Equivariant wrapped Floer homology and symmetric periodic Reeb orbits

Abstract

The aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds. This includes non-degenerate real dynamically convex star-shaped hypersurfaces in ${\mathbb {R}}^{2n}$ which are invariant under complex conjugation. As a result, we give a partial answer to the Seifert conjecture on brake orbits in the contact setting.

Remarks on the systoles of symmetric convex hypersurfaces and symplectic capacities

“…We shortly review a construction of wrapped Floer homology which is an open string analogue of symplectic homology. We refer to [3,20] for details.…”

“…Denote the set of contractible, as an element of π 1 ( W , L), Hamiltonian 1-chords by P L (H). We associate the index |x| = −µ(x) − n 2 ∈ Z for each non-degenerate contractible 1-chord in P L (H), where µ(x) is the Maslov index defined in [20,Definition 2.3]. Assume that c 1 (T W ) = 0 and π 1 (L) = 0 for well-definedness of µ(x).…”

“…We expect that the argument of [18] applies to all compact convex domains K ⊂ R 2n invariant under a linear anti-symplectic involution ρ, implying that c HW (K, ρ) = ℓ sym min (∂K, ρ). (v) Analogously to Gutt-Hutchings [16], we can construct, using positive equivariant wrapped Floer homology defined in [20], a sequence of real symplectic capacities c 1 ≤ c 2 ≤ • • • < ∞ for real Liouville domains. They satisfy all the conditions for real symplectic capacities, but the monotonicity.…”

“…Instead, they satisfy the restricted monotonicity as the wrapped Floer homology capacity. Using a Gysin-type exact sequence in wrapped Floer theory (see [20,Proposition 3.27] and [9, Proposition 2.9]), it is not hard to see that c 1 (W, λ, ρ) ≤ c HW (W, λ, ρ) for every real Liouville domain (W, λ, ρ).…”

Fredholm Theory for Pseudoholomorphic Curves with Brake Symmetry

“…Similar to Floer homology in symplectic manifolds, contact homology can be defined in contact manifolds ˚Corresponding author Email addresses: beijiachow@gmail.com (Beijia Zhou), zhucf@nankai.edu.cn (Chaofeng Zhu) by counting pseudoholomorphic curves. Pseudoholomorphic curves can also be used to study the brake orbit, we refer to the works of Frauenfelder and Kang [8] and J.Kim, S.Kim and Kwon [9].…”