Intermediate dimensions of infinitely generated attractors
Intermediate dimensions of infinitely generated attractors
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DOI：
10.1090/tran/8766
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发表时间：
2021-04
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通讯作者：
Amlan Banaji;J. Fraser
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作者：
Amlan Banaji;J. Fraser
研究了由可数无穷个收缩组成的迭代函数系统极限集的维数理论。我们主要关注的是中间维：依赖于参数$\theta \in[0,1]$的维族，它插入在Hausdorff维和box维之间。我们的主要结果是在所有宫缩都是适形的情况下。在自然分离的条件下，证明了极限集的中间维数是极限集的Hausdorff维数和收缩不动点集的中间维数的最大值。这是建立在Mauldin和Urba 'nski关于Hausdorff和upper box维度的工作基础上的。我们给出了几个（通常是反直觉的）将我们的工作应用于投影的维度、分数布朗图像和一般的H\ \ old图像。这些应用程序适用于经过充分研究的示例，例如具有限制条目的实数或复连分数展开的数字集。我们也得到了几个没有假设一致性或任何分离条件的结果。我们用拓扑压力函数证明了无限生成吸引子的Hausdorff、box和中间维的一般上界。我们还证明了“一般”无限迭代函数系统的极限集具有等于环境空间维的盒维和中间维，其中“一般”可以表示“全测度”或“共测度”。
We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of contractions. Our primary focus is on the intermediate dimensions: a family of dimensions depending on a parameter $\theta \in [0,1]$ which interpolate between the Hausdorff and box dimensions. Our main results are in the case when all the contractions are conformal. Under a natural separation condition we prove that the intermediate dimensions of the limit set are the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This builds on work of Mauldin and Urba\'nski concerning the Hausdorff and upper box dimension. We give several (often counter-intuitive) applications of our work to dimensions of projections, fractional Brownian images, and general H\"older images. These applications apply to well-studied examples such as sets of numbers which have real or complex continued fraction expansions with restricted entries. We also obtain several results without assuming conformality or any separation conditions. We prove general upper bounds for the Hausdorff, box and intermediate dimensions of infinitely generated attractors in terms of a topological pressure function. We also show that the limit set of a 'generic' infinite iterated function system has box and intermediate dimensions equal to the ambient spatial dimension, where 'generic' can mean either 'full measure' or 'comeagre.'