An atlas adapted to the Toda flow

TL;DR

This paper develops a new coordinate system called an atlas for the Toda flow on flag manifolds associated with non-compact real semisimple Lie algebras, revealing linearity and Morse-Smale properties, and uncovering new matrix theory features.

Contribution

It introduces an atlas that simplifies the Toda flow to linear form and proves Morse-Smale behavior for general non-compact real semisimple Lie algebras.

Findings

Toda flow is linear in new coordinates

The flow on full flags is Morse-Smale

New features of classical matrix theory are described

Abstract

We introduce an atlas adapted to the Toda flow on the manifold of full flags of any non-compact real semisimple Lie algebra, and on its Hessenberg-type submanifolds. In our local coordinates the Toda flow becomes linear. We use these new coordinates to show that the Toda flow on the manifold of full flags is Morse-Smale, which generalizes the main result of \cite{CSS1} to arbitrary non-compact real semisimple Lie algebras. As a byproduct we describe new features of classical constructions in matrix theory.