Lyapunov spectrum of Markov and Euclid trees

© 2017 IOP Publishing Ltd & London Mathematical Society. We study the Lyapunov exponents Λ(x) for Markov dynamics as a function of path determined by x ∈ 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 MarkovHurwitz set X of the most irrational numbers the corresponding function Λ X is monotonically increasing and in the Farey parametrization is convex.