A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb {R}^{d}$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor, which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of $\mathbb {R}^{d}$ . In this paper we give an example where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.

中文翻译：

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具有两个最大维数不变测度的强不可约仿射迭代函数系统

Hutchinson 的一个经典定理断言，如果一个迭代函数系统作用于$\mathbb {R}^{d}$通过相似性并满足开集条件，则它承认一个独特的自相似测度，其豪斯多夫维数等于吸引子的维数。在吸引子的度量类别中，作为移位不变度量在编码空间上的投影而出现，这种自相似度量是最大维数的唯一度量。在仿射迭代函数系统的上下文中，如果亲和力的线性部分共享一个共同的不变子空间，或者更一般地说，如果它们保持适当子空间的有限联合，则可能存在最大维数的多个移位不变测量。$\mathbb {R}^{d}$. 在本文中，我们给出了一个示例，其中存在多个最大维数的不变度量，即使亲和性的线性部分不保留适当子空间的有限联合。