Karlsson–Nussbaum Conjecture on Convex Boundary Attractors for Hilbert Metric Nonexpansive Maps

Prove the Karlsson–Nussbaum conjecture: For any bounded convex domain D in a finite-dimensional real vector space endowed with Hilbert’s projective metric (D, d_H), and any fixed-point-free nonexpansive mapping f: D → D, establish the existence of a convex set Ω ⊆ ∂D such that for every x ∈ D, all accumulation points ω_f(x) of the orbit O(x, f) lie in Ω.

Background

The conjecture arises from work showing that for fixed-point-free nonexpansive maps acting on the Hilbert metric of a bounded convex domain, the attractor is star-shaped with respect to a boundary point. Karlsson and Noskov’s result on star-shapedness led to the stronger assertion that there exists a convex subset of the boundary capturing all orbit accumulation points.

This paper extends Denjoy–Wolff-type results and provides related attractor properties (e.g., star-shapedness and containment in ch(ch(ξ))) for various geodesic and quasi-geodesic settings, but does not resolve the convexity assertion of the Karlsson–Nussbaum conjecture, which the authors explicitly note remains a major unresolved problem.