论文标题
自我标记集的Hausdorff尺寸
The Hausdorff dimension of self-projective sets
论文作者
论文摘要
给定有限集$ \ Mathcal {a} \ subseteq \ mathrm {sl}（2，2，\ mathbb {r}）$我们研究吸引子$ k_ \ mathcal {a a} $的迭代功能系统的尺寸，由$ \ m m iathcal的投射操作引起的迭代功能系统。特别是，我们通过表明$ k_ \ mathcal {a} $的hausdorff尺寸由最低1和关键指数给出，假设$ k_ \ mathcal {a} $尤其是概括了Solomyak和Takahashi的最新结果。我们的方法结合了迭代功能系统和莫比乌斯半群的理论的技术，并使我们能够讨论Hausdorff维度的连续性，以及Furstenberg量度的支持的维度。
Given a finite set $\mathcal{A} \subseteq \mathrm{SL}(2,\mathbb{R})$ we study the dimension of the attractor $K_\mathcal{A}$ of the iterated function system induced by the projective action of $\mathcal{A}$. In particular, we generalise a recent result of Solomyak and Takahashi by showing that the Hausdorff dimension of $K_\mathcal{A}$ is given by the minimum of 1 and the critical exponent, under the assumption that $\mathcal{A}$ satisfies certain discreteness conditions and a Diophantine property. Our approach combines techniques from the theories of iterated function systems and Möbius semigroups, and allows us to discuss the continuity of the Hausdorff dimension, as well as the dimension of the support of the Furstenberg measure.
