Why Escape Is Faster Than Expected

The paper defines escape rates for conditionally invariant measures and introduces p_i := μ_i(M^1_i) = lim_{n→∞} (μ_i(M^n_i))^{1/n} (its existence is guaranteed in their setting), then proves the inequality ∑_i μ(E_i) ρ_i ≥ −ln(∑_i μ(E_i) p_i) by noting ρ_i = −ln p_i when p_i>0 and applying Jensen’s inequality to −ln, treating p_i=0 as ρ_i=∞ for trivial cases. This is stated explicitly around equations (5)–(6) with the Jensen step and the p_i=0 contingency handled in the same way as the model does. The candidate solution reproduces exactly this reasoning: identifies ρ_i = −ln p_i from the root-limit definition of escape rate (the paper’s (2)–(3)), dispatches the p_i=0 cases, and applies Jensen to f(x)=−ln x with weights μ(E_i). Hence both the paper and the model are correct and essentially the same proof. Key statements can be seen where the authors set p_i and state the goal inequality (6), and then invoke Jensen’s inequality, including the p_i=0 discussion ; the definition of escape rate in terms of the root limit is given in (2)–(3) .