A multifractal analysis for escaping trajectories on free groups

Abstract We consider a free group extension of a subshift of finite type σ : Σ → Σ , and consider three sets of points in Σ to which the corresponding trajectories on the free group escape to a given point in the Gromov boundary of the free group in three different senses. Assuming the topological transitivity of the extended shift, we provide a common explicit positive lower bound for the Hausdorff dimensions of these three sets. We also show that this lower bound is equal to the Hausdorff dimensions of these three sets when certain symmetries are satisfied in addition. Moreover, we apply our results to geodesic flows on free group covers of Schottky surfaces, and show that there exists a dimension gap between the three sets we considered and the entire escaping set.