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

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