The Linear Escape Limit Set

If G is any Kleinian group, we show that the dimension of the limit set Λ is always equal to either the dimension of the bounded geodesics or the dimension of the geodesics that escape to infinity at linear speed. Suppose G is a discrete group of isometries on hyperbolic space B, n ≥ 2. The limit set Λ ⊂ Sn−1 is defined to be the accumulation set of the G-orbit of 0 ∈ B. A point x ∈ Λ can be associated to the radial segment that ends at x, which in turn projects to a geodesic ray γ (based at z0, the projection of 0) in the quotient M = B/G. We then write Λ as the disjoint union Λc ∪Λe, where Λc (the “conical limit set”) corresponds to γ’s that return to some compact set at arbitrarily large times and Λe (the “escaping limit set”) corresponds to γ’s that eventually leave every compact set. Obviously, dim(Λ) = max(dim(Λc), dim(Λe)) (where dim denotes Hausdorff dimension). The purpose of this note is to show that this equality is still true if we replace both Λc and Λe by certain subsets. Let Λb (the “bounded limit set”) be the subset of Λc corresponding to γ’s that remain bounded for all time. Parametrize geodesic rays by hyperbolic arclength and for 0 < α < 1, let Λα correspond to geodesic rays γ such that lim inf t distM (γ(t), z0) t > α, and let Λ` = ⋃ 0<α<1 Λα denote the “linear escape limit set”. Related sets have been considered by Lundh in [2] (Λ \Λα = L( 1 1+α ) where L is as in Definition 3.14 of [2]). Theorem 1. For any discrete group G, dim(Λ) = max(dim(Λb), dim(Λ`)). In other words, the dimension of Λ is determined either by the geodesic rays that stay bounded for all time or by those that escape to ∞ at the fastest possible speed. This is somewhat surprising since neither of these behaviors is “typical” in general. For example, if n = 2 and M is a finite area Riemann surface that is not compact, then Λc will have full Lebesgue measure but Λb will have measure zero (e.g., see [5]). Received by the editors May 22, 2002 and, in revised form, October 30, 2002. 2000 Mathematics Subject Classification. Primary 30F35.