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A multifractal analysis for escaping trajectories on free groups

Ziyu Liu

TL;DR

This work develops a multifractal analysis for escaping trajectories in free-group extensions of subshifts of finite type, defining three escaping sets toward a boundary point $x$ and measuring their $u$-dimensions. The authors introduce a restricted Poincaré series $Z(p|g)$ and its critical exponent $\delta(1_\Gamma)$, proving a universal positive lower bound $\dim_u(\mathcal{E}^*_x)\ge\dim_u(\mathcal{E}_x)\ge\dim_u(\mathcal{UE}_x)\ge\delta(1_\Gamma)$, and establishing equality under symmetry assumptions between the extension and the potential. The approach combines symbolic dynamics, group extension techniques, and properties of restricted Poincaré series, yielding explicit dimension statements for escaping sets and linking them to a random-walk-like Poincaré exponent. The results are then applied to geodesic flows on free-group covers of Schottky surfaces, demonstrating a dimension gap: the escaping sets have dimension $\delta(N)$, strictly smaller than the full limit set dimension $\delta(G)$ of the ambient Schottky group. Overall, the paper connects random-walk transience, multifractal spectra, and hyperbolic geometry through a unified symbolic framework.

Abstract

We consider a free group extension of a subshift of finite type $σ:Σ\rightarrowΣ$, 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.

A multifractal analysis for escaping trajectories on free groups

TL;DR

This work develops a multifractal analysis for escaping trajectories in free-group extensions of subshifts of finite type, defining three escaping sets toward a boundary point and measuring their -dimensions. The authors introduce a restricted Poincaré series and its critical exponent , proving a universal positive lower bound , and establishing equality under symmetry assumptions between the extension and the potential. The approach combines symbolic dynamics, group extension techniques, and properties of restricted Poincaré series, yielding explicit dimension statements for escaping sets and linking them to a random-walk-like Poincaré exponent. The results are then applied to geodesic flows on free-group covers of Schottky surfaces, demonstrating a dimension gap: the escaping sets have dimension , strictly smaller than the full limit set dimension of the ambient Schottky group. Overall, the paper connects random-walk transience, multifractal spectra, and hyperbolic geometry through a unified symbolic framework.

Abstract

We consider a free group extension of a subshift of finite type , and consider three sets of points in 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 -dimensions of these three sets. We also show that the lower bound is equal to the -dimensions of these three sets when the extension and 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.

Paper Structure

This paper contains 16 sections, 10 theorems, 88 equations.

Table of Contents

  1. Introduction and statement of main results
  2. Main results and their motivation from random walks
  3. Application to geodesic flows on free group covers of Schottky surfaces
  4. Plan of this paper
  5. Preliminaries
  6. Symbolic dynamics
  7. Coding of Gromov boundary of free groups
  8. Homomorphic projections and group extensions of SFTs
  9. Restricted Poincaré series and its properties
  10. Proofs of Theorem \ref{['mainthm:dimspec:general']} and Theorem \ref{['mainthm:dimspec:symmetric']}
  11. Proof of Theorem \ref{['mainthm:dimspec:general']}
  12. Proof of Theorem \ref{['mainthm:dimspec:symmetric']}
  13. Application to geodesic flows on free group covers of Schottky surfaces
  14. Preliminaries on Schottky groups
  15. Proof of Theorem \ref{['mainthm:geodflow']}
  16. ...and 1 more sections

Key Result

Theorem A

Let $\sigma:\Sigma\rightarrow\Sigma$ be a topologically transitive SFT, and $\Gamma$ be a non-cyclic, finitely generated free group. Suppose that $\chi:\Sigma^*\rightarrow\Gamma$ is a projection with $\chi^{-1}(1_\Gamma)$ being transitive. Let $u:\Sigma\rightarrow(0,+\infty)$ be a Hölder continuous where $\delta(1_\Gamma):=\inf\set{p\geq 0|\sum_{\omega\in\chi^{-1}(1_\Gamma)}\exp(-pS_\omega u)<+\i

Theorems & Definitions (20)

  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 10 more