A multifractal analysis for escaping trajectories on free groups

2025/11/10 by Ziyu Liu, Liu, Ziyu
Mathematics · Physics and Astronomy · #37B10 #37C45 #37D20 #37D40 #37F32 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2511.06980

openalex publication_date 2025/11/10 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28

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. Under very mild conditions, we provide a common positive lower bound for the u-dimensions of these three sets. We also show that the lower bound is equal to the u-dimensions of these three sets when the extension and u have certain symmetries. 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.

Citations

• A non-symmetric Kesten criterion and ratio limit theorem for random walks on amenable groups
• A multifractal analysis for uniform level sets with constraints on word appearance
• Random walks on weakly hyperbolic groups
• Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension
• Hausdorff dimension and Kleinian groups

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