Poisson limit theorems for systems with product structure

We obtain a Poisson Limit for return times to small sets for product systems. Only one factor is required to be hyperbolic, while the second factor is only required to satisfy polynomial deviation bounds for ergodic sums. In particular, the second factor can be elliptic or parabolic. As an application of our main result, several maps of the form Anosov map $ \times $ another map are shown to satisfy a Poisson Limit Theorem at typical points, some even at all points. The methods can be extended to certain types of skew products, including $ T, T^{-1} $-maps of high rank.