Dimensions of Kleinian orbital sets

Thomas Bartlett, Jonathan Fraser

Research output: Contribution to journalArticlepeer-review


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 our assumption that the given set is bounded (in the hyperbolic metric) cannot be removed in general.
Original languageEnglish
Pages (from-to)267-278
Number of pages12
JournalJournal of Fractal Geometry
Issue number3/4
Early online date30 Aug 2023
Publication statusPublished - 6 Oct 2023


  • Orbital set
  • Kleinian group
  • Poincaré exponent
  • Upper box dimension
  • Limit set
  • Inhomogeneous attractor


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