Dimensions of Kleinian orbital sets

TL;DR

This paper investigates the fractal dimensions of orbital sets generated by Kleinian groups acting on hyperbolic space, establishing a formula for their upper box dimension involving the set, the group’s Poincaré exponent, and the limit set.

Contribution

It provides a general formula for the upper box dimension of orbital sets without restrictive assumptions on the Kleinian group.

Findings

The upper box dimension equals the maximum of three specific quantities.

An explicit example shows boundedness assumption cannot be dropped.

The results connect geometric group theory with fractal dimension analysis.

Abstract

Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of Euclidean space. We prove that the upper box dimension of an orbital set is given by the maximum of three quantities: the upper box dimension of the given set; the Poincar\'e exponent of the Kleinian group; and the upper box dimension of the limit set of the Kleinian group. Since we do not make any assumptions about the Kleinian group, none of the terms in the maximum can be removed in general. We show by constructing an explicit example that the (hyperbolic) boundedness assumption on $C$ cannot be removed in general.