2020/10/29 by Gabriele Benedetti, Jungsoo Kang, Benedetti, Gabriele +1 · 1 citation
Mathematics · #37J46 #53D25 #53D40 #58E10 #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2010.15462
openalex publication_date 2020/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the relationship between a homological capacity cSH+(W) for Liouville domains W defined using positive symplectic homology and the existence of periodic orbits for Hamiltonian systems on W: If the positive symplectic homology of W is non-zero, then the capacity yields a finite upper bound to the π1-sensitive Hofer-Zehnder capacity of W relative to its skeleton and a certain class of Hamiltonian diffeomorphisms of W has infinitely many non-trivial contractible periodic points. En passant, we give an upper bound for the spectral capacity of W in terms of the homological capacity cSH(W) defined using the full symplectic homology. Applications of these statements to cotangent bundles are discussed and use a result by Abbondandolo and Mazzucchelli in the appendix, where the monotonicity of systoles of convex Riemannian two-spheres in \mathbb R3 is proved.