Abelian sandpiles and algebraic models

TL;DR

This paper extends the connection between abelian sandpile models and algebraic harmonic models to more general polynomials, showing that sandpile models can serve as symbolic representations of algebraic actions with equal entropy.

Contribution

It generalizes previous results to arbitrary sandpile polynomials and demonstrates that certain sandpile models act as symbolic covers of algebraic models.

Findings

Extended the class of sandpile polynomials for which the connection holds

Proved that sandpile models can serve as equal entropy covers of algebraic models

Identified conditions under which these covers are symbolic representations

Abstract

Motivated by the coincidence of topological entropies the connection between abelian sandpiles and harmonic models was established by K. Schmidt and E. Verbitskiy (2009). The dissipative sandpile models were shown to be symbolic representations of algebraic $Z^d$-actions of the harmonic models. Both models are determined by so-called simple sandpile polynomials. We extend this result to arbitrary sandpile polynomials. Moreover, we show that any sandpile model determined by a factor of a sandpile polynomial acts as an equal entropy cover of the corresponding algebraic model. For a special class of factors these covers are shown to be symbolic representations.