RUELLE’S INEQUALITY AND PESIN’S FORMULA FOR ANOSOV GEODESIC FLOWS IN NON-COMPACT MANIFOLDS

The paper establishes (i) Ruelle’s inequality for the time-one map of an Anosov geodesic flow on a non-compact manifold under bounded-curvature hypotheses, and (ii) Pesin’s formula in the finite-volume, C1–Hölder case, by carefully overcoming non-compactness via uniform bounds on dϕ, Egorov/Oseledec selections, and a Mañé-style lower bound argument (Theorems 1.1–1.2, Sections 3, 5–6). In particular, Theorem 3.1 proves the needed integrability of log∥dϕ±1∥, and Section 6 derives the entropy lower bound yielding equality hμ(ϕ)=∫X+ dμ when μ≪Lebesgue on finite-volume SM. The candidate solution invokes the compact-case Margulis–Ruelle and Pesin/Ledrappier–Young formulas as black boxes and asserts a uniform bound on ∥Dϕ±1∥ from curvature bounds via Grönwall, without addressing the non-compact obstacles addressed in the paper and without citing the non-compact extension of Ruelle’s inequality; thus its proof is not justified in this setting, even though its final conclusions match the paper’s results. See Theorems 1.1–1.2 and the proofs in Sections 3, 5, and 6 of the paper .