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 \(\mathbb Z\) $\mathbb{Z}$ -periodic Lorentz gas and in view of future development, we establish our results in a general context of perturbation by \(\mathbb Z\) $\mathbb{Z}$ -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.