Let L be a convex superlinear Lagrangian on a closed connected manifold N . We consider critical values of Lagrangians as defined by R. Mañé in [M3]. We show that the critical value of the lift of L to a covering of N equals the infimum of the values of k such that the energy level k bounds an exact Lagrangian graph in the cotangent bundle of the covering. As a consequence, we show that up to reparametrization, the dynamics of the Euler-Lagrange flow of L on an energy level that contains supports of minimizing measures with non-zero rotation vector can be reduced to Finsler metrics. We also show that if the Euler-Lagrange flow of L on the energy level k is Anosov, then k must be strictly bigger than the critical value c u (L) of the lift of L to the universal covering of N . It follows that given k < c u (L), there exists a potential ψ with arbitrarily small C 2 -norm such that the energy level k of L + ψ possesses conjugate points. Finally we show the existence of weak KAM solutions for coverings of N and we explain the relationship between Fathi's results in [F1,2] and Mañé's critical values and action potentials.

We study the dynamics and symplectic topology of energy hypersurfaces of mechanical Hamiltonians on twisted cotangent bundles. We pay particular attention to periodic orbits, displaceability, stability and the contact type property, and the changes that occur at the Mañé critical value c . Our main tool is Rabinowitz Floer homology. We show that it is defined for hypersurfaces that are either stable tame or virtually contact, and that it is invariant under homotopies in these classes. If the configuration space admits a metric of negative curvature, then Rabinowitz Floer homology does not vanish for energy levels k > c and, as a consequence, these level sets are not displaceable. We provide a large class of examples in which Rabinowitz Floer homology is nonzero for energy levels k > c but vanishes for k < c , so levels above and below c cannot be connected by a stable tame homotopy. Moreover, we show that for strictly 1=4-pinched negative curvature and nonexact magnetic fields all sufficiently high energy levels are nonstable, provided that the dimension of the base manifold is even and different from two. 53D40; 37D40

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We show that for a simple surface with boundary the attenuated ray transform in the presence of a unitary connection and a skew-Hermitian Higgs field is injective modulo the natural obstruction for functions and vector fields. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo a gauge transformation. The proofs involve a Pestov type energy identity for connections together with holomorphic gauge transformations which arrange the curvature of the connection to have definite sign.

For M a simple surface, the nonlinear statistical inverse problem of recovering a matrix field normalΦ:M→frakturso()n from discrete, noisy measurements of the SO(n)‐valued scattering data CΦ of a solution of a matrix ODE is considered (n ≥ 2). Injectivity of the map Φ ↦ CΦ was established by Paternain, Salo, and Uhlmann in 2012. A statistical algorithm for the solution of this inverse problem based on Gaussian process priors is proposed, and it is shown how it can be implemented by infinite‐dimensional MCMC methods. It is further shown that as the number N of measurements of point evaluations of CΦ increases, the statistical error in the recovery of Φ converges to 0 in L2(M)‐distance at a rate that is algebraic in 1/N and approaches 1/N for smooth matrix fields Φ. The proof relies, among other things, on a new stability estimate for the inverse map CΦ → Φ. Key applications of our results are discussed in the case n = 3 to polarimetric neutron tomography. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC