论文标题
抛物线朱莉娅套装的Assouad类型尺寸
Assouad type dimensions of parabolic Julia sets
论文作者
论文摘要
我们证明，抛物线朱莉娅集的Assouad尺寸为$ \ max \ {1，h \} $，其中$ h $是Julia Set的Hausdorff尺寸。由于$ h $严格少于1，因此提供了Assouad和Hausdorff尺寸不同的示例。众所周知，朱莉娅套装的盒子和包装尺寸与$ h $一致，此外，$ h $可以以拓扑压力功能为特征。 Assouad维度的这种独特行为促进了对Assouad类型维度的进一步分析，包括Assouad和较低的光谱。我们计算抛物线朱莉娅集的所有Assouad类型尺寸和相关的$ H $ - 符合度量。此外，我们表明，如果朱莉娅（Julia）套件有一个火化点，那么Assouad尺寸为2。
We prove that the Assouad dimension of a parabolic Julia set is $\max\{1,h\}$ where $h$ is the Hausdorff dimension of the Julia set. Since $h$ may be strictly less than 1, this provides examples where the Assouad and Hausdorff dimension are distinct. The box and packing dimensions of the Julia set are known to coincide with $h$ and, moreover, $h$ can be characterised by a topological pressure function. This distinctive behaviour of the Assouad dimension invites further analysis of the Assouad type dimensions, including the Assouad and lower spectra. We compute all of the Assouad type dimensions for parabolic Julia sets and the associated $h$-conformal measure. Further, we show that if a Julia set has a Cremer point, then the Assouad dimension is 2.
