Averaging theorems for slow fast systems in $\mathbb{Z}$-extensions (discrete time)

We study the averaging method for flows perturbed by a dynamical system preserving an infinite measure. Motivated by the case of perturbation by the collision dynamic on the finite horizon $\mathbb{Z}$-periodic Lorentz gas and in view of future development, we establish our results in a general context of perturbation by $\mathbb{Z}$-extension over chaotic probability preserving dynamical systems. As a by product, we prove limit theorems for non-stationary Birkhoff sums for such infinite measure preserving dynamical systems.