Smooth rigidity for higher-dimensional contact Anosov flows

Andrey Gogolev; Federico Rodriguez Hertz

doi:10.3842/umzh.v75i9.7253

Andrey Gogolev Department of Mathematics, The Ohio State University, Columbus, USA
Federico Rodriguez Hertz Department of Mathematics, The Pennsylvania State University, University Park, USA

DOI: https://doi.org/10.3842/umzh.v75i9.7253

Keywords: Anosov flow, Contact flow, Rigidity, Marked length spectrum, Smooth conjugacy

Abstract

UDC 515.12

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 $C^0$ conjugate, then they are $C^{r}$ conjugate for some $r\in[1,2)$ or even $C^\infty$ conjugate under certain additional assumptions. This, for example, applies to geodesic flows on compact Riemannian manifolds of $1/4$-pinched negative sectional curvature. We can also use our result to recover Hamendstўаdt's marked length spectrum rigidity result for real hyperbolic manifolds.

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