Automata in groups and dynamics and induced systems of PDE in tropical geometry

TL;DR

This paper develops a framework connecting automata, rational dynamics, and PDE systems within tropical geometry, providing new insights into automata groups and their properties through dynamical scaling limits and uniform orbit estimates.

Contribution

It introduces a dynamical scaling limit from rational dynamics to automata in tropical geometry and applies this to analyze automata groups and the Burnside problem.

Findings

Established a dynamical scaling limit connecting rational dynamics and automata.

Provided uniform orbit estimates for these dynamics.

Linked properties of automata groups to rational and PDE dynamics.

Abstract

In this paper we develop a dynamical scaling limit from rational dynamics to automata in tropical geometry. We compare these dynamics and induce uniform estimates of their orbits. We apply these estimates to introduce a comparison analysis of theory of automata groups in geometric group theory with analysis of rational dynamics and some PDE systems. Frameworks of characteristic properties of automata groups are inherited to the corresponding rational or PDE dynamics. As an application we study the Burnside problem in group theory and translate the property as the infinite quasi-recursiveness in rational dynamics.