Flots magnétiques en courbure négative

Abstract

Consider a magnetic field on a closed Riemannian manifold of negative curvature. A geometric study provides dynamical properties of the associated flow stronger than expected for general perturbations of the geodesic flow. Under natural assumptions, a magnetic flow on a closed surface cannot be ${\mathcal C}^1$-conjugate to a geodesic flow.

Rigidity of magnetic flows for compact surfaces

“…By Theorem 4.4, the magnetic curvature is constant q 0. If q < 0, the proof of Lemma 4.5 implies that f 2 < −k, and by results of Grognet in [5], we conclude that the flow is of Anosov type. If q = 0, then k = −f 2 and the magnetic flow is conjugate to a horocycle flow (see for instance [7]).…”

“…LetJ be a magnetic Jacobi field defined in a magnetic geodesic γ , J its orthogonal projection into a canonically oriented, unit vector orthogonal to γ . As in the geodesic case, the function u = J J , satisfies (see, for instance, [5,10]) the magnetic Riccati equation u + u 2 + q(γ ) = 0.…”

W-flows on Weyl manifolds and Gaussian thermostats

“…Is the limit supported on the union of the integral curves of E? Let us stress that this scenario differs from the perturbative conditions in [Go97], [Gr99], [W00'], where the geodesic curvature of the trajectories cannot be too large. Our trajectories may have arbitrarily large geodesic curvatures and yet they form a transitive Anosov flow.…”

A Note on Hyperbolic Flows in Sub-Riemannian Structures with Transverse Symmetries

“…In this context, magnetic flows, the flows generated by special forces, were discussed more than 30 years ago by Anosov and Sinai [8]. They were studied recently by Gouda [11], Grognet [12], M. and P. Paternain [14] and M.P. Wojtkowski [16].…”