We study the contact Floer homology ${ \rm HF}_*(W, h)$ introduced by Merry-Uljarevi\'c, which associates a Floer-type homology theory to a Liouville domain $W$ and a contact Hamiltonian $h$ on its boundary. The main results investigate the behavior of ${ \rm HF}_*(W, h)$ under the perturbations of the input contact Hamiltonian $h$. In particular, we provide sufficient conditions that guarantee ${ \rm HF}_*(W, h)$ to be invariant under the perturbations. This can be regarded as a contact geometry analogue 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.
Categories: math.SG math.DS