  We study the tropical version of Markov dynamics on the Cayley cubic,
introduced by V.E. Adler and one of the authors. We show that this action is
semi-conjugated to the standard action of $SL_2(\mathbb Z)$ on a torus, and
thus is ergodic with the Lyapunov exponent and entropy given by the logarithm
of the spectral radius of the corresponding matrix.
