Good inducing schemes for uniformly hyperbolic flows, and applications to exponential decay of correlations

Given an Axiom A attractor for a $C^{1+\alpha }$ flow ($\alpha >0$), we construct a countable Markov extension with exponential return times in such a way that the inducing set is a smoothly embedded unstable disk. This avoids technical issues concerning irregularity of boundaries of Markov partition elements and enables an elementary approach to certain questions involving exponential decay of correlations for SRB measures.