Proportions of Incommensurate, Resonant, and Chaotic Orbits for Torus Maps

The paper numerically confirms a power-law fit for the 1D Arnold circle map, reporting µ(a) ≈ (1−a)^{0.314} and a refined fit µ(a) = (1−a)^{p1+p2(1−a)} with p1≈0.3139, p2≈−0.0208, and acknowledges there is no theoretical reason a single-term law should hold and that µ(1)=0 follows from rigorous results; however, it does not prove a universal exponent p* for µ(a) as a→1−. In 2D, the paper explicitly states—supported by multiple examples—that there is no universal power law as ε→εcrit, but again without a general proof. The model identifies (A) that a universal 1D exponent remains unproved as of the cutoff and (B) gives a short, rigorous contradiction argument using two decoupled families to show no single exponent can be universal in 2D. Thus the paper’s claims are empirically compelling but incomplete, while the model’s conclusions are correct and provide a proof for the 2D non-universality. See abstract and Sec. III for the 1D fits and µ(1)=0 remark, and Sec. IV (with Fig. 9) for the 2D non-universality statement.