The Teichmüller Space of a 3-Dimensional Anosov Flow

For a transitive Anosov flow $\mathrm{\Phi }$ on 3-dimensional closed manifold $M$ , we realize its Teichmüller 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 ($r\ge 1$) 3-dimensional Anosov flows on $M$, we show that ${\mathcal{A}}^r(\mathrm{\Phi })$ the path component containing $\mathrm{\Phi }$ is homotopy equivalent to the identity component of the diffeomorphism group of the manifold, namely, $${\mathcal{A}}^r(\mathrm{\Phi })\simeq {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_0^r(M)\mathrm{.}$$ 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].