论文标题
无限生成的自符号集的Assouad类型尺寸
Assouad type dimensions of infinitely generated self-conformal sets
论文作者
论文摘要
我们研究迭代功能系统的极限集的维度理论，该系统由数量无限的保形收缩组成。我们的重点是Assouad类型维度，该维度提供有关集合本地结构的信息。在自然的分离条件下，我们证明了Assouad维度的公式，并根据限制集的Hausdorff尺寸和收缩固定点集的尺寸证明了Assouad Spectrum的尖锐界限。我们用来表明边界的示例家族的Assouad光谱表现出有趣的行为，例如具有两个相变。我们的结果尤其适用于一组实际或复数的数字，这些数字持续扩展，并延伸了限制条目，以及某些抛物线吸引子。
We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of conformal contractions. Our focus is on the Assouad type dimensions, which give information about the local structure of sets. Under natural separation conditions, we prove a formula for the Assouad dimension and prove sharp bounds for the Assouad spectrum in terms of the Hausdorff dimension of the limit set and dimensions of the set of fixed points of the contractions. The Assouad spectra of the family of examples which we use to show that the bounds are sharp display interesting behaviour, such as having two phase transitions. Our results apply in particular to sets of real or complex numbers which have continued fraction expansions with restricted entries, and to certain parabolic attractors.
