论文标题
$（\ times m，\ times n）$的盒子尺寸 - 不变集
Box dimensions of $(\times m, \times n)$-invariant sets
论文作者
论文摘要
我们研究了托尔内态$（x，y）\ mapsto（m x \ text {mod} 1，\，n y \ text {mod} 1）$ for integers $ n> m \ geq 2 $。此类集合的基本示例是Bedford-Mcmullen地毯，更一般而言，不变的集合是通过关联符号空间上的subshift建模的。当这种子移位在拓扑上混合和索菲时，肯尼恩和佩雷斯的结果就可以很好地理解情况。此外，肯尼恩和佩雷斯的其他作品表明，豪斯多夫的维度通常由变化原则给出。因此，在基础偏移不是拓扑混合和索菲的情况下，我们的工作集中在盒子尺寸上。 We establish straightforward upper and lower bounds for the box dimensions in terms of entropy which hold for all subshifts and show that the upper bound is the correct value for coded subshifts whose entropy can be realised by words which can be freely concatenated, which includes many well-known families such as $β$-shifts, (generalised) $S$-gap shifts, and transitive sofic shifts.我们还提供了瞬态编码子缩影的示例，其中一般上限失败，框维实际上是由一般下限给出的。在非交易效率的SOFIC设置中，我们为框尺寸提供了一个公式，该公式通常介于一般的下限和上限之间。
We study the box dimensions of sets invariant under the toral endomorphism $(x, y) \mapsto (m x \text{ mod } 1, \, n y \text{ mod } 1)$ for integers $n>m \geq 2$. The basic examples of such sets are Bedford-McMullen carpets and, more generally, invariant sets are modelled by subshifts on the associated symbolic space. When this subshift is topologically mixing and sofic the situation is well-understood by results of Kenyon and Peres. Moreover, other work of Kenyon and Peres shows that the Hausdorff dimension is generally given by a variational principle. Therefore, our work is focused on the box dimensions in the case where the underlying shift is not topologically mixing and sofic. We establish straightforward upper and lower bounds for the box dimensions in terms of entropy which hold for all subshifts and show that the upper bound is the correct value for coded subshifts whose entropy can be realised by words which can be freely concatenated, which includes many well-known families such as $β$-shifts, (generalised) $S$-gap shifts, and transitive sofic shifts. We also provide examples of transitive coded subshifts where the general upper bound fails and the box dimension is actually given by the general lower bound. In the non-transitive sofic setting, we provide a formula for the box dimensions which is often intermediate between the general lower and upper bounds.
