Hausdorff dimension of limit sets

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.

References in corpus (3)

• Dynamics on fractals and fractal distributions
• Hausdorff dimension of limit sets
• Projections of Patterson-Sullivan measures and the Mohammadi-Oh dichotomy

Cited by in corpus (6)

• Hausdorff dimension of limit sets
• Conformality for a robust class of non-conformal attractors
• Radial projections along chains
• The Yamabe operator and invariants on octonionic contact manifolds and convex cocompact subgroups of ${\mathrm{F}}_{4(-20)}$
• On McMullen's algorithm for the Hausdorff dimension of Complex Schottky Groups
• Dimension gap and variational principle for Anosov representations