  This is the first paper of a series in which we plan to study spectral
asymptotics for sub-Riemannian Laplacians and to extend results that are
classical in the Riemannian case concerning Weyl measures, quantum limits,
quantum ergodicity, quasi-modes, trace formulae.Even if hypoelliptic operators
have been well studied from the point of view of PDEs, global geometrical and
dynamical aspects have not been the subject of much attention. As we will see,
already in the simplest case, the statements of the results in the
sub-Riemannian setting are quite different from those in the Riemannian one.
Let us consider a sub-Riemannian (sR) metric on a closed three-dimensional
manifold with an oriented contact distribution. There exists a privileged
choice of the contact form, with an associated Reeb vector field and a
canonical volume form that coincides with the Popp measure. We establish a
Quantum Ergodicity (QE) theorem for the eigenfunctions of any associated sR
Laplacian under the assumption that the Reeb flow is ergodic. The limit measure
is given by the normalized Popp measure.This is the first time that such a
result is established for a hypoelliptic operator, whereas the usual Shnirelman
theorem yields QE for the Laplace-Beltrami operator on a closed Riemannian
manifold with ergodic geodesic flow.To prove our theorem, we first establish a
microlocal Weyl law, which allows us to identify the limit measure and to prove
the microlocal concentration of the eigenfunctions on the characteristic
manifold of the sR Laplacian. Then, we derive a Birkhoff normal form along this
characteristic manifold, thus showing that, in some sense, all 3D contact
structures are microlocally equivalent. The quantum version of this normal form
provides a useful microlocal factorization of the sR Laplacian. Using the
normal form, the factorization and the ergodicity assumption, we finally
establish a variance estimate, from which QE follows.We also obtain a second
result, which is valid without any ergodicity assumption: every Quantum Limit
(QL) can be decomposed in a sum of two mutually singular measures: the first
measure is supported on the unit cotangent bundle and is invariant under the sR
geodesic flow, and the second measure is supported on the characteristic
manifold of the sR Laplacian and is invariant under the lift of the Reeb flow.
Moreover, we prove that the first measure is zero for most QLs.
