  A high dimensional dynamical system is often studied by experimentalists
through the measurement of a relatively low number of different quantities,
called an observation. Following this idea and in the continuity of
Boshernitzan's work, for a measure preserving system, we study Poincar\'e
recurrence for the observation. The link between the return time for the
observation and the Hausdorff dimension of the image of the invariant measure
is considered. We prove that when the decay of correlations is super
polynomial, the recurrence rates for the observations and the pointwise
dimensions relatively to the push-forward are equal.
