Limit Theorems for Self-Intersecting Trajectories in \(\mathbb {Z}\)-Extensions

We investigate the asymptotic properties of the self-intersection numbers for \(\mathbb {Z}\) $\mathbb{Z}$ -extensions of chaotic dynamical systems, including the \(\mathbb {Z}\) $\mathbb{Z}$ -periodic Lorentz gas and the geodesic flow on a \(\mathbb {Z}\) $\mathbb{Z}$ -cover of a negatively curved compact surface. We establish a functional limit theorem.