Smooth rigidity for codimension one Anosov flows

We introduce the matching functions technique in the setting of Anosov flows. Then we observe that simple periodic cycle functionals (also known as temporal distance functions) provide a source of matching functions for conjugate Anosov flows. For conservative codimension one Anosov flows $\varphi^t\colon M\to M$, $\dim M\ge 4$, these simple periodic cycle functionals are $C^1$ regular and, hence, can be used to improve regularity of the conjugacy. Specifically, we prove that a continuous conjugacy must, in fact, be a $C^1$ diffeomorphism for an open and dense set of codimension one conservative Anosov flows.