Topological entropy of a magnetic flow and the growth of the number of trajectories

Final version. To appear in Discrete and Continuous Dynamical Systems - A.


Introduction
Topological entropy measures how complicated the global orbit structure of a dynamical system is, in terms of the exponential growth rate of the number of orbit segments that can be distinguished with arbitrarily fine, but finite, precision. It is important to relate this conjugacy invariant to dynamical or geometrical characteristic elements of the specific type of dynamical system under study, in order to have more tools to understand the structure of such systems. For the geodesic flow of a C ∞ metric on a manifold M , Mañé [7] proved formulae relating the topological entropy of the flow to the growth rate of: (A) the average on the unit sphere bundle of M of the determinant of the differential of the flow on the vertical distribution V (θ) = ker(dπ) θ , for π : SM → M the usual projection (B) the average of the number of geodesics connecting any two points on the manifold M .
Mañé's proof of this Riemannian result is symplectic, as the geodesic flow is Hamiltonian for the kinetic energy on T M endowed with a natural symplectic structure. One of the key points in the proof is the fact that the Lagrangian vertical distributionV (θ) = ker(dπ) θ , whereπ : T M → M is the standard projection, "is always twisted by the flow in the same direction" (see Paternain [9]).
This twist property of the geodesic flow for the vertical distribution, is an instance of the optical property of a Hamiltonian flow for a Lagrangian distribution, introduced by Bialy and Polterovich [1]. For optical Hamiltonian flows, an appropiate generalization of (A) was proved by Niche [8].
An important family of optical Hamiltonian flows is that of magnetic flows (or twisted geodesic flows). Let (M, ·, · ) be a smooth closed Riemannian manifold and Ω a closed 2-form on M . The tangent bundle T M is a symplectic manifold with symplectic structure ω = ω L + π * Ω, where ω L is the pullback of the standard symplectic form of T * M by the metric. The magnetic flow φ t is the The result in this note is the following version of (B) for magnetic flows.

Remarks
• The sum of the strong stable and strong unstable bundles of an Anosov flow provides a Hölder continuous, codimension one, invariant distribution of hyperplanes, transversal to the vector field, so our second formula applies to magnetic Anosov flows.
• Another important class of magnetic flows which admit a codimension one, transversal, invariant distribution is that of contact type magnetic flows. In dimension two and for M = T 2 , the magnetic flow has contact type for high energy levels and, provided that the magnetic field Ω is symplectic, for low energy levels. In dimensions greater than two the magnetic flow is contact type for high energy values if and only if the magnetic field Ω is exact. The distributions associated to these contact forms are C ∞ .
• Related results about the entropy of magnetic flows can be found in Burns and Paternain [4], Grognet [6] and Paternain and Paternain [10].

Setting and proofs
As we stated in the introduction, the magnetic flow is the Hamiltonian flow of the kinetic energy on the symplectic manifold (T M, ω = ω L + π * Ω), for ω L the pullback of the canonical symplectic form of T * M by the metric and Ω a closed 2-form. The Lorentz force is the linear map Y : Then the magnetic vector field X can be expressed in T T M as Identifying the spaces as follows