Abundance of Infinite Switching

The paper proves Theorems A and B by deriving a first-return map G_λ with a logarithmic singularity in the angular component and a height update of the form (y + λΦ2)^δ, then reducing to a singular 1D limit h_a(x) = x + ξ + a − K_ω ln|Φ2(x)| and applying results for nonuniformly expanding circle maps with log singularities; it also shows abundant infinite switching for a positive-measure set of parameters. In contrast, the candidate solution replaces the height update by an additive form Aρ^δ + λ h(θ) and invokes classical Wang–Young rank-one theory, which the paper explicitly states does not directly apply in this setting due to the unbounded derivative and non-compactness of the singular limit. The candidate’s proof therefore relies on an inapplicable framework and an incorrect normal form for the return map’s second component, even though the end conclusions align with the paper’s statements.