Symmetrical Symplectic Capacity with Applications

TL;DR

This paper introduces the concept of symmetrical symplectic capacity and proves the existence of symmetric closed characteristics on certain energy surfaces, advancing symplectic geometry theory.

Contribution

It defines symmetrical symplectic capacity and establishes a new existence result for symmetric closed characteristics on symmetric energy surfaces.

Findings

Existence of at least one symmetric closed characteristic on prescribed symmetric energy surfaces.

Introduction of symmetrical symplectic capacity as a new theoretical tool.

Application to brake orbits and S-invariant brake orbits.

Abstract

In this paper, we first introduce the concept of symmetrical symplectic capacity for symmetrical symplectic manifolds, and by using this symmetrical symplectic capacity theory we prove that there exists at least one symmetric closed characteristic (brake orbit and $S$-invariant brake orbit are two examples) on prescribed symmetric energy surface which has a compact neighborhood with finite symmetrical symplectic capacity.