The Teichm\"uller Space of a 3-Dimensional Anosov Flow

TL;DR

This paper characterizes the Teichmüller space of 3D transitive Anosov flows, proves path-connectedness of their orbit-equivalence space, and explores homotopy and rigidity properties.

Contribution

It introduces a new realization of the Teichmüller space for these flows and establishes their topological and rigidity properties.

Findings

Teichmüller space realized as a product of two function spaces

Path-connectedness of the orbit-equivalence space in dimension 3

Homotopy equivalence of the flow space component to the identity component of the diffeomorphism group

Abstract

For a transitive Anosov flow $\Phi$ on 3-dimensional closed manifold $M$ , we realize its Teichm\"uller space in the sense of smooth orbit-equivalence classes as a product of two function spaces. As an application, we show the path-connectedness of the orbit-equivalence space of 3-dimensional transitive Anosov flows which gives a positive answer of Potrie [53, Question 1] in dimension 3. Further, in the space of $C^r$-smooth ($r\geq 1$) 3-dimensional Anosov flows on $M$, we show that $\mathcal{A}^r(\Phi)$ the path component containing $\Phi$ is homotopy equivalent to the identity component of the diffeomorphism group of the manifold, namely, \[ \mathcal{A}^r(\Phi)\simeq {\rm Diff}^r_0(M). \] Moreover, we show the rigidity of time-preserving conjugacy for 3-dimensional transitive Anosov flows admitting $C^1$-smooth strong stable foliations, which gives partial answer of Gogolev-Leguil- Rodriguez Hertz [27, Question 2.8].