On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
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DOI：
10.1090/tran/7334
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发表时间：
2016-02
影响因子：
1.3
通讯作者：
I. Morris;Pablo Shmerkin
中科院分区：
文献类型：
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作者：
I. Morris;Pablo Shmerkin
在较温和的条件下，我们证明了平面自仿射集的亲和维等于原集合的自仿射真子集上的自仿射测度的Lyapunov维数的上确界。这些自仿射子集可以被选择为具有更强的分离性质，并且以其亲和性的线性部分是正矩阵的方式来选择。结合这一结果和最近在研究自仿射测度及其相关的Furstenberg测度方面的一些突破，我们得到了一个自仿射集的Hausdorff维度等于它的亲和维度的新准则。例如，应用Barany，Hochman-Solomak和Rapaport的最新结果，我们给出了Hausdorff维等于亲和维且其线性部分不满足任何控制假设的自仿射集的新的显式例子。
Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine set equals its affinity dimension. For example, applying recent results of Barany, Hochman-Solomyak and Rapaport, we provide new explicit examples of self-affine sets whose Hausdorff dimension equals its affinity dimension, and for which the linear parts do not satisfy any domination assumptions.