Lyapunov spectrum of Markov and Euclid trees

TL;DR

This paper investigates the Lyapunov exponents for Markov and Euclid trees, revealing a spectrum from 0 to the natural logarithm of the golden ratio, with specific properties on the Markov-Hurwitz set.

Contribution

It characterizes the Lyapunov spectrum for Markov and Euclid trees and analyzes its monotonicity and convexity on the Markov-Hurwitz set.

Findings

Lyapunov spectrum is [0, ln(φ)] where φ is the golden ratio.

On the Markov-Hurwitz set, the Lyapunov function is monotonically increasing.

The Lyapunov function is convex in the Farey parametrization.

Abstract

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.