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On the almost sure spiraling of geodesics in CAT(0) spaces

Harrison Bray, Andrew Zimmer

TL;DR

This work extends Sullivan’s cusp-entrance logarithm law to quotients of rank-one CAT(0) spaces by establishing a sharp penetration law for geodesics around Morse convex subsets. The authors blend Patterson–Sullivan theory, shadows of subspaces, and a Khinchin-type dichotomy to relate penetration times to a critical-exponent gap $\delta(\Gamma)-\delta(\Gamma_0)$, yielding a precise limit $\limsup_{t\to\infty} \frac{\mathfrak p_{\mathscr{C},\epsilon}(\ell,t)}{\log t} = \frac{1}{\delta(\Gamma)-\delta(\Gamma_0)}$ for almost every geodesic. The framework covers periodic Morse flats (after thickening) and yields concrete instances, including an application to nonpositive curvature 3-manifolds where the limsup equals $1/h_{top}(M)$. Central to the argument are the Subset Shadow Lemma and a Borel–Cantelli scheme, which translate geometric growth conditions into almost-sure dynamical statements via a measure-theoretic backbone. The results deepen our understanding of geodesic spiraling in nonpositively curved spaces and provide robust tools for studying logarithmic laws in higher-rank geometric contexts.

Abstract

We prove a logarithm law-type result for the spiraling of geodesics around certain types of compact subsets (e.g. quotients of periodic Morse flats) in quotients of rank one CAT(0) spaces.

On the almost sure spiraling of geodesics in CAT(0) spaces

TL;DR

This work extends Sullivan’s cusp-entrance logarithm law to quotients of rank-one CAT(0) spaces by establishing a sharp penetration law for geodesics around Morse convex subsets. The authors blend Patterson–Sullivan theory, shadows of subspaces, and a Khinchin-type dichotomy to relate penetration times to a critical-exponent gap , yielding a precise limit for almost every geodesic. The framework covers periodic Morse flats (after thickening) and yields concrete instances, including an application to nonpositive curvature 3-manifolds where the limsup equals . Central to the argument are the Subset Shadow Lemma and a Borel–Cantelli scheme, which translate geometric growth conditions into almost-sure dynamical statements via a measure-theoretic backbone. The results deepen our understanding of geodesic spiraling in nonpositively curved spaces and provide robust tools for studying logarithmic laws in higher-rank geometric contexts.

Abstract

We prove a logarithm law-type result for the spiraling of geodesics around certain types of compact subsets (e.g. quotients of periodic Morse flats) in quotients of rank one CAT(0) spaces.

Paper Structure

This paper contains 22 sections, 39 theorems, 241 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The assumptions of Theorem \ref{['thm:main']}
  3. Khinchin-type theorem
  4. Historical remarks
  5. Structure of the paper
  6. Preliminaries
  7. Notation and conventions
  8. Key definitions
  9. Convexity
  10. Patterson--Sullivan measures
  11. Morse subsets
  12. Convex cocompact subgroups
  13. Shadows of subspaces
  14. The upper bound
  15. The lower bound
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $M$ and $\mathbb{T} \subset M$ be as above. Then for any $\epsilon>0$ sufficiently small and $\mathsf{m}$-a.e. $v \in T^1 M$, where $h_{top}(M)$ is the topological entropy of the geodesic flow on $T^1M$.

Figures (2)

  • Figure 1: The configuration of points in the proof of the implication (2) $\Rightarrow$ (3) in Theorem \ref{['thm:equiv of Morse']}.
  • Figure 2: The arrangement of points in the proof of Lemma \ref{['lem:nesting_shadows']}, with geodesics labeled by the time parameters $a_i,b_i,a_i',b_i'$.

Theorems & Definitions (76)

  • Theorem 1.1: see Section \ref{['sec:proof of thm nonpositive']}
  • Theorem 1.2: see Section \ref{['s:loglaw']}
  • Remark 1.3
  • Example 1.4
  • Definition 1.5
  • Theorem 1.6: Khinchin-type theorem, see Theorem \ref{['thm:khinchin_1']}
  • Theorem 1.7: Subset Shadow Lemma, see Theorem \ref{['thm:shadowing_flats']}
  • Remark 2.1
  • Lemma 2.2
  • proof
  • ...and 66 more