Smooth Rigidity for Higher-Dimensional Contact Anosov Flows

We apply the technique of matching functions in the setting of contact Anosov flows satisfying a bunching assumption. This allows us to generalize the 3-dimensional rigidity result of Feldman and Ornstein [Ergodic Theory Dynam. Syst., 7, No. 1, 49–72 (1987)]. Namely, we show that if two Anosov flow of this kind are C⁰ conjugate, then they are C^r conjugate for some r ∈ [1, 2) or even C^∞ conjugate under certain additional assumptions. This, e.g., applies to geodesic flows on compact Riemannian manifolds of 1/4-pinched negative sectional curvature. We can also use our result to recover Hamendstädt’s marked length spectrum rigidity result for real hyperbolic manifolds.