Smooth rigidity for higher dimensional contact Anosov flows

We apply the matching functions technique in the setting of contact Anosov flows which satisfy a bunching assumption. This allows us to generalize the 3-dimensional rigidity result of Feldman-Ornstein~\cite{FO}. Namely, we show that if two such Anosov flows are $C^0$ conjugate then they are $C^r$, conjugate for some $r\in [1,2)$ or even $C^{\mathrm{\infty }}$ conjugate under some additional assumptions. This, for example, applies to $1\mathrm{/}4$-pinched geodesic flows on compact Riemannian manifolds of negative sectional curvature. We can also use our result to recover Hamendstädt’s marked length spectrum rigidity result for real hyperbolic manifolds.