Lyapunov spectrum of Markov and Euclid trees. (English) Zbl 1432.11071
Summary: 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.
MSC:
| 11J06 | Markov and Lagrange spectra and generalizations |
| 37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |
| 37E25 | Dynamical systems involving maps of trees and graphs |
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