Floer homology of automorphisms of Liouville domains

Abstract

We introduce a combination of fixed point Floer homology and symplectic homology for Liouville domains. As an application, we detect non-trivial elements in the symplectic mapping class group of a Liouville domain. 2.Floer homology for symplectomorphism of a Liouville domain 2.1. Floer data and Admissible data.The exact symplectomorphisms form a group, denoted by Symp( W , λ/d). The functions associated to composition and inverse are given byWe define the subgroup Symp c (W, λ/d) of Symp( W , λ/d) byAn exact symplectomorphism is not required to be compactly supported. Every Hamiltonian symplectomorphism generated by a compactly supported Hamiltonian is exact. However, an exact symplectomorphism need not be Hamiltonian or even isotopic to the identity.Definition 2.2. A real number a is called admissible if it is not a period of a Reeb orbit of (M, β). Definition 2.3. Let φ ∈ Symp c (W, λ/d). Floer data for φ is a pair (H, J) of a (timedependent) Hamiltonian H t : W → R and a family J t of ω-compatible almost complex structures on W satisfying the following conditions. H and J are twisted by φ, i.e.In addition, the conditions below hold near infinity H t (x, r) = ar, (2.6) J t (x, r)ξ β = ξ β , (2.7) J t (x, r)∂ r = R β , (2.8)

Volume growth in the component of fibered twists

“…We mention that a similar result using symplectic homology can be found in [38,Corollary 1.2] and Theorem B can be regarded as the relative version of that result.…”

Maximum principles in symplectic homology

“…for large r, where h t : Σ → R is a function which is invariant under the Reeb flow of α. A similar construction was also given independently by the second author in [26], and for time-independent h a different proof was given by Fauck in his thesis [9].…”

“…As remarked above, a special case of the theorem, in which the loop ϕ t is strict, was proved in [26]. In addition, recently progress has been made in several related directions by other authors: Seidel [25], Chiang, Ding and van Koert [14,15], and Barth, Geiges and Zehmisch [6].…”

“…In fact, the inclusion G c ֒→ Symp c (M, ω) is a homotopy equivalence. This is due to Biran-Giroux [7]; a proof can be found in [26,Lemma 3.3].…”

Viterbo’s transfer morphism for symplectomorphisms

“…The aim of this paper is to construct an analogue of Viterbo's transfer morphism for the groups HF * (φ, a) associated to an exact symplectomorphism φ of a Liouville domain and a slope a (see [7]). As an application we construct an asymptotic growth invariant for such symplectomorphisms.…”