环。设μmax为临界点的最大重数。目的是研究一大类有理图的庞加莱级数并建立共形测度的各态遍历和正则性质。如果 Fi 可与指数求和，其中 δPoin(J) 是 Julia 集的庞加莱指数，则存在唯一的、遍历的、非原子共形测度 ν，其指数为 δPoin(J)=HDim(J)。如果 F 与指数 α 多项式可求和，且 F 没有抛物线周期循环，则 F 相对于 ν 具有绝对连续的不变测度。这也导致了关于区间多峰映射绝对连续不变测度存在性的新结果。我们证明，如果Fi可以用指数求和，那么如果并且Fi不稳定，那么J的明可夫斯基维数严格小于2。如果 F 是多项式或 Blaschke 乘积，则 J 是保角可移除的。如果 Fi 可求和，则每个不变 Fatou 分量的边界的连通分量是局部连通的。为了研究Julia集Hausdorff维数的连续性，我们引入了一致可求性的概念。最后，我们推导了Jakobson（Benedicks-Carleson）定理的保角类比，并证明了对于来自Mandelbrot集的几乎所有点c，Julia集的Hausdorff维数关于调和测度的外部连续性。
We say that a rational functionFsatisfies the summability condition with exponent α if for every critical pointcwhich belongs to the Julia setJthere exists a positive integerncso thatandFhas no parabolic periodic cycles. Let μmaxbe the maximal multiplicity of the critical points.The objective is to study the Poincaré series for a large class of rational maps and establish ergodic and regularity properties of conformal measures. IfFis summable with exponentwhere δPoin(J) is the Poincaré exponent of the Julia set then there exists a unique, ergodic, and non-atomic conformal measure ν with exponent δPoin(J)=HDim(J). IfFis polynomially summable with the exponent α,andFhas no parabolic periodic cycles, thenFhas an absolutely continuous invariant measure with respect to ν. This leads also to a new result about the existence of absolutely continuous invariant measures for multimodal maps of the interval.We prove that ifFis summable with an exponentthen the Minkowski dimension ofJis strictly less than 2 ifandFis unstable. IfFis a polynomial or Blaschke product thenJis conformally removable. IfFis summable withthen connected components of the boundary of every invariant Fatou component are locally connected. To study continuity of Hausdorff dimension of Julia sets, we introduce the concept of the uniform summability.Finally, we derive a conformal analogue of Jakobson’s (Benedicks–Carleson’s) theorem and prove the external continuity of the Hausdorff dimension of Julia sets for almost all points c from the Mandelbrot set with respect to the harmonic measure.