Alternative proof of contact homotopy for contact Anosov flows without Vogel’s uniqueness

Develop a proof, independent of Vogel’s uniqueness theorem for contact approximations, that if a smooth Anosov flow is contact for a positive (or negative) contact structure ξ and is tangent to another positive (or negative) contact structure ξ′, then ξ and ξ′ are contact homotopic. For example, construct a contact form for ξ′ whose Reeb vector field is transverse to ξ with the correct orientation and show that linear interpolation yields a path of contact forms, thereby establishing the homotopy without invoking Vogel’s result.

Background

The paper proves that if an Anosov flow is contact for ξ and tangent to ξ′ (with matching sign), then ξ and ξ′ are contact homotopic, using Vogel’s uniqueness theorem for contact approximations of admissible foliations. The authors note that an elementary argument avoiding Vogel’s theorem would be desirable.

They outline a potential strategy based on constructing a suitable contact form whose Reeb vector field is transverse to ξ and then interpolating linearly between contact forms. However, they state that this approach did not succeed for them, leaving open the development of a direct proof independent of Vogel’s theorem.