Differentiability of the minimal average action as a function of the rotation number

Abstract

Let$\bar f$ be a finite composition of exact twist diffeomorphisms. For any real number ω, letA(ω) denote the minimal average action of$\bar f$-invariant measures with angular rotation number ω. We prove thatA(ω) is differentiable at every irrational number ω and that for generic$\bar f$ it is not differentiable at rational ω, thus verifying conjectures of S. Aubry. Moreover, we show that these results are valid for a variational principleh which satisfies the condition which we have called elsewhere (H). As a consequence, we generalize a result due to Bangert concerning geodesics on a two dimensional torus with an arbitrary, but sufficiently smooth metric.