Volterra map and related recurrences
Andrei K. Svinin
TL;DR
The paper analyzes the Volterra map defined via a Lax pair, establishing its birational action on the phase space $M_g$ and connecting it to a $(2g+1)$-order discrete equation governed by Stieltjes continued fractions. It introduces homogeneous discrete polynomials $S^k_s(n)$ to express the higher-order recurrence compactly for all $g\ge1$, and shows how invariants $c_j$ generate parameters $H_j$ through $(1+\sum_j H_j x^j)^2=1+\sum_j c_j x^j$, enabling a birational correspondence with the Volterra dynamics. A central result is that a truncated formal-series framework yields a polynomial recurrence of order $2g+1$, with explicit invariants and two-parameter families of discrete equations that exhibit the Laurent property. The paper further specializes to $g=1,2,3$ to derive Somos-5-type recurrences and related higher-order homogeneous forms, illustrating integer sequences and conjecturing structural links between invariants and recurrence coefficients. Overall, the work provides a compact, unified perspective on Volterra-type dynamics, discrete polynomials, and Laurent phenomena, with potential applications to discrete integrable systems and cluster-algebra-inspired recurrences.
Abstract
In this paper we analyze recent work \cite{Hone1} by Hone, Roberts and Vanhaecke, where the so-called Volterra map was introduced via the Lax equation that looks similar to the Lax representation for the Mumford's system \cite{Vanhaecke}. This map turns out to be birational and a corresponding dynamical system on an affine space $M_g$ of dimension $3g+1$ was associated with it. This mapping is related to some discrete equation of the order $2g+1$ associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic (elliptic) curve of genus $g\geq 1$. The authors of the paper provides examples of this equation for the simplest cases $g=1$ and $g=2$, but for higher values of $g$, corresponding equation turns out to be too cumbersome to write them out. We present an approach in which the mentioned $(2g+1)$-order equation can be written out for all values of $g\geq 1$ in a compact form. This equation is not new and can be found, for example, in \cite{Svinin3}. An essential point in our framework is the use of special class of discrete polynomials which as shown to be closely related to the Stieltjes continued fraction. On the one hand, this allows us to generalize some of the results of the work \cite{Hone1}. On the other hand, many things in this approach can be presented in a more compact and unified form. Ultimately, we believe that this allows us to give a new perspective on this topic.