Mean index for non-periodic orbits in Hamiltonian systems

TL;DR

This paper introduces a mean index for non-periodic orbits in Hamiltonian systems, generalizing the rotation number, and explores its properties, continuity, and relation to the Fredholm property of associated operators.

Contribution

It defines the mean index for non-periodic orbits in Hamiltonian systems and analyzes its properties and relation to operator theory, extending previous concepts like the rotation number.

Findings

Mean index is an interval in R, continuous on systems.

The index reduces to a point for quasi-periodic orbits.

Relation established between the mean index and Fredholm property.

Abstract

In this paper, we define mean index for non-periodic orbits in Hamiltonian systems and study its properties. In general, the mean index is an interval in R which is uniformly continuous on the systems. We show that the index interval is a point for a quasi-periodic orbit. The mean index can be considered as a generalization of rotation number which defined by Johnson and Moser in the study of almost periodic Schrodinger operators. Motivated by their works, we study the relation of Fredholm property of the linear operator and the mean index at the end of the paper.