Sullivan’s conjecture on generic sublinear orbit approach in hyperbolic space

Establish that for any discrete group action G on the hyperbolic space H^n of divergence type, with Patterson–Sullivan measure mu_o on the boundary ∂H^n, the set of boundary points xi ∈ ∂H^n for which the geodesic ray γ = [o, xi] satisfies lim_{t→∞} d(γ(t), G o)/t = 0 is generic with respect to mu_o.

Background

The paper proves that, for proper statistically convex-cocompact actions with contracting elements, sublinearly Morse directions form a full measure set for Patterson–Sullivan measures on horofunction boundaries, extending earlier results from CAT(0) and Teichmüller settings.

In the introduction, the authors recall Sullivan’s conjecture in the classical setting of discrete groups acting on hyperbolic space: for divergence-type actions, the Patterson–Sullivan measure should be supported generically on boundary points whose geodesic rays approach the orbit sublinearly. They note partial confirmations: Sullivan proved the conjecture when the Bowen–Margulis measure on the geodesic flow is finite, and the genericity of regularly contracting rays was established in negative curvature under finite BM measure, which would imply the conjecture.

The authors further observe that, for divergence-type actions, the Patterson–Sullivan measure is supported on conical limit points by the Hopf–Tsuji–Sullivan dichotomy; the conjecture thus predicts finer generic support on “sublinear” limit points without assuming SCC actions or finiteness of the Bowen–Margulis measure.