Hamiltonian perturbations in contact Floer homology

Abstract

We study the contact Floer homology \(\mathrm{HF}_*(W, h)\) introduced by Merry–Uljarević in [21], which associates a Floer-type homology theory with a Liouville domain W and a contact Hamiltonian h on its boundary. The main results investigate the behavior of \(\mathrm{HF}_*(W, h)\) under the perturbations of the input contact Hamiltonian h. In particular, we provide sufficient conditions that guarantee \(\mathrm{HF}_*(W, h)\) to be invariant under the perturbations. This can be regarded as a contact geometry analog of the continuation and bifurcation maps along the Hamiltonian perturbations of Hamiltonian Floer homology in symplectic geometry. As an application, we give an algebraic proof of a rigidity result concerning the positive loops of contactomorphisms for a wide class of contact manifolds.