Degenerate Reeb orbits

Question

I am reading about contact homology and ECH, and realized that I do not see what goes wrong with the definition of these theories, if one takes into count degenerate Reeb orbits. In general, I would really appreciate if someone can explain why non-degeneracy is an important requirement.

Answer 1

I assume here that $\lambda$ is degenerate but not too badly, i.e. such that its orbits are smooth manifolds (possibly with boundary) and $d\lambda$ has constant rank along them (along with another mild condition), then Morse theory $\longrightarrow$ Morse-Bott theory.

"Morse-Bott approach to contact homology" (thesis of Bourgeois) shows you how to handle the degenerate scenario (and Chris Wendl's thesis will explain more of the Fredholm theory), noting that finite-energy J-holomorphic curves will converge to (possibly degenerate) orbits. Everything's fine because for generic contact forms we have nondegeneracy, so you can always perturb things. In particular, we can take a sequence of nondegenerate contact forms $e^{f_k}\lambda$ converging to the degenerate one.

With nondegeneracy, the orbits are isolated and that helps greatly with computations (and constructions where you modify structures on neighborhoods of the orbits). In particular, for computing Conley-Zehnder indices. Without any modifications to the current definition of ECH, nondegeneracy of the contact form is required for the chain complex to be well-defined (up to a fixed action, we get finitely many orbits for the generators).