Foliated and compactly supported isotopies of regular neighborhoods

The paper proves a compactly supported, parametric Alexander-type linearization near an F-saturated B and uses it to deform leaf-preserving diffeomorphisms to ones linear near B. In the non-foliated case this is Theorem 4.1.1, which constructs a deformation G from Dinv(M,B) to Dinv(M,B,p) and proves both W∞,∞- and S∞,∞-continuity; the foliated version (Theorem 5.1.5) adds the hypotheses (a) U is F-homogeneous and (b) U ∩ (closure(ω) \ ω) ⊂ B, ensuring G preserves leaves for all t and, crucially, at t = 0 via a limiting argument inside U . The candidate solution proposes exactly this Alexander-type conjugation/linearization and then asserts continuity and a deformation retraction onto the linear-near-B subgroup; this closely tracks the paper’s construction of H (eqs. (1.2)–(1.3)) and its induced G (eq. (4.2)), and invokes the same two conditions (a)–(b) to prove leaf preservation at t=0 . The model’s justification of continuity in strong Whitney topology is briefer than the paper’s explicit Step 2 argument for S∞,∞-continuity in Theorem 4.1.1, but the stated conclusion matches the paper’s results exactly . Overall, the model restates the paper’s method and conclusions with minor expository differences, hence both are correct with substantially the same proof idea.