Deformations of Real Rational Dynamics in Tropical Geometry

In this paper we study analytic properties of orbits given by real rational functions. We introduce some comparison methods which allow us to compare the real rational dynamics with automata given by (max, +) functions, passing through a kind of scale transform in tropical geometry. Such a scale transform gives a one-to-one correspondence of presentations between automata and real rational functions. We study invariant properties of the real rational dynamics under change of presentations of automata.