Tropical Markov dynamics and Cayley cubic

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 $S{L}_2(\mathbb{Z})$ 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.