  Pesin sets are measurable sets along which the behavior of a matrix cocycle
above a measure preserving dynamical system is explicitly controlled. In
uniformly hyper-bolic dynamics, we study how often points return to Pesin sets
under suitable conditions on the cocycle: if it is locally constant, or if it
admits invariant holonomies and is pinching and twisting, we show that the
measure of points that do not return a linear number of times to Pesin sets is
exponentially small. We discuss applications to the exponential mixing of
contact Anosov flows, and counterexamples illustrating the necessity of
suitable conditions on the cocycle.
