Unique ergodicity of the horocyclic flow on nonpositively curved surfaces

The paper proves two claims: (A) for any horocyclic parametrization on a compact, oriented, nonflat, nonpositively curved surface, there is a unique invariant probability giving full mass to Σ0 (the union of horocycles containing a rank–1 geodesic-flow recurrent vector); and (B) if there are no flat strips, then the horocyclic flow is uniquely ergodic on T^1M. These statements appear explicitly as Theorem A and Theorem B in the introduction and abstract, and are developed via a careful reparametrization theory (Beboutoff–Stepanoff) and a Margulis parametrization defined on Σ0 and, in the no–flat–strip case, on all of T^1M, together with an adaptation of Coudène’s equicontinuity argument to the nonuniformly hyperbolic setting of nonpositive curvature (see Theorem A/B, the reparametrization result, Corollary 3.6 and Theorem 3.7, and Section 4 including Proposition 4.1, Theorem 4.2, and Lemma 4.3) . By contrast, the model’s solution hinges on collapsing flat strips via a quotient X=T^1M/∼ defined by “bi-asymptotic geodesics,” claiming that both g_t and any horocyclic flow h_s descend to X and that a uniform renormalization law holds there. This is flawed: the proposed equivalence v∼w (“lifts tangent to bi-asymptotic geodesics”) is not preserved by the horocyclic flow, because h_s changes the forward endpoint v+, so h_s generally does not respect the equivalence classes; consequently, h_s does not descend. The model also assumes, without justification, a uniform-expansion renormalization identity and near a.e. injectivity of the quotient map on p(Σ0). None of these steps are required in the paper’s correct argument, which instead constructs and compares invariant measures under changes of parametrization and proves unique ergodicity by equicontinuity and disintegration on the boundary (see Section 3 and the adaptation in Section 4) .