Polynomial Toda maps are transfer matrices

TL;DR

This work analyzes polynomial Toda maps arising from $A(z)\in\mathcal{SL}$ acting on pairs of Herglotz functions by a Möbius transformation. It introduces the domain $D(A)$ on which such maps preserve the Herglotz property and focuses on the polynomial class $\mathcal{P}$, connecting to transfer matrices and canonical systems. The main contributions are a complete factorization theorem for $A\in\mathcal{P}$ into a product of elementary factors $1+p_j(z)J P_j$ with unique structure, and a rigidity result showing that, unless degenerate, any $A$ with nonempty domain must be a transfer matrix (or its inverse); together these classify polynomial Toda maps. The results bridge polynomial cocycles, transfer matrices of canonical systems, and Toda/KdV-type integrable flows, clarifying when polynomial Toda maps are generated by transfer matrices. The analysis relies on Jordan form reductions, projection-based factorizations, and asymptotic properties of Herglotz functions, highlighting a sharp contrast with the non-polynomial case where richer Toda dynamics can occur.

Abstract

We consider entire matrix functions $A(z)$ taking values in $\operatorname{SL}(2,\mathbb C)$. These map pairs of Herglotz functions by acting pointwise as linear fractional transformations. The main examples of such Toda maps are provided by transfer matrices of differential and difference operators and by the cocycles associated with the classical integrable systems (Toda, KdV, etc.) on these operators. Here we consider polynomial matrix functions $A(z)$. We describe these in terms of a factorization, and we then prove that if $A$ induces a Toda map, then $A$ is essentially a transfer matrix.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2