A Piecewise Smooth λ-Lemma

The paper proves a piecewise-smooth λ-lemma for time-T Poincaré maps P^t_ε of Filippov systems: if a u-disk Δ in W^u(p) meets W^s(p) transversely away from the switching set, then iterates of Δ accumulate on any target u-disk D in W^u(p), with C1-approximation when D avoids the switching set and C0-approximation otherwise. This is exactly Theorem A in the paper and its proof outline: reduce to a smooth neighborhood B of the hyperbolic fixed point p disjoint from the switching set, prove a local inclination lemma there by graph-transform/slope estimates, then transport the approximation to the desired D using the piecewise-smooth regularity of P on fixed-orbit-itinerary sets and continuity across switching, relying on the homeomorphism property and diffeomorphism off Ω ∪ P^{-1}(Ω) (Proposition 5). The candidate solution mirrors these steps: (i) itinerary-wise C^r regularity via impact-time implicit functions, (ii) a local λ-lemma in B using cone/graph transform, (iii) bringing Δ into B along W^s, and (iv) propagating C1/C0 closeness depending on whether D meets the switching set. The key ingredients and case split match the paper’s statements and proof structure, so both are correct and essentially the same argument. See Theorem A and its proof plan, the definition and properties of P^t_ε, and the piecewise-smooth diffeomorphism/homeomorphism results used in the reduction and propagation steps , and the local slope/expansion estimates underpinning the inclination lemma .