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The marked length spectrum of Anosov manifolds. (English) Zbl 07097501
Summary: In all dimensions, we prove that the marked length spectrum of a Riemannian manifold $$(M,g)$$ with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a new stability estimate quantifying how the marked length spectrum controls the distance between the isometry classes of metrics. In dimension $$2$$ we obtain similar results for general metrics with Anosov geodesic flows. We also solve locally a rigidity conjecture of Croke relating volume and marked length spectrum for the same category of metrics. Finally, by a compactness argument, we show that the set of negatively curved metrics (up to isometry) with the same marked length spectrum and with curvature in a bounded set of $$C^\infty$$ is finite.

##### MSC:
 53C24 Rigidity results 53C22 Geodesics in global differential geometry 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37C27 Periodic orbits of vector fields and flows
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