Derivation of Invariant Varieties of Periodic Points from Singularity Confinement in the case of Toda Map

TL;DR

This paper introduces a new algorithm to derive invariant varieties of periodic points in integrable maps, extending previous methods to cases with fewer invariants, and applies it to the 3 point Toda map.

Contribution

A novel algorithm for deriving IVPPs in integrable maps with fewer invariants than previously possible, demonstrated on the Toda map.

Findings

Derived IVPPs for the 3 point Toda map.

Extended applicability of the derivation method to cases with p between d/2 and d-2.

Validated the new algorithm through application to the Toda map.

Abstract

In our previous work we have shown that the invariant varieties of periodic points (IVPP) of all periods of the 3 dimensional Lotka-Volterra map can be derived, iteratively, from the singularity confinement (SC). The method developed there can be applied to any integrable maps of dimension $d$ only when the number of the invariants $p$ equals to $d-1$. We propose, in this note, a new algorithm of the derivation which can be used in the cases ${d\over 2}\le p\le d-2$. Applying this algorithm to the 3 point Toda map, we derive a series of its IVPP's.