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Bishop-Jones' Theorem and the ergodic limit set

Nicola Cavallucci

Abstract

For a proper, Gromov-hyperbolic metric space and a discrete, non-elementary, group of isometries, we define a natural subset of the limit set at infinity of the group called the ergodic limit set. The name is motivated by the fact that every ergodic measure which is invariant for the geodesic flow on the quotient metric space is concentrated on geodesics with endpoints belonging to the ergodic limit set. We refine the classical Bishop-Jones' Theorem proving that the packing dimension of the ergodic limit set coincides with the critical exponent of the group.

Bishop-Jones' Theorem and the ergodic limit set

Abstract

For a proper, Gromov-hyperbolic metric space and a discrete, non-elementary, group of isometries, we define a natural subset of the limit set at infinity of the group called the ergodic limit set. The name is motivated by the fact that every ergodic measure which is invariant for the geodesic flow on the quotient metric space is concentrated on geodesics with endpoints belonging to the ergodic limit set. We refine the classical Bishop-Jones' Theorem proving that the packing dimension of the ergodic limit set coincides with the critical exponent of the group.

Paper Structure

This paper contains 10 sections, 13 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Gromov-hyperbolic spaces
  3. Gromov boundary
  4. Visual metrics
  5. Hausdorff and Packing dimensions
  6. Definitions of Hausdorff and Packing dimensions
  7. Visual dimensions
  8. Limit sets of discrete groups of isometries
  9. Bishop-Jones' Theorem
  10. An interpretation of the ergodic limit set

Key Result

Theorem 1.1

Let $X$ be a proper, $\delta$-hyperbolic metric space and let $\Gamma < \textup{Isom}(X)$ be non-elementary and discrete. Then

Theorems & Definitions (19)

  • Theorem 1.1: BJ97, Pau97, DSU17
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma 2.1: Cav21ter
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4: Cav21ter
  • Lemma 4.1
  • proof
  • ...and 9 more