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 starshaped hypersurfaces in 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.Contents * are Reeb chords 3.9. Computing positive equivariant wrapped Floer homology 4. On the minimal number of symmetric periodic Reeb orbits 4.1. Index of symmetric periodic Reeb orbits 4.2. Real dynamical convexity 4.3. Index increasing property 4.4. The common index jump theorem 4.5. Symmetric periodic Reeb orbits on displaceable hypersurfaces References