A refinement of the Hofer-Zehnder theorem on the existence of closed trajectories near a hypersurface

The Hofer-Zehnder theorem states that almost every hypersurface in a thickening of a hypersurface $S$ in a symplectic manifold $(M,\omega )$ carries a closed characteristic provided that $S$ bounds a compact submanifold and $(M,\omega )$ has finite capacity. We show that it is enough to assume that the thickening of $S$ has finite capacity.