  We discuss the algebraic dynamics on Markov cubics generated by Vieta
involutions, in the tropicalized setting. It turns out that there is an
invariant subset of the tropicalized Markov cubic where the action by Vieta
involutions can be modeled by that of $(\infty,\infty,\infty)$-triangle
reflection group on the hyperbolic plane.
  This understanding of the tropicalized algebraic dynamics produces some
results on Markov cubics over non-archimedean fields, including the existence
of the Fatou domain and finitude of orbits with rational points having prime
power denominators.
