A Bangert-Hingston Theorem for Starshaped Hypersurfaces

TL;DR

This paper proves that on certain starshaped hypersurfaces in cotangent bundles, the number of Reeb orbits grows at least logarithmically with the period, extending the Bangert-Hingston theorem to new settings.

Contribution

It establishes a logarithmic lower bound on the growth of Reeb orbits for starshaped hypersurfaces under specific topological and symmetry conditions.

Findings

Reeb orbit count grows at least logarithmically with period

Extension of Bangert-Hingston theorem to starshaped hypersurfaces

Conditions involve non-trivial first Betti number and $S^1$-action

Abstract

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