Lyapunov spectrum of Markov and Euclid trees

Abstract

We study the Lyapunov exponents $\Lambda (x)$ for Markov dynamics as a function of path determined by $x\in \mathbb{R}{P}^1$ on a binary planar tree, describing the Markov triples and their "tropical" version - Euclid triples. We show that the corresponding Lyapunov spectrum is $[0,\ln \phi ]$, where $\phi$ 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.

A comment on the Hausdorff dimension of the support of $\Lambda (x)$ and reference to Jarnik's work are added

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