Smooth rigidity for 3-dimensional volume preserving Anosov flows and weighted marked length spectrum rigidity

Abstract

Let $X_1^t$ and $X_2^t$ be volume preserving Anosov flows on a 3-dimensional manifold $M$. We prove that if $X_1^t$ and $X_2^t$ are $C^0$ conjugate then the conjugacy is, in fact, smooth, unless $M$ is a mapping torus of an Anosov automorphism of ${\mathbb{T}}^2$ and both flows are constant roof suspension flows. We deduce several applications. Among them is a new result on rigidity of Anosov diffeorphisms on ${\mathbb{T}}^2$ and a new "weighted" marked length spectrum rigidity result for surfaces of negative curvature.

V2: 27 pages, one figure. The weighted marked length spectrum rigidity is improved to allow weights which are not necessarily positive. We now have an optimal version of this result

Cited by in corpus (2)

• Marked length spectrum rigidity for Anosov surfaces
• Smooth orbit equivalence rigidity for dissipative geodesic flows