On the regularity of the Anosov splitting for twisted geodesic flows

Let $M$ denote a closed Riemannian manifold whose geodesic flow is Anosov. Given a real number $\lambda$ and a smooth one form $\theta$, consider the twisted geodesic flow obtained by twisting the canonical symplectic structure by the lift of $\lambda d\theta$ to the tangent bundle of $M$. For $\lambda$ in a certain open interval around the origin the twisted flow remains Anosov. We show that the Anosov splitting of the twisted geodesic flow is never of class $C^{1}$ unless $\lambda=0$.

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Published 1 January 1997