Relative Hofer-Zehnder capacity and positive symplectic homology

TL;DR

This paper explores the connection between positive symplectic homology and Hamiltonian dynamics, establishing bounds on capacities and the existence of periodic orbits in Liouville domains, with applications to cotangent bundles.

Contribution

It introduces a homological capacity based on positive symplectic homology and relates it to classical capacities and periodic orbit existence, providing new bounds and insights.

Findings

Positive symplectic homology non-zero implies finite upper bound on Hofer-Zehnder capacity.

Certain Hamiltonian diffeomorphisms have infinitely many non-trivial contractible periodic points.

Upper bounds for spectral capacity in terms of homological capacity.

Abstract

We study the relationship between a homological capacity $c_{\mathrm{SH}^+}(W)$ 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 $c_{\mathrm{SH}}(W)$ 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 $\mathbb R^3$ is proved.