A Bangert-Hingston Theorem for Starshaped Hypersurfaces

Let $Q$ be a closed manifold with non-trivial first Betti number that admits a non-trivial $S^1$-action, and $\mathrm{\Sigma }\subseteq T^{\ast }Q$ a non-degenerate starshaped hypersurface. We prove that the number of geometrically distinct Reeb orbits of period at most $T$ on $\mathrm{\Sigma }$ grows at least logarithmically in $T$.