Contracting isometries and differentiability of the escape rate

TL;DR

This work analyzes how the drift $l(\mu)$ and asymptotic entropy $h(\mu)$ of random walks on groups acting with contracting isometries vary with the driving measure, deriving Lipschitz continuity and differentiability formulas for the drift, and establishing entropy continuity under weak moment-type assumptions. It develops a generalized pivoting framework (adapting Gouëzel’s method) within spaces exhibiting contracting BGIP axes, producing high-probability semi-aligned configurations via long Schottky sets. The authors extend regularity results to CAT(-1) type settings, obtaining $C^{\infty}$-style regularity for the drift in suitable geometries, and connect these results to a broad class of spaces including Teichmüller space, Outer space, CAT(0) and CAT(-1) spaces, and relatively hyperbolic groups. The paper also develops the squeezing formalism and a sublinear entropy-growth argument to prove entropy continuity for measures with finite time-one entropy, even without strong moment conditions, and it provides detailed constructions of WPD Schottky sets to support the theory. These results unify and extend prior regularity findings (analyticity, $C^{1}$, and $C^{2}$-type regularity) across diverse geometric contexts, enhancing our understanding of random walks in non-positively curved and hyperbolic-like spaces.

Abstract

Let $G$ be a countable group whose action on a metric space $X$ involves a contracting isometry. This setting naturally encompasses groups acting on Gromov hyperbolic spaces, Teichm{ü}ller space, Culler-Vogtmann Outer space and CAT(0) spaces. We discuss continuity and differentiability of the escape rate of random walks on $G$. For relatively hyperbolic groups, CAT(-1) groups and CAT(0) cubical groups, we further discuss analyticity of the escape rate. Finally, assuming that the action of $G$ on $X$ is weakly properly discontinuous (WPD), we discuss continuity of the asymptotic entropy of random walks on $G$.

Figures (1)

• Figure 1: Four points and a geodesic in an $\mathbb{R}$-tree.

Theorems & Definitions (54)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Definition 2.1: Metric space
• Definition 2.2: Quasigeodesics
• Remark 2.3
• Definition 2.8: choi2022random1
• proof