Bishop-Jones' Theorem and the ergodic limit set

TL;DR

This paper introduces the ergodic limit set for groups acting on Gromov-hyperbolic spaces and proves that its packing dimension equals the group's critical exponent, linking geometric and dynamical properties.

Contribution

It defines the ergodic limit set and proves that its packing dimension matches the critical exponent, extending Bishop-Jones' classical theorem.

Findings

The ergodic limit set is a natural subset of the limit set at infinity.

The packing dimension of the ergodic limit set equals the critical exponent.

The result refines the classical Bishop-Jones' Theorem.

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.