Examples of integrable sub-Riemannian geodesic flows

We exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. Introduction Consider a distribution on a manifold M, i.e. subbundle of the tangent bundle Π ⊂ TM . Non-holonomic Riemannian metric is a Riemannian metric g ∈ SΠ on this bundle. We call the pair (Π, g) sub-Riemannian structure. A curve γ : [0, 1] → M is called horizontal if γ̇ is a section of Π. We denote the space of horizontal curves joining x to y by H(x, y). A theorem of RashevskyChow ([R]) states that if M is connected and Π is completely non-holonomic thenH(x, y) is always non-empty. By completely non-holonomic we mean distribution Π, such that the module D Π of order ≤ N self-commutators (of various kinds) of sections of Π is equal to the module D(M) of all vector fields for some big N . From now on we consider only completely non-holonomic distributions. For horizontal curves we calculate its length lg(γ) = ∫ 1 0 ‖γ̇‖gdt and this produces sub-Riemannian distance (metric) on M by dg(x, y) = inf γ∈H(x,y) lg(γ). A curve γ ∈ H is called geodesic if it realizes the minimum sub-distance for any two of its close points. The description of the most geodesics (normal ones) is given by the Euler-Lagrange variational principle. There is a Hamiltonian reformulation of this principle, due to Pontrjagin and co-authors [PBGM], which allows to consider the geodesic flow as the usual Hamiltonian flow on T M . There appear occasionally geodesics of different kind – abnormals – which are not governed by the Pontrjagin principle for γ, but depend on the distribution Π only. However if we consider contact distributions Π, i.e. distributions such that for any non-zero section α of the bundle Ann(Π) ⊂ T M we have α∧ (dα) 6= 0 for m = 2n+1 (in particular m = dimM is odd), then all geodesics are normal. As in the standard theory of geodesics we say the metric g is integrable if the Hamiltonian flow of this metric is integrable on T M in the Liouville sense,