Abstract. A contact manifold admittting a supporting contact form without contractible Reeb orbits is called hypertight. In this paper we construct a Rabinowitz Floer homology associated to an arbitrary supporting contact form for a hypertight contact manifold Σ, and use this to prove versions of conjectures of Sandon [San13] and Mazzucchelli [Maz15] on the existence of translated points and invariant Reeb orbits, and to show that positive loops of contactomorphisms give rise to non-contractible Reeb orbits.
Abstract. In this article we extend results of Grove and Tanaka [GT76,GT78,Tan82] on the existence of isometry-invariant geodesics to the setting of Reeb flows and strict contactomorphisms. Specifically, we prove that if M is a closed connected manifold with the property that the Betti numbers of the free loop space Λ(M ) are asymptotically unbounded then for every fibrewise star-shaped hypersurface Σ ⊂ T * M and every strict contactomorphism ϕ : Σ → Σ which is contact-isotopic to the identity, there are infinitely many invariant Reeb orbits.
In this article we extend results of Grove and Tanaka [GT76,GT78,Tan82] on the existence of isometry-invariant geodesics to the setting of Reeb flows and strict contactomorphisms. Specifically, we prove that if M is a closed connected manifold with the property that the Betti numbers of the free loop space Λ(M ) are asymptotically unbounded then for every fibrewise star-shaped hypersurface Σ ⊂ T * M and every strict contactomorphism ϕ : Σ → Σ which is contact-isotopic to the identity, there are infinitely many invariant Reeb orbits. * ( ϕ) and HF * ( ϕ) := HF +∞ * ( ϕ), and we always tacitly assume that the endpoints of the action windows do not belong to Spec( L c ), even if this is not explicitly said.The filtered Floer homology is stable under sufficiently small perturbations. This allows us to extend the definition of HF (a,b) * ( ϕ) to the case where the admissible ϕ is not necessarily non-degenerate. Namely, after making a C ∞ -small perturbation, one obtains a new admissible path ϕ ′ that is non-degenerate. The aforementioned stability property implies that one can unambiguously define HF (a,b) Given a < b and a ′ < b ′ such that a < a ′ and b < b ′ , there is a well defined map HF (a,b) * ( ϕ) → HF (a ′ ,b ′ ) * ( ϕ). We now use these maps to define the positive growth rate.
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