Periodic geodesics for contact sub-Riemannian 3D manifolds

The paper establishes Theorem 0.1—existence of closed sub-Riemannian geodesics that spiral toward a nondegenerate closed Reeb orbit Γ and length asymptotics lk = 2√(πkT0) + lower-order terms—by constructing a Birkhoff normal form near Γ, reducing the geodesic flow to a twist Poincaré map PC(θ,I) = (θ + 2T0/I^2, I) + O(I^∞) on an invariant cylinder, and applying the Poincaré–Birkhoff theorem to produce periodic points; the integrable case directly yields a complete expansion, and the generic case follows from an O(I^∞)-small perturbation of the integrable model . The candidate solution reaches the same conclusion via a fast–slow averaging picture in a Martinet tube: the fast horizontal controls trace Larmor cycles with per-cycle s-drift Δs(J) ≈ π/J^2 and period τ(J) ≈ 2π/J; enforcing closure kΔs(Jk)=T0 gives Jk∼√(πk/T0) and lk = kτ(Jk)=2√(πkT0)+…, matching the paper’s asymptotics (the paper’s closure condition is 2T0/I^2=2kπ, equivalent to J=1/I) . While the model’s outline omits a rigorous construction of the invariant cylinder and the twist/fixed-point step needed in the nonintegrable case, its mechanism and asymptotics agree with the paper’s; thus both are correct, with substantially different proof strategies.