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)

In the setting of symplectic manifolds which are convex at infinity, we use a version of the Aleksandrov maximum principle to derive uniform estimates for Floer solutions that are valid for a wider class of Hamiltonians and almost complex structures than is usually considered. This allows us to extend the class of Hamiltonians which one can use in the direct limit when constructing symplectic homology. As an application, we detect elements of infinite order in the symplectic mapping class group of a Liouville domain and prove existence results for translated points.

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By a polygonization of a finite point set S in the plane we understand a simple polygon having S as the set of its vertices. Let B and R be sets of blue and red points, respectively, in the plane such that B ∪ R is in general position, and the convex hull of B contains k interior blue points and l interior red points. Hurtado et al. found sufficient conditions for the existence of a blue polygonization that encloses all red points. We consider the dual question of the existence of a blue polygonization that excludes all red points R. We show that there is a minimal number K = K(l), which is polynomial in l, such that one can always find a blue polygonization excluding all red points, whenever k ≥ K. Some other related problems are also considered.

Based on the contact Hamiltonian Floer theory established in [23] that applies to any (admissible) contact Hamiltonian system (M, ξ = ker α, h), where h is a contact Hamiltonian function on a Liouville fillable contact manifold (M, ξ = ker α), we associate a persistence module to (M, ξ, h), called a gapped module, that is parametrized only by a partially ordered set. It enables us to define various numerical Floer-theoretic invariants. In this paper, we focus on the contact spectral invariants and their applications. Several key properties are proved, which include stability with respect to the Shelukhin-Hofer norm in [37] and a triangle inequality of the contact spectral invariant. In particular, our stability property does not involve any conformal factors; our triangle inequality is derived from a novel analysis on pair-of-pants in the contact Hamiltonian Floer homology. While this paper was nearing completion, the authors were made aware of upcoming work [12], where a similar persistence module for a contact Hamiltonian dynamics was constructed.