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Geometriae Dedicata

, Volume 191, Issue 1, pp 1–35 | Cite as

Hausdorff dimension of limit sets

  • Laurent Dufloux
Original Paper

Abstract

We exhibit a class of Schottky subgroups of \(\mathbf {PU}(1,n)\) (\(n \ge 2\)) which we call well-positioned and show that the Hausdorff dimension of the limit set \(\Lambda _\Gamma \) associated with such a subgroup \(\Gamma \), with respect to the spherical metric on the boundary of complex hyperbolic n-space, is equal to the growth exponent \(\delta _\Gamma \). For general \(\Gamma \) we establish (under rather mild hypotheses) a lower bound involving the dimension of the Patterson–Sullivan measure along boundaries of complex geodesics. Our main tool is a version of the celebrated Ledrappier–Young theorem.

Keywords

Hausdorff dimension Non-conformal repellers Complex hyperbolic geometry Dimension theory 

Mathematics Subject Classification

37C45 28A80 53D25 37D40 

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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Oulun YliopistoOuluFinland

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