Positive operators and Hausdorff dimension of invariant sets
Positive operators and Hausdorff dimension of invariant sets
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正算子和不变集的豪斯多夫维数
DOI：
10.1090/s0002-9947-2011-05484-x
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发表时间：
2012
影响因子：
1.3
通讯作者：
S. Lunel
中科院分区：
文献类型：
--
作者：
R. Nussbaum;A. Priyadarshi;S. Lunel
． 本文给出了完备完备度量空间上具有“无穷小相似”的有限压缩映射族的不变集的Hausdorff维数的定理。我们的工作概括了Mauldin和Williams（1988）的图导向结构，并在其一般背景下与Schief（1996）的结果相关，但关键的不同在于映射不必是相似的。我们利用正线性算子理论和Krein-Rutman定理的推广，将Hausdorff维描述为r (L σ) = 1时σ > 0的唯一值，其中L σ， σ >是一个自然相关的正线性算子族，r （L σ）表示L σ的谱半径。我们也指出这些结果如何可以推广到无穷小相似的可数族。这里的目的是基础的：推导出一个具有适当普遍性的基本公式，并强调正线性算子理论在这种情况下的效用。稍后的工作将探讨基本定理及其泛函解析设置在研究豪斯多夫维数问题中的作用。
. In this paper we obtain theorems which give the Hausdorff dimension of the invariant set for a finite family of contraction mappings which are “infinitesimal similitudes” on a complete, perfect metric space. Our work gen-eralizes the graph-directed construction of Mauldin and Williams (1988) and is related in its general setting to results of Schief (1996), but differs crucially in that the mappings need not be similitudes. We use the theory of positive lin- ear operators and generalizations of the Krein-Rutman theorem to characterize the Hausdorff dimension as the unique value of σ > 0 for which r ( L σ ) = 1, where L σ , σ > 0, is a naturally associated family of positive linear operators and r ( L σ ) denotes the spectral radius of L σ . We also indicate how these results can be generalized to countable families of infinitesimal similitudes. The intent here is foundational: to derive a basic formula in its proper generality and to emphasize the utility of the theory of positive linear operators in this setting. Later work will explore the usefulness of the basic theorem and its functional analytic setting in studying questions about Hausdorff dimension.