Explosion of differentiability for equivalencies between Anosov flows on 3-manifolds (Q2814398)

This article is devoted to Anosov flows obtained by suspension of Anosov diffeomorphisms on surfaces. It is proved that if a topological conjugacy between two Anosov flows, obtained from the suspension of Anosov maps on surfaces, is differentiable at a point, then the conjugacy has a smooth extension to the suspended 3-manifold.NEWLINENEWLINELet \(M\) and \(N\) be 3-dimensional closed and connected \(C^{\infty}\) Riemannian manifolds. The main result of the article is formulated in the following statement:NEWLINENEWLINETheorem. Let \(f : M \to M\) and \(g : N\to N\) be two \(C^{\infty}\) surface Anosov diffeomorphisms. Assume that there exists a topological conjugacy \(h : M \to N\) between them and, moreover, \(h\) is differentiable at a single point. Let \(c_f\) and \(c_g\) be two ceiling functions over \(M\) and \(N\) respectively. If \(c_f\) and \(c_g\) are differentiable, then the function \(\widehat{h} : M_{c_f}\to N_{c_g}\) is differentiable.NEWLINENEWLINEHere \(\widehat{h}\) is the restriction of the topological conjugacy \(h\) to \(M_{c_f}\) and \(M_{c_g}\).