Rigorous Computation of Expansion in One-Dimensional Dynamics

Part (A) of the paper proves that the minimum mean cycle weight in a graph representation G of f on I \ Δ gives a uniform exponential lower bound on |(f^n)'(x)| along orbit segments that avoid Δ, by decomposing any path into cycles plus a simple path and absorbing the latter into a multiplicative constant C; see Definition 3, inequality (5), and Proposition 1 with its proof, which defines C via min_{Γ simple}(w(Γ) − |Γ|λ) and concludes the bound in (1) . Part (B) shows that if Algorithm 4 certifies a period-n orbit entirely outside Δ and returns λ_max with |(f^n)'(x)| ≤ e^{λ_max n}, then any uniform expansion exponent λ must satisfy λ ≤ λ_max (Proposition 4) . The candidate solution reproduces these arguments essentially verbatim: (A) constructs the path, performs the same cycle/simple-path decomposition, and defines the identical constant C0; (B) uses the periodic-orbit identity and the uniform expansion inequality to derive λ ≤ (1/p) ln |(f^p)'(x)| ≤ λ_max, matching the paper’s conclusion. One minor presentation issue in the paper is that Algorithm 4, line 20, refers to a “lower bound” for (1/n) ln |(f^n)'(x)|, whereas Proposition 4 uses λ_max as an upper bound for this quantity; this appears to be a typographical inconsistency, not affecting the stated result .