Potential well in Poincaré recurrence

The paper proves pointwise (Type 2) exponential approximations for hitting and return times for n-cylinders under φ- or ψ-mixing, using the potential well ρ(A) as the scaling parameter and explicit error terms εφ, εψ. The statements and proofs are laid out clearly in Theorem 1 and Section 4: small-t bounds are derived via a product expansion with pi := µA(TA>i−1)/µ(TA>i−1), controlling |pi−ρ| by mixing and short-overlap terms (leading to |µ(TA>t) − e^{−ρµ(A)t}| ≤ const·(τ(A)µ(A) + tµ(A)ε(A)) for t ≤ fA) ; large-t bounds are obtained by a block decomposition at the scale fA=[2µ(A)]−1 with mixing, giving the advertised decaying error C·µ(A)tε(A)e^{−µ(A)t(ρ−Cε(A))} . The model’s solution uses nonstandard steps that are incorrect: it mis-defines the “cluster-begin” event E_t^+ (it inadvertently forces A^c at time t), asserts an identity E_t^+∩H_t = {σ^{−t}A; T_A>t−1} which is false (the event does not exclude a hit at t−τ), and from this derives a delayed recursion S_t = S_{t−1} − λ S_{t−τ−1} + r_t. The paper’s correct conditioning identity is µ(σ^{−i}A | T_A>i−1) = µ(A)pi with pi≈ρ, leading to a multiplicative recursion on S_t and the product representation, not a delay equation in S_{t−τ−1} . The model also bounds µ(σ^{−t}A; T_A>t−1) by λ up to O(µ(A)ε(A)) and then divides by S_{t−1} to claim a uniform hazard bound, which ignores the dependence on S_{t−1}; the paper instead works directly with pi and carefully controls |pi−ρ| by ε(A) and the short-overlap geometry P(A),R(A),nA, with εψ(A)=n µ(A(nA−g0))+ψ(n) and εφ(A)=inf_{1≤w≤nA}{(n+τ)µ(A(w))+φ(nA−w)} . In short, the paper’s proof is coherent and matches its statements; the model’s argument contains definitional and logical errors even though it reaches formulas similar in shape for the small-t regime.