Toda maps, cocycles, and canonical systems

TL;DR

This paper explores Toda flows, cocycles, and their induced maps on m-functions, applying these concepts to canonical systems and proposing twisted shifts as a generalization of the shift on Jacobi matrices.

Contribution

It introduces a new perspective on Toda flows emphasizing cocycles and extends the shift concept to twisted shifts in canonical systems.

Findings

Cocycles induce specific maps on m-functions.

Application of cocycle ideas to canonical systems.

Proposal of twisted shifts as a generalization of the shift on Jacobi matrices.

Abstract

I present a discussion of the hierarchy of Toda flows that gives center stage to the associated cocycles and the maps they induce on the $m$ functions. In the second part, these ideas are then applied to canonical systems; an important feature of this discussion will be my proposal that the role of the shift on Jacobi matrices should now be taken over by the more general class of twisted shifts.