Orbits Closeness for Slowly Mixing Dynamical Systems

The paper proves a general upper bound (Theorem 1.1) and then, under an inducing scheme satisfying (1.1) with C_{μ_F}=C_μ, upgrades this to a full limit 2/C_μ (Theorem 2.1) via a clean subsequence/ergodic-time-change argument; see the statement and proof excerpts for Theorem 1.1 and Theorem 2.1 in the uploaded PDF . By contrast, the model’s Step 1 contains a summability error: choosing r_n = n^{-(2/C_μ+ε)} and then using I_μ(r) ≤ r^{C_μ−ε} does not in general make ∑_k 2^{2k}(r_{2^k})^{C_μ−ε} summable unless an extra margin (e.g., δ>0 or a log factor) is introduced. This undermines the claimed universal upper bound proof. The model’s Step 2 also misstates a shift inequality (it uses the same n, whereas one needs to adjust n by a finite amount). The remaining steps align in spirit with the paper (inducing and time-change), but the two noted issues are substantive enough that the model’s proof, as written, is not correct, while the paper’s argument is correct.