  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.
