Smooth rigidity for higher dimensional contact Anosov flows

TL;DR

This paper extends rigidity results for contact Anosov flows from three dimensions to higher dimensions, showing that topological conjugacy implies smooth conjugacy under certain conditions, with applications to geodesic flows and hyperbolic manifolds.

Contribution

It generalizes 3D rigidity results to higher dimensions using matching functions, establishing smooth conjugacy from topological conjugacy for contact Anosov flows.

Findings

$C^0$ conjugate flows are $C^{r}$ conjugate for some $r ext{ in }[1,2)$

Under additional assumptions, flows are $C^ ext{infinity}$ conjugate

Application to $1/4$-pinched geodesic flows and hyperbolic manifolds

Abstract

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 $C^\infty$ conjugate under some additional assumptions. This, for example, applies to $1/4$-pinched geodesic flows on compact Riemannian manifolds of negative sectional curvature. We can also use our result to recover Hamendst\"adt's marked length spectrum rigidity result for real hyperbolic manifolds.