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

TL;DR

This paper proves that for certain Anosov flows on 3-manifolds, local differentiability of a conjugacy implies it extends smoothly, generalizing previous rigidity results in hyperbolic dynamics.

Contribution

It establishes a new rigidity result showing local differentiability leads to global smooth conjugacy for Anosov flows on 3-manifolds, extending prior work in lower dimensions.

Findings

Differentiability at a point implies smooth extension of conjugacy.

Generalization of Sullivan and Ferreira-Pinto results to 3-dimensional flows.

Rigidity phenomenon for Anosov flows on 3-manifolds.

Abstract

For Anosov flows obtained by suspensions of Anosov diffeomorphisms on surfaces, we show the following type of rigidity result: if a topological conjugacy between them is differentiable at a point, then the conjugacy has a smooth extension to the suspended 3-manifold. These result generalize the similar ones of Sullivan and Ferreira-Pinto for 1-dimensional expanding dynamics and also a result of Ferreira-Pinto for 2-dimensional hyperbolic dynamics.