Dimensions of Kleinian orbital sets
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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.
Bartlett , T & Fraser , J 2023 , ' Dimensions of Kleinian orbital sets ' , Journal of Fractal Geometry , vol. 10 , no. 3/4 , pp. 267-278 . https://doi.org/10.48550/arXiv.2105.11298 , https://doi.org/10.4171/jfg/139
Journal of Fractal Geometry
Copyright © 2023 European Mathematical Society. Published by EMS Press. Open Access. This work is licensed under a CC BY 4.0 license (https://creativecommons.org/licenses/by/4.0/).
DescriptionJMF was financially supported by an EPSRC Standard Grant (EP/R015104/1) and a Leverhulme Trust Research Project Grant (RPG-2019-034).
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