Lyapunov spectrum of Markov and Euclid trees

We study the Lyapunov exponents $Λ(x)$ 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 $[0, \ln φ]$, where $φ$ is the golden ratio, and prove that on the Markov-Hurwitz set $\mathbb{X}$ of the most irrational numbers the corresponding function $Λ_\mathbb{X}$ is monotonically increasing and in the Farey parametrization is convex.

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