Dimension of self-affine sets with holes
Dimension of self-affine sets with holes
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DOI：
10.5186/aasfm.2015.4007
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发表时间：
2012-06
期刊：
影响因子：
--
通讯作者：
Andrew Ferguson;T. Jordan;M. Rams
中科院分区：
文献类型：
--
作者：
Andrew Ferguson;T. Jordan;M. Rams
In this paper we compute the dimension of a class of dynamically defined non-conformal sets. Let $X\subseteq\mathbb{T}^2$ denote a Bedford-McMullen set and $T:X\to X$ the natural expanding toral endomorphism which leaves $X$ invariant. For an open set $U\subset X$ we let X_U={x\in X : T^k(x)\not\in U \text{for all}k}. We investigate the box and Hausdorff dimensions of $X_U$ for both a fixed Markov hole and also when $U$ is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through $U$, while the Hausdorff dimension depends on the escape rate of the measure of maximal dimension.
