Volume asymptotics on rank one manifolds of nonpositive curvature

The paper proves the Margulis-type volume asymptotic for closed rank-one manifolds of nonpositive curvature and constructs a continuous limit c(x). In particular, it establishes an orbital-counting asymptotic for a_t(x,y)=#{γ: γy ∈ B(x,t)} with e^{-ht} a_t(x,y) → (1/h) c(x,y), where c(x,y) is given by an explicit double integral involving Patterson–Sullivan data on S_x and S_y; integrating over a fundamental domain F yields bt(x) ∼ (1/h) e^{ht} ∫_F c(x,y) dVol(y), and continuity of c follows (see the abstract and Theorem A; the summation/integration steps and the explicit formula for c(x,y) are given around the derivation of at(x,y) ∼ (1/h) e^{ht} c(x,y) and the subsequent integration to bt(x) in the text . Singular contributions are shown to be exponentially negligible (Lemma 6.1) . By contrast, the model asserts a stronger orbit-counting limit lim_{t→∞} e^{-ht} N(x,y,t) = (1/(h‖m_{BM}‖)) μ_x(∂X) μ_y(∂X) for all x,y and uses this to write c(x) as a simple product of total PS masses times Vol(F). The paper does not state such a product formula; instead it obtains a more nuanced c(x,y) depending on Busemann weights and measures on the regular sets, and then integrates over y ∈ F (see the display defining c(x,y) and the subsequent integral over F) . Thus the model’s claimed identification of the orbit-counting constant as a product of total PS masses divided by h‖m_{BM}‖ is unsupported in the paper and appears oversimplified.