A dichotomy result for closed characteristics on compact star-shaped hypersurfaces in $\mathbf{R}^{2n}$

TL;DR

This paper establishes a dichotomy for closed characteristics on compact star-shaped hypersurfaces in n, showing either exactly n or infinitely many such characteristics are elliptic, depending on their degeneracy.

Contribution

It proves a new dichotomy result for closed characteristics on star-shaped hypersurfaces, linking their elliptic nature to the number of distinct closed characteristics.

Findings

If all closed characteristics are elliptic, then either exactly n or infinitely many exist.

The result applies to non-degenerate star-shaped hypersurfaces in n.

Provides a classification criterion based on ellipticity and degeneracy.

Abstract

In this paper, we prove that if all closed characteristics on a compact non-degenerate star-shaped hypersurface $\Sigma$ in $\mathbf{R}^{2n}$ are elliptic, then either there exist exactly $n$ geometrically distinct closed characteristics, or there exist infinitely many geometrically distinct closed characteristics.