Relative Hofer–Zehnder capacity and positive symplectic homology

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 R3 is proved.