Construction of invariant measures of Lagrangian maps: minimisation and relaxation (Q5947796)

The author studies exact symplectic diffeomorphisms of the cotangent bundle \(T^*(T^d)\) of the \(d\)-dimensional torus which is defined by a \(\mathbb{Z}^d\)-periodic generating function \(h: \mathbb{R}^d\times \mathbb{R}^d\to \mathbb{R}\) with the additional property that for every \(x\in \mathbb{R}^d\) the functions \(y\to \nabla_xh(y,x)\) and \(y\to \nabla_x h(x,y)\) are diffeomorphisms of \(\mathbb{R}^d\) and that moreover the growth of \(h\) is superlinear. In analogy to by-now standard Aubry-Mather theory, for such an exact Lagrangian diffeomorphism \(F\) the space of invariant action-minimizing measures with a fixed rotation vector is investigated. This is done by looking at a (formal) semiflow defined by the system of equations \[ {du_i\over dt}=- \nabla_{u_i}h(u_{i-1}, u_i)- \nabla_{u_i} h(u_i, u_{i+1})\qquad (i\in \mathbb{Z}) \] for maps in \((\mathbb{R}^d)^{\mathbb{Z}}\). The main observation is that this equation indeed induces a semi-flow and a (formal) complete separable metric space \({\mathcal A}\) on which the map \(F\) acts as a shift transformation. Thus \(F\)-invariant Borel probability measures on \(T^*(T^d)\) correspond to shift invariant measures on \({\mathcal A}\). For such a measure \(\mu\) the action is defined by \(A(\mu)= \int_{\mathcal A} h(u_0, u_1) d\mu\) and the rotation vector by \(\rho(\mu)= \int_{\mathcal A} (x_1- x_0) d\mu\). The author shows that the space of invariant probability measures with finite action and a fixed rotation vector is not empty. The action-minimizing measures in this set are supported on the stationary points in \({\mathcal A}\), and every invariant measure which is supported on an invariant Lagrangian torus for the map \(F\) is minimizing. The author also gives a sufficient condition for the existence of ergodic invariant measures with an arbitrary fixed irrational rotation vector whose action is arbitrarily close to the minimal action for this vector.