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Equivariant wrapped Floer homology and symmetric periodic Reeb orbits
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 42 / Issue 5 / May 2022
- Published online by Cambridge University Press:
- 04 February 2021, pp. 1708-1763
- Print publication:
- May 2022
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- Article
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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.
