Relative Hofer-Zehnder capacity and positive symplectic homology

We study the relationship between a homological capacity c(SH+) (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 pi(1)-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 c(SH)(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 R-3 is proved.

Automated summary

We study the relationship between positive symplectic homology and the existence of periodic orbits for Hamiltonian systems. We provide upper bounds for positive symplectic homology and discuss their applications in cotangent bundles.