  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.
