We prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen’s construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry’s measure, which is Ahlfors regular. By extending Bowen’s construction, we create a tool for transferring the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff, box-counting and Assouad dimensions of a large class of Smale spaces.

我们证明，直到拓扑共轭，每个 Smale 空间都承认一个 Ahlfors 正则 Bowen 测度。Bowen 对马尔可夫分区的构造意味着 Smale 空间是拓扑马尔可夫链的因子。后者配备了Parry's measure，这是Ahlfors常规的。通过扩展 Bowen 的构造，我们创建了一个工具，用于将 Parry 测度的 Ahlfors 正则性转移到 Smale 空间的 Bowen 测度。我们方法的一个重要部分使用了紧凑度量空间上的近似图的改进概念。此外，我们获得了对一大类 Smale 空间的 Hausdorff、box-counting 和 Assouad 维度的新估计。