There exist transitive piecewise smooth vector fields on S^2 but not robustly transitive

The paper proves (i) an explicit three-zone, piecewise-linear Filippov PSVF on S^2 is topologically transitive and extends this to a one-parameter family Z_θ with π/6 < θ ≤ π/3, and (ii) no PSVF on S^2 with finitely many tangency points is robustly topologically transitive. The explicit construction with two switching circles at z = ±1/2, piecewise linear fields X and a rotated Y, and properties (i)–(vi) are all stated and verified in the proof of Theorem A, including the extended Filippov field on Σ and the existence of a periodic Filippov orbit, from which transitivity follows . The non-robustness result appears as Theorem C and is proved via a contradiction that uses Theorem B (sliding and escaping nonempty and connected) and a careful orbit surgery near visible tangencies to break the required connection under arbitrarily small perturbations . The candidate solution correctly reproduces (1) the construction and the mechanism establishing transitivity, and (2) the non-robustness claim. Its proof sketch for (2) uses a different but plausible perturbative mechanism (creating a sliding pseudo-equilibrium/trapping region) rather than the paper’s “break the TA→TB connection” argument. Minor deviations: the candidate describes the parameter set as an open interval (the paper gives π/6 < θ ≤ π/3), and cites an external equivalence with dense orbits that is not needed here. Overall, both are correct; the proofs are substantially aligned for (1) and different but compatible for (2).