Hamiltonian Perturbations in Contact Floer Homology

The paper proves that HF*(W,h0) ≅ HF*(W,h1) for any smooth s-family of admissible contact Hamiltonians by constructing a bifurcation isomorphism: they build Hs so that all 1-periodic orbits are the same across s, hence the chain complexes are canonically identified. Crucially, they emphasize that the standard continuation map on the completion need not even be well-defined without the monotonicity h0 ≤ h1, and may fail to be an isomorphism even when it is defined; see the explicit statement that continuation maps on Ŵ fail to be well-defined unless h0 ≤ h1, and their counterexample (Theorem 1.2) where the continuation map is not an isomorphism despite a path of admissible slopes h0 ≤ h1 existing. Their isomorphism is obtained instead via a controlled interpolation on the cylindrical end using a radial cut-off µ(r) that prevents new orbits at infinity and yields a trivial (identity) bifurcation map on chains, with continuation agreeing with this map only under extra hypotheses (Theorem 1.3, e.g., h0 positive autonomous and h0 ≤ hs). The candidate solution, by contrast, asserts that continuation maps for an arbitrary admissible homotopy are well-defined and produce inverse isomorphisms, which directly contradicts the paper’s analysis and Theorem 1.2. Therefore, while the conclusion (invariance under admissible homotopy) matches Theorem 1.1, the candidate’s proof method is incorrect in this setting. Key paper evidence: the failure of well-definedness of continuation without monotonicity and the non-isomorphism phenomenon (Theorem 1.2), the bifurcation isomorphism construction via µ(r) (Lemma 4.4 and Section 2), and the agreement between continuation and bifurcation only under additional assumptions (Theorem 1.3) .