Recoding Lie algebraic subshifts

TL;DR

This paper investigates the structure of Lie algebraic subshifts, showing conditions under which they can be recoded to have cellwise Lie brackets, and explores the complexity and classification of such structures across different groups.

Contribution

It establishes conditions for recoding Lie algebraic subshifts to have cellwise brackets and demonstrates the existence of complex brackets in certain shifts, advancing understanding of algebraic structures in symbolic dynamics.

Findings

Recodable Lie algebraic subshifts exist for virtually polycyclic groups with dense homoclinic points.

Existence of non-recodable Lie algebraic subshifts on non-torsion groups.

One-dimensional full vector shifts can support infinitely many compatible Lie brackets.

Abstract

We study internal Lie algebras in the category of subshifts on a fixed group -- or Lie algebraic subshifts for short. We show that if the acting group is virtually polycyclic and the underlying vector space has dense homoclinic points, such subshifts can be recoded to have a cellwise Lie bracket. On the other hand there exist Lie algebraic subshifts (on any finitely-generated non-torsion group) with cellwise vector space operations whose bracket cannot be recoded to be cellwise. We also show that one-dimensional full vector shifts with cellwise vector space operations can support infinitely many compatible Lie brackets even up to automorphisms of the underlying vector shift, and we state the classification problem of such brackets. From attempts to generalize these results to other acting groups, the following questions arise: Does every f.g. group admit a linear cellular automaton of infinite order? Which groups admit abelian group shifts whose homoclinic group is not generated by finitely many orbits? For the first question, we show that the Grigorchuk group admits such a CA, and for the second we show that the lamplighter group admits such group shifts.