SRB AND EQUILIBRIUM MEASURES VIA DIMENSION THEORY

The paper’s Section 3.5 defines the restricted two‑sided Bowen balls B*_{s,t}(x,r), the Carathéodory C‑structure with ξ(x,s,t) = e^{Φ(f^{−s}x,s+t)}/(s+t), and produces the outer measure m(Z) = lim_{T→∞} inf Σ (s+t)^{-1} e^{Φ(f^{−s}x,s+t) − (s+t)P(ϕ)}; Theorem 3.15 proves m is a finite, nonzero, flow‑invariant Borel measure and a scalar multiple of the unique equilibrium measure . The flow‑equivariance used in the proof is f_τ B*_{s,t}(x,r) = B*_{s+τ,t−τ}(f_τx,r) for s,t ≥ |τ| (eq. (4.38)) , and the metric‑outer‑measure criterion (Lemma 2.14) applies since diameters shrink as min(s,t)→∞ (bound (4.37)) . The candidate solution reproduces this construction and argument almost verbatim, with two minor issues: (i) a sign flip in the equivariance identity (it states s−τ,t+τ rather than s+τ,t−τ), and (ii) it informally says diameters shrink as s+t→∞ rather than requiring min(s,t)→∞. These do not alter the core proof idea, which otherwise matches the paper.