  We study the Lyapunov exponents $\Lambda(x)$ for Markov dynamics as a
function of path determined by $x\in \mathbb RP^1$ on a binary planar tree,
describing the Markov triples and their "tropical" version - Euclid triples. We
show that the corresponding Lyapunov spectrum is $[0, \ln \varphi]$, where
$\varphi$ is the golden ratio, and prove that on the Markov-Hurwitz set
$\mathbb{X}$ of the most irrational numbers the corresponding function
$\Lambda_\mathbb{X}$ is monotonically increasing and in the Farey
parametrization is convex.
