Regularity of the Anosov distributions of Euler-Lagrange deformations of geodesic flows (Q964196)

The author is interested in the regularity of the Anosov distributions of certain Euler-Lagrange flows. Let \(g\) be a negatively curved Riemannian metric of a closed \(C^{\infty}\) manifold \(M\) of dimension \(\dim (M) \geq 3\). Let \(L_{\lambda}\) be a \(C^{\infty}\) one parameter convex super linear Lagrangian on \(T\,M\) such that \(L_0 (v)= (1/2) g(u,v)\) for any \(v\in T\,M\). Denote by \(\varphi^{\lambda}\) the restriction of the Euler-Lagrange flow of \(L_{\lambda}\) on the \((1/2)\)-energy level. Then \(\varphi^{\lambda}\) is Anosov for all small enough \(\lambda\). The geometric consequences of different assumptions about the regularity of the Anosov distribution of \(\varphi^{\lambda}\) are studied. For example, in the case that the initial Riemannian metric \(g\) is real hyperbolic, it is proved that for small \(\lambda\), \(\varphi^{\lambda}\) has \(C^{\infty}\) weak stable and weak unstable distribution if and only if \(\varphi^{\lambda}\) is \(C^{\infty}\) orbit equivalent to the geodesic flow of \(g\).