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On the Hausdorff dimension of geodesics that diverge on average

Felipe Riquelme, Anibal Velozo

TL;DR

The paper establishes a precise link between the geometry of diverging-on-average geodesic directions and entropy at infinity: for complete, non-elementary pinched negatively curved manifolds, the Hausdorff dimension of the diverging-on-average radial limit set equals the entropy at infinity, δ_Γ^∞. The authors develop both an upper bound via shadow-covering and a matching lower bound through a Cantor-set construction and a mass-distribution argument, yielding HD(Λ_Γ^{∞,rad}) = δ_Γ^∞. They further relate this to hyperbolic and geometrically finite settings, providing explicit dimensions in finite-volume cases and clarifying the role of parabolic subgroups. An additional contribution shows that the entropy of infinite σ-finite invariant measures is bounded above by δ_Γ^∞, hinting at a variational principle for infinite-measure entropy. Overall, the work deepens the connections between boundary geometry, geodesic flow dynamics, and thermodynamic formalism in non-compact settings, with potential implications for understanding ends of non-compact manifolds and infinite-measure dynamics.

Abstract

In this article we prove that the Hausdorff dimension of geodesic directions that are recurrent and diverge on average coincides with the entropy at infinity of the geodesic flow for any complete, pinched negatively curved Riemannian manifold. Furthermore, we prove that the entropy of a $σ$-finite, infinite, ergodic and conservative invariant measure is bounded from above by the entropy at infinity of the geodesic flow.

On the Hausdorff dimension of geodesics that diverge on average

TL;DR

The paper establishes a precise link between the geometry of diverging-on-average geodesic directions and entropy at infinity: for complete, non-elementary pinched negatively curved manifolds, the Hausdorff dimension of the diverging-on-average radial limit set equals the entropy at infinity, δ_Γ^∞. The authors develop both an upper bound via shadow-covering and a matching lower bound through a Cantor-set construction and a mass-distribution argument, yielding HD(Λ_Γ^{∞,rad}) = δ_Γ^∞. They further relate this to hyperbolic and geometrically finite settings, providing explicit dimensions in finite-volume cases and clarifying the role of parabolic subgroups. An additional contribution shows that the entropy of infinite σ-finite invariant measures is bounded above by δ_Γ^∞, hinting at a variational principle for infinite-measure entropy. Overall, the work deepens the connections between boundary geometry, geodesic flow dynamics, and thermodynamic formalism in non-compact settings, with potential implications for understanding ends of non-compact manifolds and infinite-measure dynamics.

Abstract

In this article we prove that the Hausdorff dimension of geodesic directions that are recurrent and diverge on average coincides with the entropy at infinity of the geodesic flow for any complete, pinched negatively curved Riemannian manifold. Furthermore, we prove that the entropy of a -finite, infinite, ergodic and conservative invariant measure is bounded from above by the entropy at infinity of the geodesic flow.
Paper Structure (17 sections, 28 theorems, 116 equations, 3 figures)

This paper contains 17 sections, 28 theorems, 116 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statements of the main results
  3. Motivation and previous works
  4. Preliminaries
  5. Boundary at infinity
  6. The geodesic flow
  7. Limit sets
  8. Critical exponents and entropy
  9. Hausdorff dimension
  10. Piecewise geodesic rays
  11. Proof of Theorem \ref{['main1']}
  12. The upper bound
  13. The lower bound
  14. Construction of large collections of orbit points
  15. Construction of tree and a Cantor set at infinity
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $(\mathcal{M},g)$ be a complete, pinched negatively curved Riemannian manifold which is non-elementary. Then, where the Hausdorff dimension of the diverging on average radial limit set is computed using the Gromov-Bourdon visual metric on $\partial_\infty \widetilde{\mathcal{M}}$.

Figures (3)

  • Figure 1: Comparison triangles
  • Figure 2: Triangles and angles
  • Figure 3: The construction of ${\mathcal{T}}$

Theorems & Definitions (60)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • ...and 50 more