The authors consider the one-dimensional Cahn-Hilliard equation with an inertial term

$\u03f5{u}_{tt}$, where

$\u03f5\ge 0$ is small. It is known that this equation, endowed with proper boundary conditions, generates a strongly continuous semigroup which is dissipative. The first contribution is the proof of the existence of a connected global attractor. Then they show, under non restrictive condition, that the functions belonging to the attractor enjoy good smoothness properties. The main result is the construction of a robust family of exponential attractors

${\mathcal{M}}_{\u03f5}$, whose basins of attraction are the whole phase space, and whose fractal dimension is bounded, uniformly in

$\u03f5$.