The H\"ormander index in the finite-dimensional case

TL;DR

This paper computes the Hörmander index in finite dimensions and applies it to derive iteration inequalities and establish the near-existence of mean indices for certain autonomous Hamiltonian systems on compact symplectic manifolds.

Contribution

It provides a finite-dimensional calculation of the Hörmander index and uses it to prove new iteration inequalities and the near-existence of mean indices in specific Hamiltonian systems.

Findings

Calculated Hörmander index in finite-dimensional case.

Derived iteration inequalities for Hamiltonian systems.

Proved almost existence of mean indices in certain Hamiltonian systems.

Abstract

In this paper, we calculate H\"ormander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.