On the Myrberg Limit Sets and Bowen-Margulis-Sullivan Measures for Visibility Manifolds without Conjugate Points

The uploaded paper (Fei Liu, 2024) states and proves the Myrberg type dichotomy: µp(Lm(Γ)) > 0 is equivalent to conservativity of the BMS geodesic flow on complete uniform visibility manifolds without conjugate points satisfying Axiom 2 (Theorem 3.4). The “⇒” direction uses the inclusion Lm ⊂ Lc (Proposition 2.3) and the HTS dichotomy (Theorem 1.1), yielding µp(Lc)=1 and hence conservativity . The “⇐” direction constructs the sets L(A), applies conservativity plus ergodicity from HTS, and then a Hopf–coordinate quasi-product argument to obtain µp(Lm)=µp(L(Γ))>0, completing the equivalence . As a corollary, under conservativity one even has µp(Lm)=µp(Lc) . By contrast, the candidate solution’s Step 3 invokes exactly this corollary (stating µp(Lm)=µp(Lc) under conservativity) to prove the “⇐” direction. In the paper, that equality is deduced from the very equivalence being proved, so using it here is circular. Minor point: the model assumes Axiom 2 to prove Lm ⊂ Lc, but the paper proves this inclusion without Axiom 2, using only uniform visibility and standard boundary continuity . Hence the paper’s argument is complete and correct, while the model’s relies on a circular dependency.