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Dimensions of Kleinian orbital sets

Thomas Bartlett, Jonathan M. Fraser

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 Poincaré 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.

Dimensions of Kleinian orbital sets

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 Poincaré 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 cannot be removed in general.

Paper Structure

This paper contains 11 sections, 7 theorems, 42 equations.

Table of Contents

  1. Kleinian orbital sets
  2. Hyperbolic geometry, Kleinian groups, and orbital sets
  3. Dimension theory
  4. Inhomogeneous attractors
  5. Main results
  6. Proof of Theorem \ref{['thm:2']}
  7. Preliminary estimates and results from hyperbolic geometry
  8. Proof of Theorem \ref{['thm:2']}
  9. The lower bound
  10. The upper bound
  11. Proof of Theorem \ref{['examplethm']}

Key Result

Theorem 2.1

Let $\Gamma$ be a Kleinian group acting on $\mathbb{D}^n$ and $C$ be a non-empty bounded subset of $\mathbb{D}^n$. Then

Theorems & Definitions (10)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof