Sets of beta-expansions and the Hausdorff Measure of Slices through Fractals
Sets of beta-expansions and the Hausdorff Measure of Slices through Fractals
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发表时间：
2013-07
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通讯作者：
Tom Kempton
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作者：
Tom Kempton
我们研究自然措施集的β-扩张和切片通过自相似集。在β-展开式的背景下，这些结果使我们能够更好地理解随机β-变换的最大熵的度量，并重新解释Lindenstrauss，Peres和Schlag关于等分布的一个结果。这些应用程序中的每一个都与伯努利卷积的研究有关。在分形设置，这使我们能够了解如何分解豪斯多夫措施切片，导致条件下，几乎每一个切片通过自相似集有积极的豪斯多夫措施，推广长期已知的结果几乎无处不在的值的豪斯多夫维数。
We study natural measures on sets of beta-expansions and on slices through self similar sets. In the setting of beta-expansions, these allow us to better understand the measure of maximal entropy for the random beta-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading to conditions under which almost every slice through a self similar set has positive Hausdorff measure, generalising long known results about almost everywhere values of the Hausdorff dimension.