THE TOPOLOGICAL PRESSURE OF TRAPPED SETS IN KERR-(DE SITTER) SPACETIMES

The paper’s main statement (Theorem 1.1) asserts P_Γ(0)=0 and P_Γ(s)<0 for all s>0 under ∞–normally hyperbolic trapping, using the standard variational principle for pressure and a two-step proof: (i) the entropy term vanishes and (ii) the unstable Jacobian averages are uniformly positive. This aligns with the candidate solution’s structure. However, the paper derives h_μ(ϕ_1)=0 by invoking the Pesin entropy formula, which in the stated form applies only to absolutely continuous invariant measures, while the variational principle optimizes over all ergodic measures. This leaves a logical gap (Cor. 2.6) as written. The candidate remedy—using Ruelle’s inequality together with the observation that all Lyapunov exponents along TΓ are ≤0 under ∞–normal hyperbolicity—closes the gap cleanly and yields h_μ(ϕ_1)=0 for all invariant μ. The second step (lower bound on ∫λ^u_1 dμ via backward contraction on E^u and the additive cocycle identity) is correct and matches the paper’s proof. Therefore, the result is correct, but the paper’s argument needs a minor fix in the entropy step (replace Pesin by Ruelle) to be complete. See the paper’s variational formula and main theorem, and the proof steps in Sections 2–3 for context .