Multiplicity of closed Reeb orbits on prequantization bundles

Abstract

We establish multiplicity results for geometrically distinct contractible closed Reeb orbits of non-degenerate contact forms on a broad class of prequantization bundles. The results hold under certain index requirements on the contact form and are sharp for unit cotangent bundles of CROSS's. In particular, we generalize and put in the symplectictopological context a theorem of Duan, Liu, Long, and Wang for the standard contact sphere. We also prove similar results for non-hyperbolic contractible closed orbits and briefly touch upon the multiplicity problem for degenerate forms. On the combinatorial side of the question, we revisit and reprove the enhanced common jump theorem of Duan, Long and Wang, and interpret it as an index recurrence result.

Two closed orbits for non-degenerate Reeb flows

“…In this case, the situation is more delicate since we do have several examples of contact forms with finitely many simple closed orbits. Some works provide lower bounds for the number of simple closed orbits (some of these estimates are sharp), but, in general, these bounds request certain index conditions on the closed Reeb orbits; see [3,18,22,24,29] and references therein.…”

On the filtered symplectic homology of prequantization bundles

“…This is the assumption we will make henceforth. We will then use the linking number filtration to reprove the non-degenerate case of the contact Conley conjecture, originally established in [GGM15,GGM17], without relying on the machinery of contact homology.…”

“…Prequantization S 1 -bundles M form an interesting class of examples to study Reeb dynamics, and, in particular, the question of multiplicity of closed Reeb orbits with applications, for instance, to closed magnetic geodesics and geodesics on CROSS's; see, e.g., [GGM17]. The range of possible dynamics behavior in this case is similar to that for Hamiltonian diffeomorphisms and more limited than for all contact structures where it should be, more adequately, compared with the class of symplectomorphisms; see [GG15].…”

“…This fact has important conceptual consequences. For instance, the proof of the contact Conley conjecture for prequantization bundles with aspherical base (i.e., the existence of infinitely many simple closed Reeb orbits) from [GGM15,GGM17], which relies on the free homotopy class grading of the cylindrical contact homology, fails to directly translate to the symplectic homology framework.…”

“…One of the goals of this paper is to systematically study the filtered symplectic homology of W , equivariant and ordinary. Note that total Floer theoretic "invariants" such as the equivariant and/or positive symplectic homology of E has been calculated explicitly; see [GGM17,Oa08,Ri14]. Moreover, in many cases the total symplectic homology of E vanishes and this fact alone is sufficient for many applications.…”