SYMBOLIC DYNAMICS OF PLANAR PIECEWISE SMOOTH VECTOR FIELDS

The paper’s constructions and conjugacies (Theorem A(i–iv), Theorem B) are coherent and internally justified on the space of global trajectories, with a quotient to remove non-uniqueness and a carefully designed coding map and metric. By contrast, the candidate solution contains substantive mistakes: (1) it defines Θ_∞ allowing |x_{j+1}−x_j|≤2 including 0, but in the paper the infinite-alphabet subshift satisfies 0<|x_{j+1}−x_j|≤2; (2) it claims a direct homeomorphism s: Ω_k→shift without the crucial quotient Ω_k/∼ used in the paper to ensure injectivity; (3) it makes incorrect claims about arc lengths and time-1 transitions (e.g., for k=2 and “interior arcs of length 1/2”), and (4) adds an unsupported “logistic map” remark not present in the paper and not derived. The paper’s main arguments are sound; the model’s proof requires corrections to match the paper’s precise symbolic coding and topological framework (see Theorem A/B, Def. 15, Props. 10–12, 15–17).