正线子系统上自仿射测度下的Hausdorff维数与亲和维数相等性
动力系统
2019-03-18 v3 经典分析与常微分方程
摘要
在温和条件下，我们证明了平面自仿射集的亲和维数等于支撑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。这些自仿射子集可被选取以具有更强的分离性质，并且其仿射变换的线性部分为正矩阵。将这一结果与自仿射测度及其关联的Furstenberg测度研究中的一些近期突破相结合，我们获得了自仿射集的Hausdorff维数等于其亲和维数的新判据。例如，应用Bárány、Hochman-Solomyak和Rapaport的近期结果，我们给出了Hausdorff维数等于亲和维数且线性部分不满足任何控制假设的自仿射集的新显式例子。
引用
@article{arxiv.1602.08789,
title = {On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems},
author = {Ian D. Morris and Pablo Shmerkin},
journal= {arXiv preprint arXiv:1602.08789},
year = {2019}
}
备注
v3: added several examples and figures, corrected statement of Proposition 7.2. v2: Added some new examples to show that the SOSC cannot be relaxed to the OSC in Theorems 1.2 and 1.3. Applications of some results of Hochman-Solomyak have been updated to more accurately reflect the form which those results take in Hochman and Solomyak's recent preprint. Minor additional typographical fixes