Quantum limits of the Martinet sub-Laplacian

In this article we study the semiclassical asymptotics of the Martinet sub-Laplacian on the flat toroidal cylinder $M=\mathbb{R}\times {\mathbb{T}}^2$. We describe the asymptotic distribution of sequences of eigenfunctions oscillating at different scales prefixed by Rothschild-Stein estimates via the introduction of adapted two-microlocal semiclassical measures. We obtain concentration and invariance properties of these measures in terms of effective dynamics governed by harmonic or an-harmonic oscillators depending on the regime, and we show additional regularity properties with respect to critical points of the eigenvalues of the Montgomery family of quartic oscillators.