Sublinearly Morseness in Higher Rank Symmetric Spaces

The paper proves that, under ϕ-divergence and finiteness of the Bowen–Margulis–Sullivan (BMS) measure on Γ0\SΩ, one has μ(∂SM,θ(Γ))=1, by constructing Pθ-sublinearly Morse sequences via ergodicity of the geodesic flow, a positive-time-in-compact-part argument (Birkhoff), and explicit estimates (Propositions 6.6 and 6.9) culminating in Theorem 6.3; see the definition of the BMS measure and Hopf–Tsuji–Sullivan dichotomy, and the proof of the main theorem in Section 6 (ergodicity via Theorem 3.3, then (6.9), then Theorem 6.3), and the precise content of Theorem 3.1 giving the projective model and boundary homeomorphism . By contrast, the candidate solution incorrectly bases the construction on Poincaré recurrence to a measurable “fundamental domain” and assumes compactness to get a finite increment set E_F; finiteness of m̄ does not produce a compact fundamental domain on SΩ, and this step is unjustified. It also attributes the sublinear quasigeodesic tracking and θ-regular drift to Theorem 3.1, which does not provide those properties—these are instead established later in the paper via Section 6 (Propositions 6.6, 6.9, and Theorem 6.3) . Therefore, the paper’s argument is correct, while the model’s proof has critical gaps.