Fractal Curvature Measures and Minkowski Content for Limit Sets of Conformal Function Systems
Fractal Curvature Measures and Minkowski Content for Limit Sets of Conformal Function Systems
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共形函数系统极限集的分形曲率测度和闵可夫斯基内容
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
S. Kombrink
中科院分区：
文献类型：
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作者：
S. Kombrink
我们刻画了由共形迭代函数系统(CIF)和共形图有向马尔可夫系统(CGDMS)产生的分形集，其中存在Minkowski内容和分形曲率测度，如[Win08]所介绍的。在此基础上，我们推广了关于由相似项组成的迭代函数系统不变集的研究。对于d维欧氏空间的自共形子集，我们证明了在一定的几何条件下，局部平均Minkowski含量总是存在的，并给出了一个显式公式。如果系统是非格子系统，我们证明了局部Minkowski内容也存在，并且与它的平均形式重合。从这一一般结果出发，我们得到了关于自相似集子类的新结果，大大推广了[DKz10，LPW11]中的结论。此外，对于自相似集，我们还证明了在非晶格情况下存在分形曲率度量，并且对于晶格和非晶格系统都存在平均形式。由此，我们给出了与[Win08，WZ10，Zah11]中的证明完全不同的证明，并得到了关于分形曲率度量的另一种有用的公式。自共形集的另一个重要的子类是自相似集的共形C1+α-微分同胚象类，其中α∈(0，1].我们证明了当所考虑的自相似集的局部Minkowski内容存在时，这类图像的局部Minkowski内容是存在的。这一新结果也给出了自相似集的局部Minkowski容度与其像之间的良好关系。与R的非退化自相似子集的Minkowski含量存在当且仅当系统是非格的[LP93，Fal95，LvF06]相反，我们证明了存在格CIF的不变集，对它确实存在Minkowski含量。文中举例说明了这一令人惊讶的结果，并反驳了[Lap93]关于自共形集的猜想4。此外，我们还证明了即使在自相似集的C_1+α-微分同胚象的子类中，也存在存在Minkowski内容的格集。然而，我们证明了R中这类非退化集的分形曲率测度存在的充要条件是其基础系统是非格的。R中由解析映射组成的格CIF与由相似性组成的格系自动共轭的结果强调了这一子类的重要性。由此推论，R中由解析映射构成的CIF的非退化不变集的分形曲率测度存在的充要条件是系统是非格子系统。对于不变集为R的子集的系统，上述结果对更一般的cGDMS极限集是有效的。具体地说，我们证明了由相似组成的cGDMS的非退化极限集的Minkowski内容存在当且仅当系统是非格的，从而推广了R的自相似子集的相应结果。进一步，我们得到了非格cGDMS的极限集是Minkowski可测的，并由此验证了[Lap93]中的猜想4对于肖特基型Fuchsian群的极限集，因为它们总是非格的(例如，见[Lal89])。
We characterise fractal sets arising from conformal iterated function systems (cIFS) and conformal graph directed Markov systems (cGDMS) for which the Minkowski content and the fractal curvature measures, as introduced in [Win08], exist. With this, we generalise studies that have been carried out for invariant sets of iterated function systems consisting of similarities. For self-conformal subsets of the d-dimensional Euclidean space we show, under certain geometric conditions, that the local average Minkowski content always exists and provide an explicit formula. If the system is non-lattice, we prove that also the local Minkowski content exists and coincides with its average version. From this general result we deduce new results for the subclass of self-similar sets, which significantly generalise the statements from [DKz10, LPW11]. For self-similar sets we additionally show that the fractal curvature measures exist in the non-lattice situation and that an average version exists for both lattice and non-lattice systems. With this, we provide a substantially different proof to those presented in [Win08, WZ10, Zah11] and gain alternative useful formulae for the fractal curvature measures. Another important subclass of self-conformal sets is the class of conformal C1+α-diffeomorphic images of self-similar sets, where α ∈ (0, 1]. We show that the local Minkowski content of such an image exists, whenever the local Minkowski content of the considered self-similar set exists. This new result also yields nice relationships between the local Minkowski contents of the self-similar set and its image. In contrast to the fact that the Minkowski content of a non-degenerate self-similar subset of R exists if and only if the system is non-lattice [LP93, Fal95, LvF06], we prove that there exist invariant sets of lattice cIFS for which the Minkowski content does exist. This surprising result is illustrated with examples and disproves Conjecture 4 of [Lap93] for self-conformal sets. We additionally show that even amongst the subclass of C1+α-diffeomorphic images of self-similar sets, there exist lattice sets for which the Minkowski content exists. However, we prove that the fractal curvature measures of such non-degenerate sets in R exist if and only if the underlying system is non-lattice. The importance of this subclass is emphasised by the result that a lattice cIFS in R, which consists of analytic maps, is automatically conjugate to a lattice system consisting of similarities. From this, we infer that the fractal curvature measures of a non-degenerate invariant set of a cIFS in R consisting of analytic maps exist if and only if the system is non-lattice. The above-mentioned results for systems, whose invariant set is a subset of R, are shown to be valid for the more general class of limit sets of cGDMS. Specifically, we show that the Minkowski content of a non-degenerate limit set of a cGDMS consisting of similarities exists if and only if the system is non-lattice, providing an important generalisation of the respective result for self-similar subsets of R. Further, we obtain that limit sets of non-lattice cGDMS are Minkowski measurable and by this verify Conjecture 4 of [Lap93] for limit sets of Fuchsian groups of Schottky type, since they are always non-lattice (see for instance [Lal89]).
自相似集的分形曲率测度
DOI：
10.1515/advgeom-2012-0026
发表时间：
2013
期刊：
arXiv: Metric Geometry
影响因子：
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作者：
Winter;Zähle
通讯作者：
Zähle