Rate Distortion Dimension of Random Brody Curves

The paper proves the Ruelle-type inequality rdim(B_N,T,d,μ) ≤ ∫ψ dμ and constructs invariant measures achieving equality for any 0 ≤ c < 2(N+1)ρ(CP^N) via the variational principle for mean dimension with potential and the deformation theory of Brody curves. The candidate solution follows the same two-step blueprint: (i) apply the variational inequality sup_μ{rdim+∫φ} ≤ mdim_M with φ = −ψ, together with mdim_M(B_N,T,d,−ψ) = 0, to get the inequality; (ii) use the paper’s construction to produce measures attaining equality. The only substantive deviation is the candidate’s extra claim that the constructed measures are ergodic, which the paper does not assert and in fact leaves as an open problem (Problem 9.10). Aside from that overclaim, the approach and conclusions match the paper’s main theorems. See Theorem 5.1, (3·5), Corollary 6.5 (= Theorem 2.4), and Theorem 9.9 (= Theorem 2.5) in the paper.