Iterated function systems of affine expanding and contracting maps on the unit interval

The paper’s main theorem (covering Lp0<0, =0, >0) is consistent and supported by clear arguments using Borel–Cantelli and a carefully engineered random-walk analysis for the neutral case, and by existence of an absolutely continuous stationary measure for the two-point system in the expanding case. By contrast, the model’s Phase-2 solution makes two critical errors: (1) in the neutral case it asserts that only finitely many ‘wrap’ events occur almost surely—this is not shown in the paper and is generally false; the paper only needs and proves zero density of bad times. (2) In the expanding case it claims an explicit frequency P(ε)=(1−p0)^{tε}, which contradicts the paper’s results; for multiplicatively dependent (M,N) the paper proves a different ε-asymptotic governed by ν1 solving p0ζ^{k+ℓ}−ζ+1−p0=0, yielding P(ε)∼ε^{−ln(ν1)/ln(κ)}, not the model’s power with base 1−p0.