Level Set Structure of an Integrable Cellular Automaton

TL;DR

This paper introduces a group-theoretic approach to analyze a cellular automaton related to the discrete Toda lattice, revealing its level sets decompose into toroidal connected components, a novel structural insight.

Contribution

It provides the first demonstration that the level sets of this cellular automaton are decomposed into connected toroidal components, linking integrable systems and cellular automata.

Findings

Level sets are decomposed into connected components

Each component is topologically a torus

First such structural characterization for this automaton

Abstract

Based on a group theoretical setting a sort of discrete dynamical system is constructed and applied to a combinatorial dynamical system defined on the set of certain Bethe ansatz related objects known as the rigged configurations. This system is then used to study a one-dimensional periodic cellular automaton related to discrete Toda lattice. It is shown for the first time that the level set of this cellular automaton is decomposed into connected components and every such component is a torus.