Dimension of limit sets in variable curvature

TL;DR

The paper addresses the problem of quantifying the size of limit sets for Kleinian groups acting on complete simply connected manifolds with pinched negative curvature, focusing on non-wandering versus recurrent geodesics and linear escape. It proves a dimension formula $\mathrm{HD}(\Lambda_\Gamma)=\max\{\mathrm{HD}(\Lambda^r_\Gamma),\ \mathrm{HD}(\Lambda^l_\Gamma)\}$ and connects weakly recurrent and linear-escape dynamics to bound the radial component via $\mathrm{HD}(\Lambda^r(\kappa))\le (1+\kappa)\delta_\Gamma$, while showing that small linear-escape parameters can realize full dimension for $\Lambda^l(\alpha)$. Additionally, it constructs a torsion-free infinitely generated Fuchsian group with $\mathrm{HD}(\Lambda^\tau_\Gamma)=0$, demonstrating that transient dynamics can be dimensionally trivial even in geometrically infinite settings. Overall, the work clarifies how recurrence and escape rates govern the fractal size of limit sets in variable negative curvature and provides concrete examples with vanishing transient dimension.

Abstract

We compute the Hausdorff dimension of the limit set of an arbitrary Kleinian group of isometries of a complete simply-connected Riemannian manifold with pinched negative sectional curvatures $-b^2\leq k\leq -1$. Moreover, we construct hyperbolic surfaces with a set of non-recurrent orbits of dimension zero.

Figures (3)

• Figure 1: Comparison theorem
• Figure 2:
• Figure 3:

Theorems & Definitions (29)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Theorem 2.2: Kaimanovich
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof