Multiplicity and stability of closed characteristics on compact convex P-cyclic symmetric hypersurfaces in ${\bf R}^{2n}$

TL;DR

This paper proves the existence of at least n geometrically distinct closed characteristics on certain symmetric convex hypersurfaces in R^{2n}, confirming a longstanding conjecture and analyzing their stability and symmetry properties.

Contribution

It establishes the minimal number of closed characteristics on P-cyclic symmetric hypersurfaces and explores their symmetry and stability under specific conditions.

Findings

At least n geometrically distinct closed characteristics exist.

If finite, at least 2[ n/2 ] are non-hyperbolic.

When exactly n characteristics exist with k≥3, all are P-cyclic symmetric.

Abstract

Let $\Sigma$ be a compact convex hypersurface in ${\bf R}^{2n}$ which is P-cyclic symmetric, i.e., $x\in \Sigma$ implies $Px\in\Sigma$ with P being a $2n\times2n$ symplectic orthogonal matrix and satisfying $P^k=I_{2n}$, $ker(P^l-I_{2n})=0$ for $1\leq l< k$, where $n, k\geq2$. In this paper, we prove that there exist at least $n$ geometrically distinct closed characteristics on $\Sigma$, which solves a longstanding conjecture about the multiplicity of closed characteristics for a broad class of compact convex hypersurfaces with symmetries(cf.,Page 235 of \cite{Eke1}). Based on the proof, we further prove that if the number of geometrically distinct closed characteristics on $\Sigma$ is finite, then at least $2[\frac{n}{2}]$ of them are non-hyperbolic; and if the number of geometrically distinct closed characteristics on $\Sigma$ is exactly $n$ and $k\geq3$, then all of them are P-cyclic symmetric, where a closed characteristic $(\tau, y)$ on $\Sigma$ is called P-cyclic symmetric if $y({\bf R})=Py({\bf R})$.